{
"schema": "oa-flow-proof-locations/v2",
"description": "Source locations and prerequisite links for mathematical passages in the course. The linked lesson passages state their hypotheses and conclusions.",
"current_local_proof_bindings": [
{
"id": "OA-FLOW.SC.0",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-00",
"start_line": 8,
"end_line": 27,
"range_sha256": "a5fe9d52348394e56a90b606809b21774c96806f3bda5754bbb7c06812e81a09",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.SC.1",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-01",
"start_line": 29,
"end_line": 49,
"range_sha256": "292ec35e155772c6704a1503b7dbd39bc192ec5b66cca3ed5fdb0122df2aaa41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"PRIMITIVE-SETS-AND-EXTENDED-NONNEGATIVE-SUMS"
]
},
{
"id": "OA-FLOW.SC.2",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-02",
"start_line": 51,
"end_line": 79,
"range_sha256": "c2466cd07ec73269d920c2630eae6a14ca0be7cbadf45d369f06f8f45d045b37",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.CF.1",
"PRIMITIVE-CHOICE"
]
},
{
"id": "OA-FLOW.SC.3",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-03",
"start_line": 81,
"end_line": 109,
"range_sha256": "2b5422dab16d15b9708daeff09a581527de5123f6fb4e4411df4f006089b7797",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.2"
]
},
{
"id": "OA-FLOW.SC.4",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-04",
"start_line": 111,
"end_line": 143,
"range_sha256": "48e2d3f6ba07c752efa95f6b3930d6e12f3a02fb63664ced05b176b448ce06c0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3"
]
},
{
"id": "OA-FLOW.SC.4M",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-04m",
"start_line": 145,
"end_line": 159,
"range_sha256": "e97e2356bae19c622efcd292c24f55f8dab44cb6bcb6fdbfea2a4a5521f0c1f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.SC.5",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-05",
"start_line": 161,
"end_line": 195,
"range_sha256": "4936a40727ec878aa8d7dc622316315ada5c91691c730f6331b477141a1e8084",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.SC.6",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-06",
"start_line": 197,
"end_line": 232,
"range_sha256": "1626e0ccff1308960a057b262178840589329e4977659f89cc46be20028b34ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.SC.7",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-07",
"start_line": 234,
"end_line": 272,
"range_sha256": "b61a44358323c5a7d6ae21ab4ff459bec17d8285935200b7db9eb170254f7930",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6"
]
},
{
"id": "OA-FLOW.SC.8",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-08",
"start_line": 274,
"end_line": 299,
"range_sha256": "73c30f3d665316b2049f631c48a57b2646f90b66dd2d1abeda65db5a159fcd5a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.SC.9",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-09",
"start_line": 301,
"end_line": 320,
"range_sha256": "7645c2af9a3a260a86058ed35120247439c68b78edabe3de5d81fd732f2aef66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.SC.10",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-10",
"start_line": 322,
"end_line": 335,
"range_sha256": "42c55b9d3d9c9563d45e336089287c083431bd455859a5d16aaa2fe8ea5d02d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.GNS.2.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-2-1",
"start_line": 24,
"end_line": 42,
"range_sha256": "e18f4d71a1f5ccd9270cac8dbdb6130fe0f78083f4aa64da0bfa8b76703176f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.GNS.2.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-2-2",
"start_line": 44,
"end_line": 57,
"range_sha256": "5b0b859044eb5dea939de31047dc2c4ddff86c9179b4c448dacc4b772e36c2f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.1"
]
},
{
"id": "OA-FLOW.GNS.3.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-3-1",
"start_line": 62,
"end_line": 86,
"range_sha256": "305b21431a24d8ea174e1c33f1eb18a6bfae6f25ef6a62c35fba33076d05c6fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
]
},
{
"id": "OA-FLOW.GNS.3.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-3-2",
"start_line": 88,
"end_line": 101,
"range_sha256": "f3ff955527ed9b07b65de99e72750f7e3a822b03aef36ea5a65a2191f73d4c12",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GNS.3.3",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-3-3",
"start_line": 104,
"end_line": 128,
"range_sha256": "db472be36d8270cf939460cabc136ebd915608df903e176154a7cb604959b621",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GNS.3.4",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-corollary-3-4",
"start_line": 130,
"end_line": 135,
"range_sha256": "633396d2029a42703fedf4cc95dde31e1aa3138c45895784dabab79a5feb90a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.3"
]
},
{
"id": "OA-FLOW.GNS.4.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-4-1",
"start_line": 140,
"end_line": 177,
"range_sha256": "f8490a7a8a58bcff8519e9f8db62815edae1d34d0341657d96ff400147a158c8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.3.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
]
},
{
"id": "OA-FLOW.GNS.4.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-corollary-4-2",
"start_line": 179,
"end_line": 194,
"range_sha256": "29477c60a27c4263e7608785e487e356ef303fdaa9aab86a5b39edbbf74e72c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.4.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
]
},
{
"id": "OA-FLOW.GNS.5.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-5-1",
"start_line": 199,
"end_line": 259,
"range_sha256": "a03096e6adbc140dec97e3387bbcf55ecb4473817951c0fdbd187076617b7d92",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.4.2",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GNS.5.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-5-2",
"start_line": 262,
"end_line": 280,
"range_sha256": "a68ae6347f9ecdaf573f21e584dcb9b1a914bb8c285d60bb9b20e94e19901c74",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GNS.5.3",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-corollary-5-3",
"start_line": 283,
"end_line": 293,
"range_sha256": "c4b9c8189db263fff5d4e2541e5f247cfbda83b100f91bc48933c1b5a3d77399",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.5.1"
]
},
{
"id": "OA-FLOW.GNS.6.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-6-1",
"start_line": 298,
"end_line": 309,
"range_sha256": "43bd9b88d7b95d68a8533674a89e524bdbe1a9f89736f444b415199836377b64",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
]
},
{
"id": "OA-FLOW.GNS.6.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-6-2",
"start_line": 311,
"end_line": 325,
"range_sha256": "a6c8858b36b0a0b6c4078531201ea58b20e725f2b471f00dd081932c0549fafc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.6.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.GNS.7.1",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-7-1",
"start_line": 330,
"end_line": 349,
"range_sha256": "55f961938d320f6bc4927e44473ac66df5935c223b3f88bd9a3860efba2be254",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GNS.7.2",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-lemma-7-2",
"start_line": 351,
"end_line": 354,
"range_sha256": "02a92edb57932745f2ad51c0e78e559047b80d206456104f2c0a3745fab36858",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.GNS.7.1"
]
},
{
"id": "OA-FLOW.GNS.7.3",
"unit": "OA-FLOW-GNS",
"source_file": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8",
"reader": "OA-FLOW-GNS.html#gns-theorem-7-3",
"start_line": 357,
"end_line": 375,
"range_sha256": "00ab66886c813767bc23581b6432d009522e60cd4d6ffdc39897639702ad89a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.GNS.6.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GNS.7.2"
]
},
{
"id": "OA-FLOW.GRP.SETTING",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.setting",
"start_line": 1,
"end_line": 8,
"range_sha256": "58a9bbfc271e3c3e8b7cf72089620214683a050f299faeede0c50eafcd5552dc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.GRP.IMPORT.FOUNDATIONS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.import.foundations",
"start_line": 9,
"end_line": 14,
"range_sha256": "3cfb84847805c2c4e846aeb7ec613ce32341c4ba1d3b198b74cfd216aafad383",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9"
]
},
{
"id": "OA-FLOW.GRP.HAARCONVENTIONS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.haarconventions",
"start_line": 15,
"end_line": 51,
"range_sha256": "1c91f6472912f73517af3649aba336024311f8eeaf56ed35da8a4a2bff443a5b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.TRANSLATIONS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.translations",
"start_line": 52,
"end_line": 81,
"range_sha256": "5a832255b86b34f112f64bcbe2bb2bb74413714281018e608b25d24b5d6875d0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.VECTORINTEGRATION",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.vectorintegration",
"start_line": 82,
"end_line": 132,
"range_sha256": "5f2caa0d0ba58be262922aa194ed404f8024dace88f19b7e4e3cf1d8146cdd5c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.ALGEBRA",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.algebra",
"start_line": 133,
"end_line": 203,
"range_sha256": "c9e417574a6c499d19c7c55baec55d59ec926471523a0c2b0d401bdb63d8746a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GRP.CONTRACTIVITY",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.contractivity",
"start_line": 204,
"end_line": 262,
"range_sha256": "617c6c360474189114c1c0eca05ebcfd8883fefcd775b0f535449177c93fa324",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.INTEGRATION",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.integration",
"start_line": 263,
"end_line": 300,
"range_sha256": "393061f9aff28c15944f07cff24a9b6f844914821145c0214c32bb799fea0553",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.GRP.RECOVERY",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.recovery",
"start_line": 301,
"end_line": 358,
"range_sha256": "6726112f3a940ff92b95edefd4cfa4d18e7f6d1d6f74a78206ddbcedfbc32238",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.CONTRACTIVITY",
"OA-FLOW.GRP.INTEGRATION"
]
},
{
"id": "OA-FLOW.GRP.INTERTWINERS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.intertwiners",
"start_line": 359,
"end_line": 378,
"range_sha256": "bc0baa221e2ff07d3b09344a7771bf27ee1fab108eff78cae34d488b8a36f1d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTEGRATION"
]
},
{
"id": "OA-FLOW.GRP.REGULAR",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.regular",
"start_line": 379,
"end_line": 390,
"range_sha256": "3c88a7b46e7c34d45f8ef0898b93755b21516653a49757e9981b4c04544df81f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GRP.CSTARQUOTIENTS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.cstarquotients",
"start_line": 391,
"end_line": 450,
"range_sha256": "0198a086a7cf14d30096ae2da7967b6d86b644aee8761b30dd269682313f09de",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.COMPLETIONS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.completions",
"start_line": 451,
"end_line": 490,
"range_sha256": "ba1fd0d6b61b60740dc56313675d6a2b27d7be8dd7c4839e029f33e0fc6c70e7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.REGULAR",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.CONTRACTIVITY",
"OA-FLOW.GRP.CSTARQUOTIENTS"
]
},
{
"id": "OA-FLOW.GRP.MODELS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.models",
"start_line": 491,
"end_line": 570,
"range_sha256": "f056ab6610f7bd4333af8f31d22fba40a470b4bb9750b9ca5e8724c327986778",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.FREEGROUPGAP",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.freegroupgap",
"start_line": 523,
"end_line": 570,
"range_sha256": "536f69cc5c0137899f9b3e215a35fa20539c2a267751761537f4b451d00825d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.GRP.PROBLEMS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.problems",
"start_line": 571,
"end_line": 662,
"range_sha256": "de364228f72213ad4fbf0e9db0c658b5a7dc9d45d456b6d1ea278f65a0ed86ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.MODELS",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.SC.8",
"OA-FLOW.SC.4",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5"
]
},
{
"id": "OA-FLOW.GRP.ANTECEDENTS",
"unit": "OA-FLOW-L24",
"source_file": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7",
"reader": "OA-FLOW-L24.html#oa-flow.grp.antecedents",
"start_line": 663,
"end_line": 668,
"range_sha256": "4186a8be53b23564c257d40d6bdafc0f22c4817b6cb389e62c0f5114ce086465",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.CCOV.SETTING",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.setting",
"start_line": 1,
"end_line": 8,
"range_sha256": "6626560a2d36ad7eaceaf7a088b8f1cf451253d6d1084665c9570ebb6439817e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.CCOV.IMPORT.FOUNDATIONS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.import.foundations",
"start_line": 9,
"end_line": 14,
"range_sha256": "f700285070074843b04207cc5e14205cabdbc64a824218c936f58a5aec0d5e33",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.IMPORT.FOUNDATIONS",
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.GNS.3.4",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GNS.7.2",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.5.2",
"OA-FLOW.GNS.5.3",
"OA-FLOW.GNS.6.1",
"OA-FLOW.GNS.6.2",
"OA-FLOW.GNS.7.3",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FND-WT-02",
"OA-FND-WT-03",
"OA-FND-WT-06",
"OA-FND-HS-04"
]
},
{
"id": "OA-FLOW.CCOV.ALGEBRA",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.algebra",
"start_line": 15,
"end_line": 82,
"range_sha256": "6dd5a9377e9e4f57f6eeeb20bed06bb39513da03338da4b9f5b2edcd9f7aff63",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.CSTARQUOTIENTS"
]
},
{
"id": "OA-FLOW.CCOV.INTEGRATION",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.integration",
"start_line": 83,
"end_line": 115,
"range_sha256": "8f28ada0214f05ca0b6cc841755b877cce31492f587c11570798364981c59629",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.CCOV.COEFFICIENTRECOVERY",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.coefficientrecovery",
"start_line": 116,
"end_line": 145,
"range_sha256": "0c5b6e8adc42464e589e03e62e22c59b9bc77ab49db4380bd03a64742d203606",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.GRP.CONTRACTIVITY",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.GROUPRECOVERY",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.grouprecovery",
"start_line": 146,
"end_line": 179,
"range_sha256": "ff0a0be73d50b1d0b403d5bca1565797d0639ad0af964a7c267fd4242cad8f97",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.COEFFICIENTRECOVERY",
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.CCOV.INTEGRATION"
]
},
{
"id": "OA-FLOW.CCOV.REGULAR",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.regular",
"start_line": 180,
"end_line": 220,
"range_sha256": "4a09a0ca5291c1d8eb2652dfaa6a53ddd323bd1267c0aa9d6ff34d28d658ea06",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.INTEGRATION",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.CCOV.MATRIXNORMS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.matrixnorms",
"start_line": 221,
"end_line": 230,
"range_sha256": "d14f6b2b220ff27519673a9f13296274c2360b261e58bbcda6dacf3060b640cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.CSTARQUOTIENTS",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.REGULARINDEPENDENCE",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.regularindependence",
"start_line": 231,
"end_line": 259,
"range_sha256": "47a11fdbf5d66fceb7dba63ead72fa2f60f4852850fa7e63ce8c5e495fa29d6c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.MATRIXNORMS",
"OA-FLOW.CCOV.REGULAR"
]
},
{
"id": "OA-FLOW.CCOV.COMPLETIONS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.completions",
"start_line": 260,
"end_line": 399,
"range_sha256": "92324887f1de005ff9436fb8a2d71f61dac65f12958c17e647cb48ad38ae5cef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.REGULARINDEPENDENCE",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.MULTIPLIERS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.multipliers",
"start_line": 289,
"end_line": 318,
"range_sha256": "f658a514fb6e2acbd4e944e90402ef281fa9999658254d7d1adb9b4226e1d51b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CCOV.REGULAR",
"OA-FLOW.CCOV.INTEGRATION"
]
},
{
"id": "OA-FLOW.CCOV.COMPARISON",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.comparison",
"start_line": 319,
"end_line": 399,
"range_sha256": "141063cf2d031f1ff7b5bbfe38388cb02528be42b990af83796adde8d330a253",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.REGULARINDEPENDENCE",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.INVARIANTSTATE",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.invariantstate",
"start_line": 400,
"end_line": 430,
"range_sha256": "b4ab738e342aa9b360cbb54f9f18cf0473a70a6cdf692d19b43294ff600a3846",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.EXTREMALITY",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.extremality",
"start_line": 431,
"end_line": 485,
"range_sha256": "9b7aedea28eecb2e9a58e1945d69bc3f1583f29601456c6f3e1e40c0bc8b9844",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.INVARIANTSTATE",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS"
]
},
{
"id": "OA-FLOW.CCOV.FINITEORBITS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.finiteorbits",
"start_line": 513,
"end_line": 522,
"range_sha256": "1f36a1295ef9fca99faf4c0d4a3b1d6c39508f0b011e459283ff110547319eac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.EXTREMALITY"
]
},
{
"id": "OA-FLOW.CCOV.TRANSFORMATIONS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.transformations",
"start_line": 523,
"end_line": 595,
"range_sha256": "f7e14528177886275e5fd9962a493e99d31def70a8cf623021592bf58ce4ae03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.COMPLETIONS",
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.CCOV.INTEGRATION",
"OA-FLOW.CCOV.REGULAR",
"OA-FLOW.CCOV.IMPORT.FOUNDATIONS",
"OA-FND-HS-04",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.CCOV.PROBLEMS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.problems",
"start_line": 640,
"end_line": 673,
"range_sha256": "f5269a741a59622e90044f996121e32c9e3917aca3af55a70b16f0507b1c39b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CCOV.REGULAR",
"OA-FLOW.CCOV.TRANSFORMATIONS",
"OA-FLOW.CCOV.COMPARISON",
"OA-FLOW.CCOV.EXTREMALITY"
]
},
{
"id": "OA-FLOW.CCOV.ANTECEDENTS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.antecedents",
"start_line": 674,
"end_line": 678,
"range_sha256": "a2e8bebf074a173777e1fbf0035a63f7466b0b8af906d32fe80f24e2261fe142",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.XGAPS.CSTAR.HARMONIC",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#oa-flow.xgaps.cstar.harmonic",
"start_line": 120,
"end_line": 127,
"range_sha256": "bd1b91acab4e826bae23f1348fbe4b47566ad2f8fcd61a314237cfb0279e62b4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4"
]
},
{
"id": "OA-FLOW.XGAPS.FOURIER",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.fourier",
"start_line": 17,
"end_line": 31,
"range_sha256": "b19ae890f2fbda9d7d53cae49e402c14a91da6b3439d6bf20b8fc7d42ebee461",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.HARMONIC"
]
},
{
"id": "OA-FLOW.XGAPS.AMENABILITY.REPAIR",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.amenability.repair",
"start_line": 34,
"end_line": 43,
"range_sha256": "d19af0e661090a87c27acb5fb5883ee385057d0907d96d2997cb6e142589c882",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.COMPARISON"
]
},
{
"id": "OA-FLOW.XGAPS.AMENABILITY.ABELIAN",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.amenability.abelian",
"start_line": 46,
"end_line": 61,
"range_sha256": "24770b24d60c831e3a710e90e4adee2478ab8928ce86e6774d46c7161694099a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.HARMONIC",
"OA-FLOW.XGAPS.AMENABILITY.REPAIR",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H2"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.DUAL",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.dual",
"start_line": 65,
"end_line": 98,
"range_sha256": "c9c02e94c7d9b1a08397ce2edb518088c2cfc6a3889fc3d93b9ec2994497d683",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.CCOV.COMPLETIONS",
"OA-FLOW.CCOV.MULTIPLIERS",
"OA-FLOW.XGAPS.FOURIER"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.BIDUALACTION",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#l138-returned-action",
"start_line": 281,
"end_line": 302,
"range_sha256": "7378980985f9400c7c48b4332abea7216ea05feeba014f3c4af89ccd90085755",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.DUAL",
"OA-FLOW.XGAPS.CSTAR.TRANSFORM",
"OA-FLOW.XGAPS.CSTAR.KERNELCLOSURE",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 302,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L138.md",
"line": 5,
"through_line": 7,
"description": "Arbitrary nonunital coefficient algebra and locally compact Hausdorff abelian group; the specified earlier harmonic proofs apply."
}
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.TAKAI",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.takai",
"start_line": 305,
"end_line": 307,
"range_sha256": "8f25b3530609c167917318301725594f2ef4878f8e75042a2096fef632aafe81",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.TRANSFORM",
"OA-FLOW.XGAPS.CSTAR.KERNELCLOSURE",
"OA-FLOW.XGAPS.CSTAR.DUAL",
"OA-FLOW.XGAPS.AMENABILITY.ABELIAN",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.BIDUALACTION"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.TRANSFORM",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.transform",
"start_line": 102,
"end_line": 181,
"range_sha256": "1e33fefe2f85ff792ece37f7d641f43005e0d8655c0a9cd4a84a75aabc8aeda3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.HARMONIC",
"OA-FLOW.XGAPS.AMENABILITY.ABELIAN",
"OA-FLOW.XGAPS.CSTAR.DUAL",
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.7.3",
"OA-FLOW.CF.10",
"OA-FLOW.CCOV.REGULAR",
"OA-FLOW.CCOV.REGULARINDEPENDENCE",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.KERNELCLOSURE",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.kernelclosure",
"start_line": 185,
"end_line": 280,
"range_sha256": "e712051cb6424c1d217ad60ba61a6d222f0a256f4c6fcfbd8d3f27ae96e2b69d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.TRANSFORM",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.CSTARQUOTIENTS",
"OA-FLOW.CCOV.INTEGRATION"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.CYCLIC",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.cyclic",
"start_line": 311,
"end_line": 348,
"range_sha256": "d06ceca6c4f65653238ad40ece03ce2b9464406681b4838b6f9794ff2b2c1481",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.TAKAI",
"OA-FLOW.XGAPS.CSTAR.DUAL",
"OA-FLOW.CCOV.COMPLETIONS"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.DISCRETE",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#l138-discrete",
"start_line": 349,
"end_line": 389,
"range_sha256": "6c88545eccff0753cdec47d3c9d5d55f0e6c4dbe91dc550af422fd95f89657b4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.TRANSFORM",
"OA-FLOW.XGAPS.CSTAR.KERNELCLOSURE",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 389
},
{
"id": "OA-FLOW.XGAPS.CSTAR.SOLUTIONS",
"unit": "OA-FLOW-L138",
"source_file": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5",
"reader": "OA-FLOW-L138.html#oa-flow.xgaps.cstar.solutions",
"start_line": 394,
"end_line": 412,
"range_sha256": "270f33e4e74711d71bd1f024f146a316ddb21e5123abd1f485fa4107af8b6085",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.CYCLIC",
"OA-FLOW.XGAPS.CSTAR.TAKAI",
"OA-FLOW.XGAPS.CSTAR.HARMONIC"
]
},
{
"id": "OA-FLOW.GI.SETTING",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.setting",
"start_line": 9,
"end_line": 44,
"range_sha256": "5f64c21fa1c730872313adb307942914afec8242a265bf2356e0ed6aee0f416f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4"
]
},
{
"id": "OA-FLOW.GI.FOURIER",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.fourier",
"start_line": 45,
"end_line": 112,
"range_sha256": "18dfe83598034dcf49af53ac0f97aa901ffa91e29f85e046a6d06ff67583dd1a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.SETTING"
]
},
{
"id": "OA-FLOW.GI.FILTERS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.filters",
"start_line": 113,
"end_line": 202,
"range_sha256": "f3c8e3ee13bbd454adfc87005bd42b0da43b8a9cbf7d03f4e3aa4c25d4bfd0fb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.SETTING"
]
},
{
"id": "OA-FLOW.GI.DIAGONAL",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.diagonal",
"start_line": 203,
"end_line": 253,
"range_sha256": "82272d09f955b0b0fb7114fc047c8500001cfc99e65fb3018cb8e717e331373f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.FOURIER"
]
},
{
"id": "OA-FLOW.GI.TAILS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.tails",
"start_line": 254,
"end_line": 290,
"range_sha256": "db131568686940f00783e1d3828242d3a108473d20229589bab9db8f1c132f29",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.SETTING"
]
},
{
"id": "OA-FLOW.GI.TRANSFER",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.transfer",
"start_line": 291,
"end_line": 353,
"range_sha256": "ec651285430c222dce2bbb604869b7bff9057edcd7d09ee32c89304f98bb966b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.FILTERS"
]
},
{
"id": "OA-FLOW.GI.INNERNESS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.innerness",
"start_line": 354,
"end_line": 386,
"range_sha256": "02f6e65f9e0cfc8f3be7d7523c383194b2fedbc5fe4a98c79541aa1420c9e7bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.GI.SETTING"
]
},
{
"id": "OA-FLOW.GI.NORMALITY",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.normality",
"start_line": 387,
"end_line": 405,
"range_sha256": "6c105485e613d18c0794666cdada333b1b7cbaea242eead332288905dadd016f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.INNERNESS"
]
},
{
"id": "OA-FLOW.GI.GENERATORSPECTRUM",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.generatorspectrum",
"start_line": 406,
"end_line": 444,
"range_sha256": "49511635826223833260d97dc5f9f5604734fab6e659363668c5b0a8e7dea0f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.FOURIER"
]
},
{
"id": "OA-FLOW.GI.MATRIX",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.matrix",
"start_line": 445,
"end_line": 476,
"range_sha256": "762cd179ee44c607e197d0f72973863eacfda63c0ecf8f31e7f28186c79a6f26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.INNERNESS",
"OA-FLOW.GI.GENERATORSPECTRUM"
]
},
{
"id": "OA-FLOW.GI.PROBLEMS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.problems",
"start_line": 477,
"end_line": 498,
"range_sha256": "b2e3839c5d931a1cb131eddfb42049b725cf5b51095ebf3d20eac74a3ab65d3b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.MATRIX",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.INNERNESS"
]
},
{
"id": "OA-FLOW.GI.ANTECEDENTS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.antecedents",
"start_line": 1022,
"end_line": 1026,
"range_sha256": "dff7cd0b873e8c9555b82af8e7600af1d001c25c2c61103f82f225b0db0b7c20",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null
},
{
"id": "OA-FLOW.FSCORE.INPUT",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.input",
"start_line": 13,
"end_line": 41,
"range_sha256": "5f0b2341f5a80786c29ec308ec1fd022de43869ab6ccd7dea92f6e6fac313b9b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.REGULAR",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.regular",
"start_line": 44,
"end_line": 109,
"range_sha256": "1b3d03147e045b772092316dc77e37d306b7e633750fed59d43afd6fbaf938c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.SCALAR",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.scalar",
"start_line": 251,
"end_line": 307,
"range_sha256": "eb2632c6f8b6cc6700a2208425d54a51ed8e91b6f924f48a24202309e7d69ce0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.SANDWICH",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.sandwich",
"start_line": 335,
"end_line": 364,
"range_sha256": "7ea96c61d05925eb471105757f328343967f281fc6dd061df238c6b45c3256a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.AVERAGE",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.average",
"start_line": 251,
"end_line": 307,
"range_sha256": "eb2632c6f8b6cc6700a2208425d54a51ed8e91b6f924f48a24202309e7d69ce0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.COEFFICIENT",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.coefficient",
"start_line": 368,
"end_line": 404,
"range_sha256": "5628ddbd84194dc63b62e832d48ee95ccc3b51f514ffa564da52604bc48b3367",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.VECTOR",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.vector",
"start_line": 311,
"end_line": 331,
"range_sha256": "f3f8a241285674f286b80e3040b9ef8cdd3fb3bb0ef27b154383d4dd571f0f45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.CORNERS",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.corners",
"start_line": 112,
"end_line": 178,
"range_sha256": "5f1db88c47fb9ff4c7405365ce058f6363d2a73e40b103ee50d837adbd740efd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.TRACE",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.trace",
"start_line": 182,
"end_line": 222,
"range_sha256": "f8711391f548a10d67e57d62dab9cb8463c6d7558b019fb109ead47b70b489bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.DENSITY",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.density",
"start_line": 227,
"end_line": 453,
"range_sha256": "0e4676f710ed9810649db6100dcb6596f2bf0eca164aafe7a2565e7171546aff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.RESOLVENT",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.resolvent",
"start_line": 408,
"end_line": 443,
"range_sha256": "f008ac9f4f94145971b8d54b4a70d80531e36f8caa6769e45e91f83e4b5e2650",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.FSCORE.SCOPE",
"unit": "OA-FLOW-L159",
"source_file": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886",
"reader": "OA-FLOW-L159.html#oa-flow.fscore.scope",
"start_line": 457,
"end_line": 461,
"range_sha256": "325282123ea80e27ebb4c06d8a36921de63a99e1fb0068f1128016d476fca8a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.2",
"OA-FLOW.ST.3"
]
},
{
"id": "OA-FLOW.SF.SF0",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sf0",
"start_line": 11,
"end_line": 24,
"range_sha256": "735126020685a0d3ce831bf6a70397ac882ef4f6dab5b3c0c4183d9256763348",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
],
"range_bytes": 2083
},
{
"id": "OA-FLOW.SF.SB0",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb0",
"start_line": 25,
"end_line": 46,
"range_sha256": "4540e65a5d21cdeb0d594a121882839120b4d0c586b44ad2049cc42ef4f6373f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
],
"range_bytes": 3235
},
{
"id": "OA-FLOW.SF.SB1",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb1",
"start_line": 47,
"end_line": 105,
"range_sha256": "f1daa0bedf63f02f56f5dbd01d2b1f4cd632e6b2342e4569cabbc90a838b5ff7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0"
],
"range_bytes": 3544
},
{
"id": "OA-FLOW.SF.SB2",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb2",
"start_line": 106,
"end_line": 166,
"range_sha256": "026b2c6b95b3f3cf1bd089347739be606b8e2eb99cc529041936d14c83e0c79f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1"
],
"range_bytes": 6768
},
{
"id": "OA-FLOW.SF.SB3",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb3",
"start_line": 167,
"end_line": 196,
"range_sha256": "33db1979e405687efe08482de740875fb2ec6b6b57f715378c37e67f229bff1c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2"
],
"range_bytes": 3133
},
{
"id": "OA-FLOW.SF.SB4",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb4",
"start_line": 197,
"end_line": 233,
"range_sha256": "592bba56d9c6e5114869180704d183e753bddd5331c02a2a24d2be087f376ec4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3"
],
"range_bytes": 5284
},
{
"id": "OA-FLOW.SF.SB5",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb5",
"start_line": 234,
"end_line": 248,
"range_sha256": "a75765726f82bf034ab0d598370caeab79703e03e818aa24fbe75cddae170f6a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4"
],
"range_bytes": 2405
},
{
"id": "OA-FLOW.SF.SB6",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sb6",
"start_line": 249,
"end_line": 274,
"range_sha256": "a3fa7dc42de3b3cb1c97d440d7e1a60817c50a86810261b456647b387d73285d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5"
],
"range_bytes": 5727
},
{
"id": "OA-FLOW.SF.SF1",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sf1",
"start_line": 275,
"end_line": 335,
"range_sha256": "d670b211c4d6df725aa12aa015eda2c3df0d1e144a7ff3f319b032e9ee1eabd7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
],
"range_bytes": 5009
},
{
"id": "OA-FLOW.SF.SF2",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sf2",
"start_line": 336,
"end_line": 375,
"range_sha256": "fe9c3cd3b27c1ec45fd3e0f534634e05c18340d7a50d28fe81e40cc7dc84df5a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1"
],
"range_bytes": 2162
},
{
"id": "OA-FLOW.SF.SF3",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sf3",
"start_line": 376,
"end_line": 399,
"range_sha256": "1b4a18d79968b7fdfe2b48bbc98c3c2618c9afa974df4a6fcac55097c9298658",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2"
],
"range_bytes": 1502
},
{
"id": "OA-FLOW.SF.SF4",
"unit": "OA-FLOW-SF",
"source_file": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f",
"reader": "OA-FLOW-SF.html#oa-flow.sf.sf4",
"start_line": 400,
"end_line": 435,
"range_sha256": "448f06708b9df083e43c8c76da7ed2e844eabf6269ebb37a335dc64a910c78f5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3"
],
"range_bytes": 5548
},
{
"id": "OA-FLOW.FF.1",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.1",
"start_line": 15,
"end_line": 48,
"range_sha256": "1cd9f228277bb22804e35b05fc5bc4a40ef10c7bcfa6246f7700fcb356f4595b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.FF.2",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.2",
"start_line": 49,
"end_line": 94,
"range_sha256": "11672a1ce6019c362eed37da50275c5c2f5ae09d271e55fd0eed76dfb2c20b24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1"
]
},
{
"id": "OA-FLOW.FF.3",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.3",
"start_line": 95,
"end_line": 152,
"range_sha256": "955a32f23be7f9d77e9a529daa8ddf09cc898b79d56123feac32d73ddee8939c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2"
]
},
{
"id": "OA-FLOW.FF.4",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.4",
"start_line": 153,
"end_line": 164,
"range_sha256": "1e7b0eb160a8c4bd7123670184c939713bd42b01513cabdc961361efcacd684b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.3"
]
},
{
"id": "OA-FLOW.FF.5",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.5",
"start_line": 165,
"end_line": 204,
"range_sha256": "9129a1c5f59f5057e1d3b3157aaf12839d4c608b785e632c462b7a76ae711bb2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.4"
]
},
{
"id": "OA-FLOW.FF.6",
"unit": "OA-FLOW-FF",
"source_file": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36",
"reader": "OA-FLOW-FF.html#oa-flow.ff.6",
"start_line": 205,
"end_line": 253,
"range_sha256": "813f97020d870284a42b9feb4fb1e9b36e6f9755265aaacf61a173859f2cd142",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.5"
]
},
{
"id": "OA-FLOW.WH02.1",
"unit": "OA-FLOW-WH02",
"source_file": "src/OA-FLOW-WH02.md",
"source_sha256": "5ced6463ce46f108a0ce9ee23a88a83bdcd554ff2a342207870d20092cf9446f",
"reader": "OA-FLOW-WH02.html#oa-flow.wh02.1",
"start_line": 9,
"end_line": 23,
"range_sha256": "972ac063b8c9fec9e38878d3a52cc924f7a1b0ce11baa41ab4ec6999d0e69e47",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.GNS.3.2",
"OA-FLOW.WF.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6"
]
},
{
"id": "OA-FLOW.WH02.2",
"unit": "OA-FLOW-WH02",
"source_file": "src/OA-FLOW-WH02.md",
"source_sha256": "5ced6463ce46f108a0ce9ee23a88a83bdcd554ff2a342207870d20092cf9446f",
"reader": "OA-FLOW-WH02.html#oa-flow.wh02.2",
"start_line": 24,
"end_line": 60,
"range_sha256": "dc34e2e70f2fa3a604d63149563afd53fcc4573778b4d5d7b1d265a372155905",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.GNS.3.2",
"OA-FLOW.WF.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6"
]
},
{
"id": "OA-FLOW.WH02.3",
"unit": "OA-FLOW-WH02",
"source_file": "src/OA-FLOW-WH02.md",
"source_sha256": "5ced6463ce46f108a0ce9ee23a88a83bdcd554ff2a342207870d20092cf9446f",
"reader": "OA-FLOW-WH02.html#oa-flow.wh02.3",
"start_line": 61,
"end_line": 86,
"range_sha256": "fe5d07456f734fd3cb5502b538e65a0a834fe2f595eda4aad74b3ef5702ad8ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.GNS.3.2",
"OA-FLOW.WF.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6",
"OA-FLOW.WH02.2"
]
},
{
"id": "OA-FLOW.WH02.4",
"unit": "OA-FLOW-WH02",
"source_file": "src/OA-FLOW-WH02.md",
"source_sha256": "5ced6463ce46f108a0ce9ee23a88a83bdcd554ff2a342207870d20092cf9446f",
"reader": "OA-FLOW-WH02.html#oa-flow.wh02.4",
"start_line": 87,
"end_line": 91,
"range_sha256": "e951a9a8bf8aac7a224adc6e2244a00f1e81d95df2fd400e05b8b190e661269f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.GNS.3.2",
"OA-FLOW.WF.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6"
]
},
{
"id": "OA-FLOW.WH04.1",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.1",
"start_line": 7,
"end_line": 33,
"range_sha256": "364e3375ec97bcecab61692710f947f0d479a435dd6cdb9a9122ab07ed41f73d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.BD.3"
],
"range_bytes": 2240
},
{
"id": "OA-FLOW.WH04.2",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.2",
"start_line": 34,
"end_line": 60,
"range_sha256": "1db8a21f5a505d11ff476672afde78b4b696e0522c765c79e43d889a580b2e89",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.BD.3"
],
"range_bytes": 1630
},
{
"id": "OA-FLOW.WH04.3",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.3",
"start_line": 61,
"end_line": 74,
"range_sha256": "8cf043ff6b31d713b542dba16b2c85d73d2d4f59f163c8ebdd499d5772364dd3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.BD.3"
],
"range_bytes": 949
},
{
"id": "OA-FLOW.WH04.4",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.4",
"start_line": 75,
"end_line": 94,
"range_sha256": "9fde4095a7eb2738ad9d957fb7db3427e140b5c932333e1702531ab3a10f6028",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.BD.3"
],
"range_bytes": 1611
},
{
"id": "OA-FLOW.WH04.5",
"unit": "OA-FLOW-WH04-WEIGHT",
"source_file": "src/OA-FLOW-WH04-WEIGHT.md",
"source_sha256": "6411579ccc1046c13fdbf05980c8ea503cc715d6410484c8d00bc3f0fbdbede1",
"reader": "OA-FLOW-WH04-WEIGHT.html#oa-flow.wh04.5",
"start_line": 7,
"end_line": 28,
"range_sha256": "7fdc1a41f17e2426adc80919ecf11ceb1efd0394a5d97d9f992c0775889336f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.WF.4",
"OA-FLOW.WF.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.WR.6"
]
},
{
"id": "OA-FLOW.WH04.6",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.6",
"start_line": 95,
"end_line": 110,
"range_sha256": "942ffccbc97cd49adfb4c5fe8a7a3529b10ad4d3655906f6918c04d9f36a060a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.1"
],
"range_bytes": 2607
},
{
"id": "OA-FLOW.WH04.7",
"unit": "OA-FLOW-WH04",
"source_file": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505",
"reader": "OA-FLOW-WH04.html#oa-flow.wh04.7",
"start_line": 111,
"end_line": 115,
"range_sha256": "854b56fee8ae83d0db2e7c964773c3d433065b7a476a81b0f3a1167e8f134a33",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4"
],
"range_bytes": 1055
},
{
"id": "OA-FLOW.MF-SB.1",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.1",
"start_line": 7,
"end_line": 21,
"range_sha256": "270fca6e0364adf412914e52efb6c7c641ca66496b1e8df9bd81bb0e2745bae9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"range_bytes": 1447
},
{
"id": "OA-FLOW.MF-SB.2",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.2",
"start_line": 24,
"end_line": 43,
"range_sha256": "5c2f19fa52462c2f5746c14ad6066e795756eb2d3b993f4f4d3f25e596d9a388",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF4"
],
"range_bytes": 2001
},
{
"id": "OA-FLOW.MF-SB.3",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.3",
"start_line": 46,
"end_line": 104,
"range_sha256": "9264a54c268cc1209ff76f657fbbfd48e630569d2e2466d2deaa113767d960cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF4"
],
"range_bytes": 4546
},
{
"id": "OA-FLOW.MF-SB.4",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.4",
"start_line": 107,
"end_line": 137,
"range_sha256": "0b4aebfb0a44e9aa726dee7fbbb4ef921c2362960bbace14f6e1d5e8eb745264",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF4",
"OA-FLOW.MF-SB.2"
],
"range_bytes": 2270
},
{
"id": "OA-FLOW.MF-SB.5",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.5",
"start_line": 140,
"end_line": 149,
"range_sha256": "a1ba527f66adb72eb7c82fb196fa06fb9784589872c0a66059e32abb8c0db819",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF4",
"OA-FLOW.MF-SB.2",
"OA-FLOW.MF-SB.3",
"OA-FLOW.MF-SB.4"
],
"range_bytes": 723
},
{
"id": "OA-FLOW.MF-SB.6",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.6",
"start_line": 162,
"end_line": 164,
"range_sha256": "0d3e71bbb0eaa82a5438b402ab4c208e3dab1cb1096e0b255f2334bfa492d561",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"range_bytes": 280
},
{
"id": "OA-FLOW.MF06.1",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.1",
"start_line": 7,
"end_line": 26,
"range_sha256": "422df11ae43200cb0d6bf651d5c9d37dd42df10f784a090ecbfb017b21035ee0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": []
},
{
"id": "OA-FLOW.MF06.2",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.2",
"start_line": 27,
"end_line": 82,
"range_sha256": "5101b61f376954088a37f73b25486908df210866edab237360a6cef793d92fce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": []
},
{
"id": "OA-FLOW.MF06.3",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.3",
"start_line": 209,
"end_line": 280,
"range_sha256": "64880d4d407c739ec0dabaccd287b6f2bb41354fff0c327f435e511ffdc8e841",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.HA-R.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.HA-R.5",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 280,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.4",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.4",
"start_line": 281,
"end_line": 304,
"range_sha256": "c12ad83fda58d59e21a8950017091be78e2c954dcf7d2ac6d25b53cdffc91abc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.SF.SB4",
"OA-FLOW.MF06.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 304,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.5",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.5",
"start_line": 305,
"end_line": 373,
"range_sha256": "b8a63930ad6d431f3b38bd3313d30d5d9229b062eb5ba2a2a43766eabbde20f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.HA-R.2",
"OA-FLOW.WH04.3",
"OA-FLOW.SF.SB4",
"OA-FLOW.MF-SB.2",
"OA-FLOW.MF06.3",
"OA-FLOW.MF06.4",
"OA-FLOW.MF06.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 373,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.6",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.6",
"start_line": 135,
"end_line": 208,
"range_sha256": "e9040c367b6a22571cad9755180465c41b1df4d591a858e55a650616284ffbef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.CF.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.MF-SB.2",
"OA-FLOW.MF-SB.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 208,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.7",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.7",
"start_line": 374,
"end_line": 433,
"range_sha256": "5f9490a7a8a883a7d9191caa21a8da72873a83262b0a954aebf2d178c2624746",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.HA-R.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.MF-SB.2",
"OA-FLOW.MF-SB.4",
"OA-FLOW.MF06.5",
"OA-FLOW.MF06.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 433,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.8",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.8",
"start_line": 434,
"end_line": 472,
"range_sha256": "549e3787fc4d66df0042c503bdc26d383b2fd21675912a29be05b75eb79269ca",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.MF06.POLAR",
"OA-FLOW.MF06.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 472,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MF06.md",
"line": 27,
"through_line": 82,
"description": "Arbitrary left Hilbert algebra, full completion, entire bounded-vector spaces and full polar domains; opening theorem statement is context, not a proof premise."
}
]
},
{
"id": "OA-FLOW.MF06.9",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.9",
"start_line": 473,
"end_line": 518,
"range_sha256": "48cd1a48649980e06084a011f7e98d05308beca33a133851735e448843d6412f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MF06.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 518,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.MF06.10",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.10",
"start_line": 519,
"end_line": 523,
"range_sha256": "07cd35fcb5b1c867583fbd55c1b39128b1af7204ab65cf00beea356650cac7e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": []
},
{
"id": "OA-FLOW.L157.1",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.1",
"start_line": 18,
"end_line": 54,
"range_sha256": "5a4e75d3bf25d5461911d6555d34c062be842a8b5a15866613a929710e71627b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 208,
"range_bytes": 1541
},
{
"id": "OA-FLOW.L157.2",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.2",
"start_line": 55,
"end_line": 123,
"range_sha256": "4c175b3a979526d427bfc99c14d57c74a1d49fcccff8d9277b84848e3e372a6a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
],
"range_bytes": 2508
},
{
"id": "OA-FLOW.L157.3",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.3",
"start_line": 124,
"end_line": 189,
"range_sha256": "5a355f85582a3474aacce9eaf168d00da6b1645aeaa051da79a090b5948d2bb3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L157.2"
],
"range_bytes": 2316
},
{
"id": "OA-FLOW.L157.4",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.4",
"start_line": 190,
"end_line": 214,
"range_sha256": "3f682bf11f4eb44cb08d3a894a8a0adcefa1bbb6a58e86b611f47dbf22dc09e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L157.2",
"OA-FLOW.L157.3"
],
"range_bytes": 1602
},
{
"id": "OA-FLOW.L157.5",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.5",
"start_line": 215,
"end_line": 255,
"range_sha256": "d540403f0e78d2f342d347bf32cf682836baf45cf048f331808f3644d71aa089",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L157.2",
"OA-FLOW.L157.3",
"OA-FLOW.L157.4",
"OA-FLOW.BC.4"
],
"range_bytes": 2472
},
{
"id": "OA-FLOW.L157.6",
"unit": "OA-FLOW-L157",
"source_file": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3",
"reader": "OA-FLOW-L157.html#oa-flow.l157.6",
"start_line": 256,
"end_line": 307,
"range_sha256": "7be1fae0dd11f7c12a68654e0c89a7507830158da83f202bdc07d40cbcdc01ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L157.2",
"OA-FLOW.L157.3",
"OA-FLOW.L157.4",
"OA-FLOW.L157.5"
],
"range_bytes": 2477
},
{
"id": "OA-FLOW.L158.1",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.1",
"start_line": 18,
"end_line": 50,
"range_sha256": "9f79c45de32387c9dbfeb4f8c80fd61b92a2dc9af732d92835b0fca5375e8848",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 267,
"range_bytes": 1580
},
{
"id": "OA-FLOW.L158.2",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.2",
"start_line": 51,
"end_line": 74,
"range_sha256": "da4d76c8a1c78b5247f4ea3bac657bbe170a364b0c4b754a1a502f328894dd22",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
],
"range_bytes": 1444
},
{
"id": "OA-FLOW.L158.3",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.3",
"start_line": 75,
"end_line": 127,
"range_sha256": "2f306329a6266c29d8afe5af0f7fcb856383869db49f6ca56ed3e39ea9251801",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2"
],
"range_bytes": 3693
},
{
"id": "OA-FLOW.L158.4",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.4",
"start_line": 128,
"end_line": 224,
"range_sha256": "3c80387ef8f567836a75dadd3d6b4cc0210fa2fa430b0be7ec43778e73efe3ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2",
"OA-FLOW.L158.3"
],
"range_bytes": 4892
},
{
"id": "OA-FLOW.L158.5",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.5",
"start_line": 225,
"end_line": 252,
"range_sha256": "fd84876c11ec146632fd2e62dc86c4dc006ed6dd0ba2a7f151b6c671478a34c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2",
"OA-FLOW.L158.3",
"OA-FLOW.L158.4"
],
"range_bytes": 2247
},
{
"id": "OA-FLOW.L158.6",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.6",
"start_line": 253,
"end_line": 275,
"range_sha256": "aee1938802b5a78f2e54a6dc1aae77e59e5a46ab06c95a4e565d692858f16138",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2",
"OA-FLOW.L158.3",
"OA-FLOW.L158.4",
"OA-FLOW.L158.5"
],
"range_bytes": 3753
},
{
"id": "OA-FLOW.L158.7",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.7",
"start_line": 276,
"end_line": 339,
"range_sha256": "994ed16616e2266e7a9d2b840e562d0b65997d7655e5da3778048c70bc3fee2a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2",
"OA-FLOW.L158.3",
"OA-FLOW.L158.4",
"OA-FLOW.L158.5",
"OA-FLOW.L158.6"
],
"range_bytes": 3898
},
{
"id": "OA-FLOW.L158.8",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.8",
"start_line": 340,
"end_line": 378,
"range_sha256": "89b9c6616f471fc3e39501a57083f9bc5ddcbf463a789cf2a3af9709b5f81fce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.L158.2",
"OA-FLOW.L158.3",
"OA-FLOW.L158.4",
"OA-FLOW.L158.5",
"OA-FLOW.L158.6",
"OA-FLOW.L158.7"
],
"range_bytes": 2417
},
{
"id": "OA-FLOW.L158.9",
"unit": "OA-FLOW-L158",
"source_file": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a",
"reader": "OA-FLOW-L158.html#oa-flow.l158.supported",
"start_line": 379,
"end_line": 469,
"range_sha256": "1a374f713f3346e94a55dba8521b3e37f79dc33fb6dbc53a3abfa712ecd2a5c3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L158.2",
"OA-FLOW.L158.3",
"OA-FLOW.L158.4",
"OA-FLOW.L158.5",
"OA-FLOW.L158.6",
"OA-FLOW.L158.7",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
],
"proof_available_after_line": 469,
"range_bytes": 8627
},
{
"id": "OA-FLOW.CF.1",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.1",
"start_line": 9,
"end_line": 34,
"range_sha256": "2c107a68ff1c30e21b9076b501042618bd072a041790de4fa81e104631e74a3c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"range_bytes": 6171
},
{
"id": "OA-FLOW.CF.2",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.2",
"start_line": 35,
"end_line": 67,
"range_sha256": "a013577520a459db3fc34e7886675a107c56a402919cbb023f181ccfd74e312a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
],
"range_bytes": 3105
},
{
"id": "OA-FLOW.CF.3",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.3",
"start_line": 68,
"end_line": 81,
"range_sha256": "f493838211134830f997e903090ad778cb4109f73423c22bf32910cd49fc87f3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2"
],
"range_bytes": 2206
},
{
"id": "OA-FLOW.CF.4",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.4",
"start_line": 82,
"end_line": 93,
"range_sha256": "1abca590712d5be32b3231bbf25034a7c1a168b2fe7d6da37084819c16780436",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3"
],
"range_bytes": 2892
},
{
"id": "OA-FLOW.CF.5",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.5",
"start_line": 94,
"end_line": 107,
"range_sha256": "6bc4de3e5f107f3c95850989f1ac10d011caff4002dd885fa0d7e2db3c6b17ea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4"
],
"range_bytes": 1884
},
{
"id": "OA-FLOW.CF.6",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.6",
"start_line": 108,
"end_line": 130,
"range_sha256": "3edfd2104bcddcfbca4dc5623a58fda686b32064473e542d7569b662d9efcbc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5"
],
"range_bytes": 3324
},
{
"id": "OA-FLOW.CF.7",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.7",
"start_line": 131,
"end_line": 153,
"range_sha256": "b955b2edb8e1cdc9f6c01996794bafc1e5fce01d196b4e128430c565a88382bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6"
],
"range_bytes": 2168
},
{
"id": "OA-FLOW.CF.8",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.8",
"start_line": 154,
"end_line": 187,
"range_sha256": "8b64e9fcdcc16cefba44ab3bd0647df360f1c64096f20324114e66432e7e57e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
],
"range_bytes": 3319
},
{
"id": "OA-FLOW.CF.9",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.9",
"start_line": 188,
"end_line": 268,
"range_sha256": "e3e27464fb285c0170592a94c530df82cbfe9529f3b42ea9c80a7eca647b1dc7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 5929
},
{
"id": "OA-FLOW.CF.10",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.10",
"start_line": 269,
"end_line": 312,
"range_sha256": "72a3556178e5a0ed18e86e4d0c035c2927225e2b43a9b5c44b873fe6338b0ca0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9"
],
"range_bytes": 3434
},
{
"id": "OA-FLOW.CF.11",
"unit": "OA-FLOW-CF",
"source_file": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357",
"reader": "OA-FLOW-CF.html#oa-flow.cf.11",
"start_line": 313,
"end_line": 321,
"range_sha256": "9a164b470d3e6b4a36532267d6ab617e30a23d34c8092980fa046dd55514231e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": false,
"range_bytes": 1435
},
{
"id": "OA-FLOW.L34.1",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.1",
"start_line": 16,
"end_line": 32,
"range_sha256": "4cf83a877738b538b80c175a53ef4b23751668f207c8c72887fffac528d8effb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.L34.2",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.2",
"start_line": 33,
"end_line": 53,
"range_sha256": "95fc1ec7c59b095e6acef1fad442c1faa3673a34133b1c16e1cf43a899a3e3ad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1"
]
},
{
"id": "OA-FLOW.L34.3",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.3",
"start_line": 54,
"end_line": 91,
"range_sha256": "d2eb33d81a9e4bf24d5cf98ebd6673cc00a0f4b7d045506ec879bedb1d5a559b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2"
]
},
{
"id": "OA-FLOW.L34.4",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.4",
"start_line": 92,
"end_line": 159,
"range_sha256": "36e5a4c0f41102a000a053b77bedbaf3d368a1422813fdfb44b0361f77f6f9f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3"
]
},
{
"id": "OA-FLOW.L34.5",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.5",
"start_line": 160,
"end_line": 193,
"range_sha256": "38f7f9a8e953719f976accc010b3303aec1bdf44c12c36523fcb113e2b1ec205",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4"
]
},
{
"id": "OA-FLOW.L34.5a",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.5a",
"start_line": 194,
"end_line": 209,
"range_sha256": "2448193aac92e0b5f433844b3dc26fec3463e0722096cc4d3cb27eedcb8f4f45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5"
]
},
{
"id": "OA-FLOW.L34.6",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.6",
"start_line": 210,
"end_line": 237,
"range_sha256": "e4eda215f360d2b8e2e00c3ba1058b4855bf03ccc4ec1c8c300cd98474f4d60b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a"
]
},
{
"id": "OA-FLOW.L34.7",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.7",
"start_line": 238,
"end_line": 272,
"range_sha256": "d80e79b67c5a6539824c221f9b55c7018ebd7b38c1019b038a87b12ff300ce4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6"
]
},
{
"id": "OA-FLOW.L34.8",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.8",
"start_line": 273,
"end_line": 311,
"range_sha256": "11969904396d963267c11b940c8fec9b4204300d7834ae50334573d3f075ac0e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6",
"OA-FLOW.L34.7"
]
},
{
"id": "OA-FLOW.L34.8a",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.8a",
"start_line": 312,
"end_line": 399,
"range_sha256": "ba81b21d3fb182a28426c9b03c1d05cd067949d06888f54bbf57fc848de033e5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6",
"OA-FLOW.L34.7",
"OA-FLOW.L34.8"
]
},
{
"id": "OA-FLOW.L34.9",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.9",
"start_line": 400,
"end_line": 451,
"range_sha256": "52be1d4462b2bfb596434a64b2cc2ec120e03589f77c00327c8256092ff48e91",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6",
"OA-FLOW.L34.7",
"OA-FLOW.L34.8",
"OA-FLOW.L34.8a"
]
},
{
"id": "OA-FLOW.L34.10",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.10",
"start_line": 452,
"end_line": 493,
"range_sha256": "1bcc948c133bfed4a9f9aa60eca3910af7a683febf7e028377d6c59a413d03a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6",
"OA-FLOW.L34.7",
"OA-FLOW.L34.8",
"OA-FLOW.L34.8a",
"OA-FLOW.L34.9"
]
},
{
"id": "OA-FLOW.L34.Appendix-A",
"unit": "OA-FLOW-L34",
"source_file": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6",
"reader": "OA-FLOW-L34.html#oa-flow.l34.appendix-a",
"start_line": 494,
"end_line": 550,
"range_sha256": "575e62d130268924091cf16e2758fda61ed143f1001a025bbb4266ee0d33086f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.L34.1",
"OA-FLOW.L34.2",
"OA-FLOW.L34.3",
"OA-FLOW.L34.4",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.L34.6",
"OA-FLOW.L34.7",
"OA-FLOW.L34.8",
"OA-FLOW.L34.8a",
"OA-FLOW.L34.9",
"OA-FLOW.L34.10"
]
},
{
"id": "OA-FLOW.MF-SB.7",
"unit": "OA-FLOW-MF-SB",
"source_file": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46",
"reader": "OA-FLOW-MF-SB.html#oa-flow.mf-sb.7",
"start_line": 152,
"end_line": 159,
"range_sha256": "a5e5c89ccacad43f1bff1033ec14ca92adb2203856f920fffac64009bb2ff0d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MF-SB.3"
],
"may_be_used_as_earlier_proved_premise": false,
"range_bytes": 1349
},
{
"id": "OA-FLOW.CP.1",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.1",
"start_line": 9,
"end_line": 27,
"range_sha256": "1c592fbfd0817845a2190fd536a7c4343d850c99759ad8a7a990ae573c9126c0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0"
]
},
{
"id": "OA-FLOW.CP.2",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.2",
"start_line": 30,
"end_line": 63,
"range_sha256": "22251dfd5fa1fd948f8205e126a3627994742731d34a5980fac20441cbacc780",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1"
]
},
{
"id": "OA-FLOW.CP.3",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.3",
"start_line": 66,
"end_line": 93,
"range_sha256": "6b3b2b54240b94571c6202e0a72d61ae39a36ba3ef89321596ed2744b72ea57e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2"
]
},
{
"id": "OA-FLOW.CP.4",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.4",
"start_line": 96,
"end_line": 125,
"range_sha256": "c4d77b8618e265357a09e4e814be8925ecd19a3f2ab53b81fa0c3734d0716075",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3"
]
},
{
"id": "OA-FLOW.CP.5",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.5",
"start_line": 128,
"end_line": 159,
"range_sha256": "618dfa9bd66e9dc96cfea36baa169d5aeacc5e224ad58f303b3b6c9e0d1f2844",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4"
]
},
{
"id": "OA-FLOW.CP.6",
"unit": "OA-FLOW-CP",
"source_file": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a",
"reader": "OA-FLOW-CP.html#oa-flow.cp.6",
"start_line": 162,
"end_line": 190,
"range_sha256": "6948ab1f87cb34b375648f9d3556f0ebf95218a06873d8480d8398c35eef86e8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5"
]
},
{
"id": "OA-FLOW.CI.1",
"unit": "OA-FLOW-CI",
"source_file": "src/OA-FLOW-CI.md",
"source_sha256": "776a46025ddc560e38207993b85be6f616ed42530a2a4754b18c84a5cdf9d401",
"reader": "OA-FLOW-CI.html#oa-flow.ci.1",
"start_line": 11,
"end_line": 36,
"range_sha256": "381caaa6ae0f535ef670f9e713e64ac5d4e0ae9e7f87de85f616fcb1890b4a7a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0"
]
},
{
"id": "OA-FLOW.CI.2",
"unit": "OA-FLOW-CI",
"source_file": "src/OA-FLOW-CI.md",
"source_sha256": "776a46025ddc560e38207993b85be6f616ed42530a2a4754b18c84a5cdf9d401",
"reader": "OA-FLOW-CI.html#oa-flow.ci.2",
"start_line": 39,
"end_line": 70,
"range_sha256": "000577f6fa3116acf7ede744eb97becf092bfead468b9993e27d73e21f3c3ea2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.CI.3",
"unit": "OA-FLOW-CI",
"source_file": "src/OA-FLOW-CI.md",
"source_sha256": "776a46025ddc560e38207993b85be6f616ed42530a2a4754b18c84a5cdf9d401",
"reader": "OA-FLOW-CI.html#oa-flow.ci.3",
"start_line": 73,
"end_line": 140,
"range_sha256": "a36e4d8ea71aa2215c1a49a9c4dfc0a2cae0e174888c429cfbd5ec7159cd97a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.CI.4",
"unit": "OA-FLOW-CI",
"source_file": "src/OA-FLOW-CI.md",
"source_sha256": "776a46025ddc560e38207993b85be6f616ed42530a2a4754b18c84a5cdf9d401",
"reader": "OA-FLOW-CI.html#oa-flow.ci.4",
"start_line": 143,
"end_line": 169,
"range_sha256": "d71b595d4ce2e40f1cf76d9722c46a8f5b7047b98a729f843bec5cea0a3f4443",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.BD.1",
"unit": "OA-FLOW-BD",
"source_file": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e",
"reader": "OA-FLOW-BD.html#oa-flow.bd.1",
"start_line": 11,
"end_line": 25,
"range_sha256": "3a24dfa76eaf5fafb3791ee7aa3253b447ef13d07436180ba66c1e381a60af47",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0"
]
},
{
"id": "OA-FLOW.BD.2",
"unit": "OA-FLOW-BD",
"source_file": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e",
"reader": "OA-FLOW-BD.html#oa-flow.bd.2",
"start_line": 28,
"end_line": 44,
"range_sha256": "dfb0e875ee080143411024aa978e2b70e4d533b7bbb4504146f9b7d498c57a77",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.BD.3",
"unit": "OA-FLOW-BD",
"source_file": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e",
"reader": "OA-FLOW-BD.html#oa-flow.bd.3",
"start_line": 47,
"end_line": 78,
"range_sha256": "74e5139aa883d8d6b83ed1bf036f47a98081eefbc8d025b37a1b6ca79908613d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.BD.4",
"unit": "OA-FLOW-BD",
"source_file": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e",
"reader": "OA-FLOW-BD.html#oa-flow.bd.4",
"start_line": 81,
"end_line": 91,
"range_sha256": "8c0798d04c87ab1a88ed9d617527338d34c0c917c8e2a84a76e81a5cdaa31a85",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3"
]
},
{
"id": "OA-FLOW.BD.5",
"unit": "OA-FLOW-BD",
"source_file": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e",
"reader": "OA-FLOW-BD.html#oa-flow.bd.5",
"start_line": 94,
"end_line": 107,
"range_sha256": "5e7e23ed0da865c50cda6918ceac7f99c964b9f3e78f9cdaac220b8649be9eb5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BD.4",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.HA-R.1",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.1",
"start_line": 11,
"end_line": 22,
"range_sha256": "ea1f300de5df28956479d4b04b96e7118dc9581855a018486324a71727bfff8d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.1",
"OA-FLOW.BD.1"
]
},
{
"id": "OA-FLOW.HA-R.2",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.2",
"start_line": 25,
"end_line": 66,
"range_sha256": "6a0d788d02d9506f9b76a18436ed042089ca1e8a2ddb7bb80361e3bf0b9a8596",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.CI.1"
]
},
{
"id": "OA-FLOW.HA-R.3",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.3",
"start_line": 69,
"end_line": 100,
"range_sha256": "a66a0f7f1e034ccef81268c7eab4f54a8e2be1e23e6b230fb0f9d2a41a8c75e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.CI.1",
"OA-FLOW.BD.1"
]
},
{
"id": "OA-FLOW.HA-R.4",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.4",
"start_line": 103,
"end_line": 127,
"range_sha256": "b76dc0845a80658aae6cc4286c1f1c26e34e5cf23ab69412952059d2d0c5e5f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.5",
"OA-FLOW.CI.1",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.HA-R.5",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.5",
"start_line": 130,
"end_line": 221,
"range_sha256": "9b595e469d3456adeecf42431b113946a0b561b3871db4298951e2ce86d6e82c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.HA-R.6",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.6",
"start_line": 224,
"end_line": 236,
"range_sha256": "9c654f64624ac920647c44b715472b2316f8d8cbcd2324f7422e9691a0359dc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.BD.3"
]
},
{
"id": "OA-FLOW.HA-R.7",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.7",
"start_line": 239,
"end_line": 264,
"range_sha256": "5d8004f65e6dfa8ec0cd143dd567b71c8c0f66c115b5ae77c6538dc16ff68d35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.HA-R.8",
"unit": "OA-FLOW-HA-R",
"source_file": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611",
"reader": "OA-FLOW-HA-R.html#oa-flow.ha-r.8",
"start_line": 267,
"end_line": 277,
"range_sha256": "14a0b7a788faf91b3069b96501e4f12686dac0fb9601a9645ff4b0005c5d03ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7"
]
},
{
"id": "OA-FLOW.ST.1",
"unit": "OA-FLOW-ST12",
"source_file": "src/OA-FLOW-ST12.md",
"source_sha256": "700c9ed27a668738ba8289d3ccdede950f41ecd77847e7e1d6af4efa239d5599",
"reader": "OA-FLOW-ST12.html#oa-flow.st.1",
"start_line": 13,
"end_line": 29,
"range_sha256": "8d3b2178fd84dc1bd07bff4cb8a2244185f0829bc2e5c71ae30db6f88621a9e2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CP.6"
],
"range_bytes": 2034
},
{
"id": "OA-FLOW.ST.2",
"unit": "OA-FLOW-ST12",
"source_file": "src/OA-FLOW-ST12.md",
"source_sha256": "700c9ed27a668738ba8289d3ccdede950f41ecd77847e7e1d6af4efa239d5599",
"reader": "OA-FLOW-ST12.html#oa-flow.st.2",
"start_line": 32,
"end_line": 54,
"range_sha256": "14a884bb7a297cbf3df57331c7b219ccd912c85e1c8fb712532a34249c09b86d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.GNS.3.2",
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"range_bytes": 3226
},
{
"id": "OA-FLOW.ST.3",
"unit": "OA-FLOW-ST",
"source_file": "src/OA-FLOW-ST.md",
"source_sha256": "f3b9b64d9615a66347e42ce1d2590e918fa6dd163260173cad30bf1c8b5898c5",
"reader": "OA-FLOW-ST.html#oa-flow.st.3",
"start_line": 16,
"end_line": 41,
"range_sha256": "86e174ee195b390c6cb55a5b456a9e9788ddcf8204eefbc329f67aca7ffa43e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CI.4",
"OA-FLOW.HA-R.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.COEFFICIENT"
],
"range_bytes": 2746
},
{
"id": "OA-FLOW.ST.4",
"unit": "OA-FLOW-ST",
"source_file": "src/OA-FLOW-ST.md",
"source_sha256": "f3b9b64d9615a66347e42ce1d2590e918fa6dd163260173cad30bf1c8b5898c5",
"reader": "OA-FLOW-ST.html#oa-flow.st.4",
"start_line": 42,
"end_line": 46,
"range_sha256": "4c56d158154a402ee7c7cfe75a16c638c3d63ac30c3026b50c884936ae8a1304",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CI.4",
"OA-FLOW.HA-R.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.MF06.8",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.WR.6",
"OA-FLOW.WF.4",
"OA-FLOW.EW.5",
"OA-FLOW.MW.4"
],
"range_bytes": 995
},
{
"id": "OA-FLOW.WF.1",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.1",
"start_line": 13,
"end_line": 44,
"range_sha256": "9383fa3a44515654fd0cb9f865bba1812ab73f7ee2335aa731b8d675cba0f693",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7"
],
"range_bytes": 1698
},
{
"id": "OA-FLOW.WF.2",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.2",
"start_line": 47,
"end_line": 69,
"range_sha256": "f7e5758238f44fc3cc0ee35750ff1ba92ab30577cef2fc86b5e24d15f92cec31",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.BD.2",
"OA-FLOW.CF.4"
],
"range_bytes": 2527
},
{
"id": "OA-FLOW.WF.3",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.3",
"start_line": 72,
"end_line": 116,
"range_sha256": "0610cff40dff67c33b4f78ba5774ca243a92370b4c0752619debbd99b9904aa7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.HA-R.4",
"OA-FLOW.CF.7"
],
"range_bytes": 2050
},
{
"id": "OA-FLOW.WF.4",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.4",
"start_line": 119,
"end_line": 166,
"range_sha256": "20351f15786efac0d200a72595aeeef6f643b0ce03c74eea2fcc615fa8697390",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.1",
"OA-FLOW.WF.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.SF.SF0"
],
"range_bytes": 2573
},
{
"id": "OA-FLOW.WF.5",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.5",
"start_line": 169,
"end_line": 194,
"range_sha256": "e6a874e2e2ca3d26b9e368eb29b49a0a649377067b2e207ba67d55d7265419a9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.BD.3",
"OA-FLOW.BD.5",
"OA-FLOW.CF.4"
],
"range_bytes": 2306
},
{
"id": "OA-FLOW.WF.6",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.6",
"start_line": 197,
"end_line": 237,
"range_sha256": "4cb8b4bbec4b924e3d0f50a853265ef73d6d44476e40136156fac0daf6a82c76",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.1",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.CF.10",
"OA-FLOW.HA-R.7"
],
"range_bytes": 2317
},
{
"id": "OA-FLOW.WF.7",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.7",
"start_line": 240,
"end_line": 247,
"range_sha256": "43ca7e259929700ee805469c99f8c6316c2896c45074724fb22619fbca4c16ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6"
],
"range_bytes": 1697
},
{
"id": "OA-FLOW.WF.8",
"unit": "OA-FLOW-WF",
"source_file": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8",
"reader": "OA-FLOW-WF.html#oa-flow.wf.8",
"start_line": 250,
"end_line": 265,
"range_sha256": "6c81820b8218eef7b4e2347bbd2ccb849741b80ad5e54210cc6a8785b5dfad70",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WF.3",
"OA-FLOW.WF.5",
"OA-FLOW.BD.5"
],
"range_bytes": 3347
},
{
"id": "OA-FLOW.ST.COEFFICIENT",
"unit": "OA-FLOW-STC",
"source_file": "src/OA-FLOW-STC.md",
"source_sha256": "6905b695aad18e71172e326d2d77f4572ea4519444280bbfbe2fe9b380dd7dfe",
"reader": "OA-FLOW-STC.html#oa-flow.st.coefficient",
"start_line": 9,
"end_line": 26,
"range_sha256": "6bab22c7ac0336d4f067928a41f2bd331a3fc3d0da03576936b0b6c92efce50c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.GW.1",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.1",
"start_line": 13,
"end_line": 59,
"range_sha256": "fce79101059acae7dbbc36719b4684c29b4f3b9066a7f22bc9d7664632596e95",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.GW.2",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.2",
"start_line": 62,
"end_line": 86,
"range_sha256": "83eb7b8ac4a293a34a2bb8e15c5a0023a9a15a8c1cb9644e986ee2740a009a1b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1"
]
},
{
"id": "OA-FLOW.GW.3",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.3",
"start_line": 89,
"end_line": 120,
"range_sha256": "578efab29ddc09b93df78cf17f5608baa0299e0e98c8f7e548330f1917d75b59",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.GW.4",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.4",
"start_line": 123,
"end_line": 167,
"range_sha256": "2f3e555a1997c4669997c64b14ebe234bd3f1cc9a8ec85daea080a05fd1bb6a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.WF.2",
"OA-FLOW.BD.5",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.GW.5",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.5",
"start_line": 170,
"end_line": 204,
"range_sha256": "8f35e602e3a3e7726a4ef484689669e8b6bc3a235701fac9af2fb09735a9ec1b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.WF.2"
]
},
{
"id": "OA-FLOW.GW.6",
"unit": "OA-FLOW-GW",
"source_file": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2",
"reader": "OA-FLOW-GW.html#oa-flow.gw.6",
"start_line": 207,
"end_line": 218,
"range_sha256": "ab21808dd2562de7443afe8d82a4d3de5ecb71b8954357b41465d6958f113b03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.5"
]
},
{
"id": "OA-FLOW.NF.1",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.1",
"start_line": 13,
"end_line": 35,
"range_sha256": "1fbd28217415430959ceb7da37fac1f9db2fcec9a779059551758dbe6f1ca327",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.2",
"OA-FLOW.WF.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.NF.2",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.2",
"start_line": 36,
"end_line": 52,
"range_sha256": "557da7ee1a73a5e4de4cd6a4f7dde8e20d342e25a001ebe5b29c5dffeae92838",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.1",
"OA-FLOW.CF.1",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.ST.COEFFICIENT"
]
},
{
"id": "OA-FLOW.NF.3",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.3",
"start_line": 53,
"end_line": 121,
"range_sha256": "10aaf72d0c9bd29678bfe33b0f0807c955ff594b337d9cbab55328651c02df14",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.FF.5",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.NF.4",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.4",
"start_line": 122,
"end_line": 164,
"range_sha256": "dcd4b5a71dc1a986256affa87ce0fd515826834ef47c8c51795ea931ab33649f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.CF.1",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2",
"OA-FLOW.BD.5"
]
},
{
"id": "OA-FLOW.NF.5",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.5",
"start_line": 165,
"end_line": 189,
"range_sha256": "242411a6a444ca64d1762cc0b0506f4b55863e54b92664bbba10f8ee91fc7a3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.NF.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.NF.6",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.6",
"start_line": 190,
"end_line": 219,
"range_sha256": "30cf344004bd0e6a490585e9ed2bdfb179617e843f529ccffa571cc347f671f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.NF.4",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2",
"OA-FLOW.BD.5"
]
},
{
"id": "OA-FLOW.NF.7",
"unit": "OA-FLOW-NF",
"source_file": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27",
"reader": "OA-FLOW-NF.html#oa-flow.nf.7",
"start_line": 220,
"end_line": 256,
"range_sha256": "e3c743092abab6bf78120a563160647456996522ae1ce506506826f01466ca79",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.3",
"OA-FLOW.NF.4"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H0",
"unit": "OA-FLOW-TOPOLOGY",
"source_file": "src/OA-FLOW-TOPOLOGY.md",
"source_sha256": "bc8a5db2cd33669d0b4eed82a56f622b185f2f5900e2fcc73ae71dfac23c6cd8",
"reader": "OA-FLOW-TOPOLOGY.html#oa-flow.xgaps.cstar.h0",
"start_line": 4,
"end_line": 44,
"range_sha256": "f4dc694826e41b0069f0d5faa3e0548e078fa300a252f70458269ca0f037ff3e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H1",
"unit": "OA-FLOW-HARMONIC",
"source_file": "src/OA-FLOW-HARMONIC.md",
"source_sha256": "543b5cee0579fecf19b41edeb0c8d80f4299932fe4126a2f11d4aaae0a035ef2",
"reader": "OA-FLOW-HARMONIC.html#oa-flow.xgaps.cstar.h1",
"start_line": 9,
"end_line": 68,
"range_sha256": "ac2ca4b8fc6755f447756178ba0f61e7e39f357ff83c171258db6d5c52842e1f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.9"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H2",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#oa-flow.xgaps.cstar.h2",
"start_line": 9,
"end_line": 45,
"range_sha256": "dba868874d31f1176e205e0bf7726a12399ae73843c2b861da1ff1ea338c3b8d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.SC.0",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.9",
"OA-FLOW.SP.0",
"OA-FLOW.SP.2",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H3",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#oa-flow.xgaps.cstar.h3",
"start_line": 46,
"end_line": 98,
"range_sha256": "60a1179d853683a168a8a418d56ecde920aa021310b7c1981b17631fe3d35b91",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.CF.5",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.9",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H4",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#oa-flow.xgaps.cstar.h4",
"start_line": 99,
"end_line": 119,
"range_sha256": "366d7200932c44d9136922974779f421746f9c01c8e10ecf66626cba8d3115f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.9",
"OA-FLOW.SP.2",
"OA-FLOW.SP.3"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.H5",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#oa-flow.xgaps.cstar.h5",
"start_line": 120,
"end_line": 127,
"range_sha256": "bd1b91acab4e826bae23f1348fbe4b47566ad2f8fcd61a314237cfb0279e62b4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.9"
]
},
{
"id": "OA-FLOW.XGAPS.CSTAR.INVERSION",
"unit": "OA-FLOW-HARMONIC-LATE",
"source_file": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722",
"reader": "OA-FLOW-HARMONIC-LATE.html#harmonic-integrable-inversion",
"start_line": 129,
"end_line": 268,
"range_sha256": "bdb278a7a560114c20d6b996b2215aa1d1e9ede9cc6279ed6a5c44b18099c5f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.HR.3",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 268,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HARMONIC-LATE.md",
"line": 129,
"through_line": 146,
"description": "Arbitrary LCA group with paired dual Haar measures. Negative forward Fourier transform. Original kernel is L1; its actual transform is L2 in the first theorem and L1 in the inversion consequence. No separability or sigma compactness is assumed."
},
{
"source": "src/OA-FLOW-HARMONIC-LATE.md",
"line": 129,
"through_line": 268,
"description": "Complete common-domain identification via compact-tail estimates for the neighbourhood net, summable-error comparison of L1 and L2 limits, full inverse integral, uniform continuity and equality at continuity points."
}
]
},
{
"id": "OA-FLOW.HR.TOPOLOGY",
"unit": "OA-FLOW-TOPOLOGY",
"source_file": "src/OA-FLOW-TOPOLOGY.md",
"source_sha256": "bc8a5db2cd33669d0b4eed82a56f622b185f2f5900e2fcc73ae71dfac23c6cd8",
"reader": "OA-FLOW-TOPOLOGY.html#oa-flow.hr.topology",
"start_line": 4,
"end_line": 44,
"range_sha256": "f4dc694826e41b0069f0d5faa3e0548e078fa300a252f70458269ca0f037ff3e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.HR.1",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-01",
"start_line": 15,
"end_line": 54,
"range_sha256": "459f1b3a750f06a525e3f68479f41802564ec7d9f9503f38b104d64e4cc52340",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY"
]
},
{
"id": "OA-FLOW.HR.2",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-02",
"start_line": 58,
"end_line": 118,
"range_sha256": "070320e01e5031cafaf4ff7d0917c6142cd17207bca0b0de3e0f0631074764b5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.1"
]
},
{
"id": "OA-FLOW.HR.3",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-03",
"start_line": 122,
"end_line": 144,
"range_sha256": "6da9a0598b269f09beac7c758c74d11e40d18d0a16bc5a7149b4258a5526545c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.1",
"OA-FLOW.HR.2"
]
},
{
"id": "OA-FLOW.HR.4",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-04",
"start_line": 148,
"end_line": 165,
"range_sha256": "935d6e8cbbbd7ca74a0f5373f7f01fb6dd3e20d83812c7379d41b229b94047b7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.1"
]
},
{
"id": "OA-FLOW.HR.5",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-05",
"start_line": 169,
"end_line": 241,
"range_sha256": "36ea8ff094329dde4577760e2ab593ca744f57eb4222aeb0d8eff09c69911fee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.1",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3"
]
},
{
"id": "OA-FLOW.HR.6",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-06",
"start_line": 245,
"end_line": 276,
"range_sha256": "a30f25aa64b469d24e0a62572ad4134299ab67ac38e4259ad31cdd2c65deb4ba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.4"
]
},
{
"id": "OA-FLOW.HR.7",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-07",
"start_line": 280,
"end_line": 321,
"range_sha256": "14b89e8fc8bb48ed27ab6d44aa53b3a76f1a55d1fe5c8e0df4725f1fbcfc4a8a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6"
]
},
{
"id": "OA-FLOW.HR.8",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-08",
"start_line": 325,
"end_line": 340,
"range_sha256": "f41ed7ad29410c617e2e214565b30d413b0adb64398f851eb799a28e6aea4534",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.6"
]
},
{
"id": "OA-FLOW.HR.9",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-09",
"start_line": 344,
"end_line": 376,
"range_sha256": "af21f041cd524d650049ea1cbb8207aa7437eff1ed1b919990f3492653a07871",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.8"
]
},
{
"id": "OA-FLOW.HR.10",
"unit": "OA-FLOW-HR",
"source_file": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5",
"reader": "OA-FLOW-HR.html#hr-10",
"start_line": 380,
"end_line": 398,
"range_sha256": "162d51a7b746a71c48557f59d9235e5ca5d8e3e012c451c5cb1f6488b8291510",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9"
]
},
{
"id": "OA-FLOW.SP.0",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.0",
"start_line": 1,
"end_line": 47,
"range_sha256": "2971996f663e2b16d35933c890fda10d9b9fc37c096bf35ad52a7ec4d19a087b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.SC.10",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.REGULAR",
"OA-FLOW.GRP.COMPLETIONS"
],
"range_bytes": 8208
},
{
"id": "OA-FLOW.SP.1",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.1",
"start_line": 51,
"end_line": 100,
"range_sha256": "b7e5bb9e79b7c2fd6338cc75d73ebe1b0331a3bb0d8c391627306851a3cd9d92",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.0",
"OA-FLOW.CF.6",
"OA-FLOW.CF.10",
"OA-FLOW.HR.2",
"OA-FLOW.HR.6",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.INTEGRATION"
],
"range_bytes": 4515
},
{
"id": "OA-FLOW.SP.2",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.2",
"start_line": 104,
"end_line": 165,
"range_sha256": "854310b22f9f9f54f132a84ec652d4b2983e7ae025b62a99c4528f0eefdc27e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.0",
"OA-FLOW.SP.1",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.7",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.TRANSLATIONS"
],
"range_bytes": 5116
},
{
"id": "OA-FLOW.SP.3",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.3",
"start_line": 169,
"end_line": 226,
"range_sha256": "3856de41528927753c63d17f97139e21505b0a06468bf8bae3b24df285cb3ee9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.0",
"OA-FLOW.SP.1",
"OA-FLOW.SP.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.9",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.CF.10",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.REGULAR"
],
"range_bytes": 4650
},
{
"id": "OA-FLOW.SP.4",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.4",
"start_line": 230,
"end_line": 256,
"range_sha256": "c5d4383bf757d02196771ea79ac72a4584fd49db660b50f93396907af7c8bc71",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.0",
"OA-FLOW.SP.3",
"OA-FLOW.HR.3",
"OA-FLOW.HR.9",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"range_bytes": 1612
},
{
"id": "OA-FLOW.SP.5",
"unit": "OA-FLOW-PLANCHEREL",
"source_file": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3",
"reader": "OA-FLOW-PLANCHEREL.html#oa-flow.sp.5",
"start_line": 260,
"end_line": 277,
"range_sha256": "c18dfc62fbc725a43d0d91a5b045364679022962e43d0c307fcfbda162435f22",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.0",
"OA-FLOW.SP.1",
"OA-FLOW.SP.2",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4"
],
"range_bytes": 2059
},
{
"id": "OA-FLOW.CV.1",
"unit": "OA-FLOW-CV",
"source_file": "src/OA-FLOW-CV.md",
"source_sha256": "a50eca2ce7684dbd1cae54941b0449e3428ecd4fd524edd6a6ef81a80b3d243d",
"reader": "OA-FLOW-CV.html#oa-flow.cv.1",
"start_line": 11,
"end_line": 43,
"range_sha256": "8e30ad0335d148be889b28705da3810cdb10dd9a0232540b7315072a5656923d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.CV.2",
"unit": "OA-FLOW-CV",
"source_file": "src/OA-FLOW-CV.md",
"source_sha256": "a50eca2ce7684dbd1cae54941b0449e3428ecd4fd524edd6a6ef81a80b3d243d",
"reader": "OA-FLOW-CV.html#oa-flow.cv.2",
"start_line": 44,
"end_line": 61,
"range_sha256": "00c0d6762e8037dd182b0290704493d8821053922969ca43b183fd04a08ae5dc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CV.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.CV.3",
"unit": "OA-FLOW-CV",
"source_file": "src/OA-FLOW-CV.md",
"source_sha256": "a50eca2ce7684dbd1cae54941b0449e3428ecd4fd524edd6a6ef81a80b3d243d",
"reader": "OA-FLOW-CV.html#oa-flow.cv.3",
"start_line": 62,
"end_line": 71,
"range_sha256": "4a8ff9b180785b203113d1457f8bf798eb71e6fe6ec65eaae86f90ca64d456dc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.4",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.CV.4",
"unit": "OA-FLOW-CV",
"source_file": "src/OA-FLOW-CV.md",
"source_sha256": "a50eca2ce7684dbd1cae54941b0449e3428ecd4fd524edd6a6ef81a80b3d243d",
"reader": "OA-FLOW-CV.html#oa-flow.cv.4",
"start_line": 72,
"end_line": 107,
"range_sha256": "80dcdd5afe040125bbc0ed8560e41c3604403f54a4575196674d613ebd942b15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CV.1",
"OA-FLOW.CV.3",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.EW.1",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.1",
"start_line": 1,
"end_line": 54,
"range_sha256": "a69f97de92f714687c28ccb7f619316556b50ccda5bafbaa8bf24cc720f58c9f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.CF.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.EW.2",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.2",
"start_line": 55,
"end_line": 125,
"range_sha256": "e40ea3ab400f7991af47a6671937220b958b787035c5975930bd15ec1bbabd6e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.1",
"OA-FLOW.NF.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6"
]
},
{
"id": "OA-FLOW.EW.3",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.3",
"start_line": 126,
"end_line": 143,
"range_sha256": "e725336cfcbdcde856e7fad1a6a56cdf7593730e3600d105ab2977dea6c218f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.2",
"OA-FLOW.CV.2",
"OA-FLOW.CV.3",
"OA-FLOW.CV.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.EW.4",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.4",
"start_line": 144,
"end_line": 204,
"range_sha256": "4c5481ba35e52b0086cfa9ce0606a33bc4ec341b0edf936ae9dbde6dd394142e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.3",
"OA-FLOW.CV.3",
"OA-FLOW.CV.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.EW.5",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.5",
"start_line": 205,
"end_line": 221,
"range_sha256": "5190c32f123a3ea9d0b71a95397ffa07f2974899536d2416c57a73f1d21ec201",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.3",
"OA-FLOW.EW.4",
"OA-FLOW.CV.1",
"OA-FLOW.BD.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.EW.6",
"unit": "OA-FLOW-EW",
"source_file": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4",
"reader": "OA-FLOW-EW.html#oa-flow.ew.6",
"start_line": 224,
"end_line": 226,
"range_sha256": "a2cd258b33182f9f970cfd23ba29dbd564f680957e33ce6d85c7f6e9c451fc29",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.WR.1",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.1",
"start_line": 1,
"end_line": 57,
"range_sha256": "3fad1577ee632b9508729dd853d9f1e8b28bb95138305171ca1d6f68fdefa2dc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.EW.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.WR.2",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.2",
"start_line": 58,
"end_line": 82,
"range_sha256": "771fc668ffd906f9e04bbf3e4a53b4ae1e7fd50936d42698d56de64602b5472f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.1",
"OA-FLOW.EW.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.WR.3",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.3",
"start_line": 83,
"end_line": 111,
"range_sha256": "c974ea6149a2c9f3b1f505cfe151cadee220da530dee1c0831eafa397e7ee4cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.5",
"OA-FLOW.ST.2",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3"
]
},
{
"id": "OA-FLOW.WR.4",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.4",
"start_line": 112,
"end_line": 163,
"range_sha256": "aaea414df9cb29899ea64fb132cf4eb20ed194ba26602a67cde844550c7be468",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.WR.3",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.WF.1"
]
},
{
"id": "OA-FLOW.WR.5",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.5",
"start_line": 164,
"end_line": 187,
"range_sha256": "cab27ae19e3495f4da9ad82d89f5a37614a83df95aadd137fbc34b38cf88889b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.4",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3"
]
},
{
"id": "OA-FLOW.WR.6",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.6",
"start_line": 188,
"end_line": 233,
"range_sha256": "f7bdcc3f51ef457d7152d92f6d660ad9a257e396b7e7354dd57a65d4a805fed5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.HA-R.7",
"OA-FLOW.NF.5",
"OA-FLOW.GW.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.WR.7",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.7",
"start_line": 234,
"end_line": 249,
"range_sha256": "9ac0d16b064f7246d762533e07c735eb322fcdede15b640c8b446a1b3f7223ca",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.6",
"OA-FLOW.CV.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.HA-R.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.WR.8",
"unit": "OA-FLOW-WR",
"source_file": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693",
"reader": "OA-FLOW-WR.html#oa-flow.wr.8",
"start_line": 250,
"end_line": 300,
"range_sha256": "304b1cff5b97e370dc0fde236cff05a8e86ff91735f8fe3659525544ef3d3b35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.2",
"OA-FLOW.WR.1",
"OA-FLOW.WR.4",
"OA-FLOW.WR.6",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.MW.1",
"unit": "OA-FLOW-MW",
"source_file": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c",
"reader": "OA-FLOW-MW.html#oa-flow.mw.1",
"start_line": 1,
"end_line": 51,
"range_sha256": "ab3069cc4be8a97185af88815e4dc2618fa862d0634402a8077de1db03e38c30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MF06.8",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.HA-R.7",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3"
]
},
{
"id": "OA-FLOW.MW.2",
"unit": "OA-FLOW-MW",
"source_file": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c",
"reader": "OA-FLOW-MW.html#oa-flow.mw.2",
"start_line": 52,
"end_line": 84,
"range_sha256": "a8a3227c7a9e96fae2cb5e7cc6696352a391a4bb011413abc171894bd87fd967",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MW.1",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.WR.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.MW.3",
"unit": "OA-FLOW-MW",
"source_file": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c",
"reader": "OA-FLOW-MW.html#oa-flow.mw.3",
"start_line": 85,
"end_line": 117,
"range_sha256": "cd49336d7450448d264b7de9d919eba566fa911deee8ef0859e052cdc651e78f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MF06.8",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.MW.4",
"unit": "OA-FLOW-MW",
"source_file": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c",
"reader": "OA-FLOW-MW.html#oa-flow.mw.4",
"start_line": 118,
"end_line": 164,
"range_sha256": "af473959f31293ea74022540b0441f61f04e3b6207f434a3968159a53e311a34",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.WR.6",
"OA-FLOW.WR.7",
"OA-FLOW.ST.2",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6"
]
},
{
"id": "OA-FLOW.MW.5",
"unit": "OA-FLOW-MW",
"source_file": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c",
"reader": "OA-FLOW-MW.html#oa-flow.mw.5",
"start_line": 165,
"end_line": 206,
"range_sha256": "5abe3b61bdac64d8989e907924211f5ed3e2fe5d02d23e8bba8ab19450a1de92",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.WR.8",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.NO.1",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.1",
"start_line": 1,
"end_line": 43,
"range_sha256": "a93f6b277e1b7e7dba89fabb8c02c201b299fca1ec8d35010efdd9856cdde637",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.1",
"OA-FLOW.NF.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.WF.2"
],
"range_bytes": 6385
},
{
"id": "OA-FLOW.NO.2",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.2",
"start_line": 44,
"end_line": 80,
"range_sha256": "ca5384b81a393f25e4cb63b53810c8f908658e6a824bf41472cd47aa56ebba15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 1679
},
{
"id": "OA-FLOW.NO.3",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.3",
"start_line": 81,
"end_line": 118,
"range_sha256": "bc8039638d895d0ac8aedaa59f56b55aa6428b1c86b2ecc315f56c11a0f9c2c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.EW.1",
"OA-FLOW.EW.2",
"OA-FLOW.EW.3",
"OA-FLOW.EW.4",
"OA-FLOW.EW.5",
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.NF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 1801
},
{
"id": "OA-FLOW.NO.4",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.4",
"start_line": 119,
"end_line": 157,
"range_sha256": "8ca7dc6f9bf28869ab77f92ac74f7775c65d3102d817218aed4f9f0e2d4f0a62",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.CV.3",
"OA-FLOW.BD.5",
"OA-FLOW.EW.1",
"OA-FLOW.EW.2",
"OA-FLOW.EW.3",
"OA-FLOW.EW.4",
"OA-FLOW.EW.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
],
"range_bytes": 2752
},
{
"id": "OA-FLOW.NO.5",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.5",
"start_line": 158,
"end_line": 195,
"range_sha256": "a94aac1824753a6a725d022ba581085c1751ae711c62e330f929d51d19097caf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.NO.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 1862
},
{
"id": "OA-FLOW.NO.6",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.6",
"start_line": 196,
"end_line": 211,
"range_sha256": "6d5e6ac73bed545a7356331ea5fd8b77598a909ab06a9456945b7960d6680ce9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.NO.4",
"OA-FLOW.NO.5",
"OA-FLOW.ST.2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.WR.4",
"OA-FLOW.HA-R.5",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.WR.6"
],
"range_bytes": 1505
},
{
"id": "OA-FLOW.NO.7",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.7",
"start_line": 212,
"end_line": 217,
"range_sha256": "6c992bede99d24458809be2ee23fa731245f0598c9249cf5f82603118c3c31d4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.NO.4",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6"
],
"range_bytes": 873
},
{
"id": "OA-FLOW.NO.8",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.8",
"start_line": 218,
"end_line": 225,
"range_sha256": "6ab7a7533b410e701a39d46932350f54b09342582fcb86f26d92b5191a60c31c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6"
],
"range_bytes": 813
},
{
"id": "OA-FLOW.NO.9",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.9",
"start_line": 226,
"end_line": 245,
"range_sha256": "e2443451f3c73abee5f15c2033f28da646d51d9079669248ba42d3f0bd69eaba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 1490
},
{
"id": "OA-FLOW.NO.10",
"unit": "OA-FLOW-NO",
"source_file": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d",
"reader": "OA-FLOW-NO.html#oa-flow.no.10",
"start_line": 246,
"end_line": 276,
"range_sha256": "e8b47d0d3cc5aa0708c9b7cae5decd6dd95f0c6bafb62a9d48ff782c7fcafd76",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6",
"OA-FLOW.WF.2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 3165
},
{
"id": "OA-FLOW.KT.1",
"unit": "OA-FLOW-KT",
"source_file": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72",
"reader": "OA-FLOW-KT.html#oa-flow.kt.1",
"start_line": 1,
"end_line": 62,
"range_sha256": "5d57c41cdf8f95cc552811a5c177ea26ee732ed1f08f42ffea9b8f34537c86cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.MW.4"
]
},
{
"id": "OA-FLOW.KT.2",
"unit": "OA-FLOW-KT",
"source_file": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72",
"reader": "OA-FLOW-KT.html#oa-flow.kt.2",
"start_line": 63,
"end_line": 104,
"range_sha256": "40ae54a82a553dbd4df3c9eab43b9f93a8c3255c9c9d5071498dd56c0fb77f73",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.1",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5"
]
},
{
"id": "OA-FLOW.KT.3",
"unit": "OA-FLOW-KT",
"source_file": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72",
"reader": "OA-FLOW-KT.html#oa-flow.kt.3",
"start_line": 105,
"end_line": 147,
"range_sha256": "c12648f43824504241780eaf1ce1d172db554ff2d8108a244d8c98b6cf403025",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.KT.4",
"unit": "OA-FLOW-KT",
"source_file": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72",
"reader": "OA-FLOW-KT.html#oa-flow.kt.4",
"start_line": 148,
"end_line": 207,
"range_sha256": "278cf93bed25a03d4b5ffbcb2f4050000511de572689411847cb3ccf6d2b5d83",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.KT.5",
"unit": "OA-FLOW-KT",
"source_file": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72",
"reader": "OA-FLOW-KT.html#oa-flow.kt.5",
"start_line": 208,
"end_line": 230,
"range_sha256": "a2a98f46d720a21357adb10a6ff814bd59754e6cddd9f276b39f166716281288",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.EW.2",
"OA-FLOW.EW.3",
"OA-FLOW.EW.5",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3"
]
},
{
"id": "OA-FLOW.KU.1",
"unit": "OA-FLOW-KU",
"source_file": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec",
"reader": "OA-FLOW-KU.html#oa-flow.ku.1",
"start_line": 1,
"end_line": 59,
"range_sha256": "43095938fb9020d417487c04c71977a0bb057152fec82f0c625e69d6558653b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.1",
"OA-FLOW.WR.2",
"OA-FLOW.NF.5",
"OA-FLOW.MW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.KU.2",
"unit": "OA-FLOW-KU",
"source_file": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec",
"reader": "OA-FLOW-KU.html#oa-flow.ku.2",
"start_line": 60,
"end_line": 97,
"range_sha256": "15f1a2dc0d40791480f40c8fe3ba59db9b75f67f10ba648f5eaf2f55b3c78fa2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KU.1",
"OA-FLOW.KT.3",
"OA-FLOW.EW.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.KU.3",
"unit": "OA-FLOW-KU",
"source_file": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec",
"reader": "OA-FLOW-KU.html#oa-flow.ku.3",
"start_line": 98,
"end_line": 203,
"range_sha256": "0baf89a4c5b5a64e6961439d2e273df2249b7edf35fba7d05fac57cf9de944bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF3",
"OA-FLOW.FF.2",
"OA-FLOW.WR.3",
"OA-FLOW.GW.5",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.KU.4",
"unit": "OA-FLOW-KU",
"source_file": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec",
"reader": "OA-FLOW-KU.html#oa-flow.ku.4",
"start_line": 204,
"end_line": 216,
"range_sha256": "d8c93f6d85ca58eb3a837023f3ffb6a15742ea00b8ec586318e5b7550706633a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.SF.SF4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.KU.5",
"unit": "OA-FLOW-KU",
"source_file": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec",
"reader": "OA-FLOW-KU.html#oa-flow.ku.5",
"start_line": 217,
"end_line": 256,
"range_sha256": "892b1fceef6c132c2d4630b3ba757d2e925d9b2e4ad6ddb5088454070e734ac5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KU.3",
"OA-FLOW.MW.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.PC.0",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc0",
"start_line": 1,
"end_line": 27,
"range_sha256": "5c95286a26cb95ac80efbe43f9807bda265d8e5672b3f7ba3b795504304f8714",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.PC.1",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc1",
"start_line": 28,
"end_line": 61,
"range_sha256": "16a7f864500006cf74d55b9f955dc028e44a9f18559bea8c5abb8dd60d9ec118",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CP.1"
]
},
{
"id": "OA-FLOW.PC.2",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc2",
"start_line": 62,
"end_line": 84,
"range_sha256": "e39e6658b76cc2cc9ff6a9bfaa4e85ad43ed897f425384383188a86acfcb68d7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.PC.3",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc3",
"start_line": 85,
"end_line": 108,
"range_sha256": "27defab61893f61c3cbb5263c0972789d8220f9436cf19ea612519867fc2e50f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
]
},
{
"id": "OA-FLOW.PC.4",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc4",
"start_line": 109,
"end_line": 132,
"range_sha256": "e9086cde9d3cc80cd19816b0bed232dae02c383279a929122c484e3494c5f1c6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.PC.5",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc5",
"start_line": 133,
"end_line": 162,
"range_sha256": "7480e71476923280784bda5bd4d11149bcd857e3d1aa2060365e023abf12af52",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.PC.6",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc6",
"start_line": 163,
"end_line": 178,
"range_sha256": "56342645fb7d38844d0706c98e8996e33fe33b70533c5889d482e146882fa23c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5"
]
},
{
"id": "OA-FLOW.PC.7",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc7",
"start_line": 179,
"end_line": 203,
"range_sha256": "f18e6cc9a83d45cd91ac2a9b600bfc65e2437a40c2f816265aefee1aa3528df6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.PC.3"
]
},
{
"id": "OA-FLOW.PC.8",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.projection.pc8",
"start_line": 204,
"end_line": 215,
"range_sha256": "95fce5acf6481a8c06c299e17d5ae4e4dffb8ffad14dc3b9901de17ee6ed3809",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.PC.9",
"unit": "OA-FLOW-PC",
"source_file": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463",
"reader": "OA-FLOW-PC.html#oa-flow.pc.9",
"start_line": 216,
"end_line": 236,
"range_sha256": "2b0a45be5c7181257e93580ba373e5fbcf6f5f0870de829e1cee0cb62ec061ef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.BC.1",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.1",
"start_line": 1,
"end_line": 71,
"range_sha256": "9ba22db30838ab7bb03966c9ecdfc7624c034e3b17394e91274ea8213f68bf9f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.BC.2",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.2",
"start_line": 72,
"end_line": 129,
"range_sha256": "1130d8a141677dc480a16e8385398ee4074648dbe0675d09a4fc1d5834f4a5af",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.BC.3",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.3",
"start_line": 130,
"end_line": 166,
"range_sha256": "02811429fc9ca13872bed2dd17bbba1f62a3f60d1937665832437128622bc55f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.MW.4",
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.BC.4",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.4",
"start_line": 167,
"end_line": 218,
"range_sha256": "4d2b610028c54deafd2df8c126933cda2c17165e58fdd5707af910c4bc8db26a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.MW.4"
]
},
{
"id": "OA-FLOW.BC.5",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.5",
"start_line": 219,
"end_line": 248,
"range_sha256": "9a4ec302b39743f922694690ad3e3d0e8e3cdc7c31f8c222f03273a882aded00",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.GW.3"
]
},
{
"id": "OA-FLOW.BC.6",
"unit": "OA-FLOW-BC",
"source_file": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274",
"reader": "OA-FLOW-BC.html#oa-flow.bc.6",
"start_line": 249,
"end_line": 317,
"range_sha256": "7d341490b9929dcf7e0e7ee99edc65a62916592e9471d2ad0ce08bc5d6df61a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.EP.1",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.1",
"start_line": 1,
"end_line": 55,
"range_sha256": "1e0e24f1ec14ed56d425b8c8eaa9b223f663a0258af484e8e4e440064e9c8d3c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.EP.2",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.2",
"start_line": 56,
"end_line": 120,
"range_sha256": "57500402065b423569e2f51c77da53008468abbf1cf92add21a0f975cd08dca8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.EP.3",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.3",
"start_line": 121,
"end_line": 158,
"range_sha256": "d9501d1d8ce379736be2087dc83ab1edb1cb8920d3df6b502037c16762d21363",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.FF.5",
"OA-FLOW.FF.6",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.EP.4",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.4",
"start_line": 159,
"end_line": 179,
"range_sha256": "7de562cf30844d5bf2f6387909a6063d44619e07f274dee0b93c17fad13d4b42",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.EP.5",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.5",
"start_line": 180,
"end_line": 223,
"range_sha256": "d96a49fc1e226f64b5f6451db46a2b3d2fb65a39085163246d90b8743abdec01",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EW.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.EP.6",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.6",
"start_line": 224,
"end_line": 279,
"range_sha256": "9e780df07e26dcc5d300cb4b74a011ec9a1061d858e532df136f4c4737a9585a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.EP.7",
"unit": "OA-FLOW-EP",
"source_file": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6",
"reader": "OA-FLOW-EP.html#oa-flow.ep.7",
"start_line": 280,
"end_line": 324,
"range_sha256": "f338dd2060b48978c9d9afddded4c98e10585ff6a60b19991523533eec3b6ee6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.AS.1",
"unit": "OA-FLOW-AS",
"source_file": "src/OA-FLOW-AS.md",
"source_sha256": "ed52cc65c52f569ec7aa09e0633eef3058b29667995fdc64afd07f9190517449",
"reader": "OA-FLOW-AS.html#oa-flow.as.1",
"start_line": 1,
"end_line": 72,
"range_sha256": "62f208a8b6d1a5a8a3d68c5b9ee269cada0667f203b3b8332c9909f88e3e0f81",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.AS.2",
"unit": "OA-FLOW-AS",
"source_file": "src/OA-FLOW-AS.md",
"source_sha256": "ed52cc65c52f569ec7aa09e0633eef3058b29667995fdc64afd07f9190517449",
"reader": "OA-FLOW-AS.html#oa-flow.as.2",
"start_line": 73,
"end_line": 86,
"range_sha256": "bcc2f062f03850d5f2b0eee555b2cc40960a6ef6169ebf691c1ec9903d5daa50",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.AS.3",
"unit": "OA-FLOW-AS",
"source_file": "src/OA-FLOW-AS.md",
"source_sha256": "ed52cc65c52f569ec7aa09e0633eef3058b29667995fdc64afd07f9190517449",
"reader": "OA-FLOW-AS.html#oa-flow.as.3",
"start_line": 87,
"end_line": 180,
"range_sha256": "9803fa5f9bbce41b962e3242947f4875c6987fc94abe8acd9bc11b3d9c5a8130",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AS.2",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.AS.4",
"unit": "OA-FLOW-AS",
"source_file": "src/OA-FLOW-AS.md",
"source_sha256": "ed52cc65c52f569ec7aa09e0633eef3058b29667995fdc64afd07f9190517449",
"reader": "OA-FLOW-AS.html#oa-flow.as.4",
"start_line": 181,
"end_line": 227,
"range_sha256": "c41eba1b1b93791a1d9071755b8f4b00c87f21c925a3ee5589723b5b6c382a90",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AS.1",
"OA-FLOW.AS.2",
"OA-FLOW.AS.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.MA.1",
"unit": "OA-FLOW-MA",
"source_file": "src/OA-FLOW-MA.md",
"source_sha256": "e674e5058e4ee80cb1875ef61256a7dd6b4747a3f91e7462c2347f162caf836a",
"reader": "OA-FLOW-MA.html#oa-flow.ma.1",
"start_line": 1,
"end_line": 34,
"range_sha256": "6d4a260ea023d7d3e22f1961b7633dc6657623f1e3570ea10268845a951f8878",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5"
]
},
{
"id": "OA-FLOW.MA.2",
"unit": "OA-FLOW-MA",
"source_file": "src/OA-FLOW-MA.md",
"source_sha256": "e674e5058e4ee80cb1875ef61256a7dd6b4747a3f91e7462c2347f162caf836a",
"reader": "OA-FLOW-MA.html#oa-flow.ma.2",
"start_line": 35,
"end_line": 111,
"range_sha256": "e7f829f8e6f07fa26aebb92682c39481a7c5451cbb7343d71a6bfa0b61a60f18",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MA.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.MA.3",
"unit": "OA-FLOW-MA",
"source_file": "src/OA-FLOW-MA.md",
"source_sha256": "e674e5058e4ee80cb1875ef61256a7dd6b4747a3f91e7462c2347f162caf836a",
"reader": "OA-FLOW-MA.html#oa-flow.ma.3",
"start_line": 112,
"end_line": 223,
"range_sha256": "a1c43a893fc2e509533431e7705f42de211e9fd7141e9cee143767bb968a1535",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MA.1",
"OA-FLOW.MA.2",
"OA-FLOW.AS.1",
"OA-FLOW.AS.2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF4",
"OA-FLOW.KT.3",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SB4",
"OA-FLOW.MW.3"
]
},
{
"id": "OA-FLOW.MA.4",
"unit": "OA-FLOW-MA",
"source_file": "src/OA-FLOW-MA.md",
"source_sha256": "e674e5058e4ee80cb1875ef61256a7dd6b4747a3f91e7462c2347f162caf836a",
"reader": "OA-FLOW-MA.html#oa-flow.ma.4",
"start_line": 224,
"end_line": 265,
"range_sha256": "ef7c4f96b82dde60ee8f6af871203506d6d7bbb428a8f2ccaf379a3838ec86f3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MA.1",
"OA-FLOW.MA.2",
"OA-FLOW.MA.3",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.AS.1",
"OA-FLOW.AS.2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.OT.1",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.1",
"start_line": 1,
"end_line": 85,
"range_sha256": "90c869822141e6bb74e99c0ce352f272072ea029b260d0ad2cc8618d514864d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.OT.2",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.2",
"start_line": 86,
"end_line": 132,
"range_sha256": "84742116f3296d7238484fd74abe11ef37b4e623477d9f85b8d5633c754d8ba4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.1",
"OA-FLOW.EP.6",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2"
]
},
{
"id": "OA-FLOW.OT.3",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.3",
"start_line": 133,
"end_line": 196,
"range_sha256": "8dd466c387367fb8747163bd957bffbfb7cda0988b65c0bce781a59e463179b7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.1",
"OA-FLOW.OT.2",
"OA-FLOW.MA.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2"
]
},
{
"id": "OA-FLOW.OT.4",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.4",
"start_line": 197,
"end_line": 280,
"range_sha256": "2b6a2d658a5d0b7f9cd219d86412d779455de48fc4b8640c98bc9b915aaf004f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.2",
"OA-FLOW.OT.3",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.OT.5",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.5",
"start_line": 281,
"end_line": 308,
"range_sha256": "397a7cf122bd7b797620351d59b9f14137de5e93b5261d9b4cceffac5615012f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.3",
"OA-FLOW.OT.4",
"OA-FLOW.MA.4",
"OA-FLOW.AS.1",
"OA-FLOW.AS.2",
"OA-FLOW.AS.3",
"OA-FLOW.AS.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.OT.6",
"unit": "OA-FLOW-OT",
"source_file": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f",
"reader": "OA-FLOW-OT.html#oa-flow.ot.6",
"start_line": 309,
"end_line": 420,
"range_sha256": "d1e8b06b2807f4aa9fdc10429e9e59bc15734f673cb32d86a0d7dc3ee4c457c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.1",
"OA-FLOW.OT.2",
"OA-FLOW.OT.3",
"OA-FLOW.OT.4",
"OA-FLOW.OT.5",
"OA-FLOW.MA.2",
"OA-FLOW.MA.3",
"OA-FLOW.MA.4",
"OA-FLOW.AS.1",
"OA-FLOW.AS.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.WS.1",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws1",
"start_line": 1,
"end_line": 43,
"range_sha256": "fb90b3ac2e61e864af852501bb1ddbde78362a30c68aa99d9de9f959803d0080",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.WS.2",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws2",
"start_line": 44,
"end_line": 82,
"range_sha256": "19c851e3d16e29648c59051d9d691ac9265b6fc585eff7a56ba449e886516112",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.1",
"OA-FLOW.NO.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.WS.3",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws3",
"start_line": 83,
"end_line": 151,
"range_sha256": "e2027fa79d81c20553fadb3881c91e63acc840329cfc21bd1d99213d40562666",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.2",
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.NO.4",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6",
"OA-FLOW.EW.5",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5"
]
},
{
"id": "OA-FLOW.WS.4",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws4",
"start_line": 152,
"end_line": 247,
"range_sha256": "a8ab17505c3838bd929469bda67140965a8c1e5377eee7d835748c0725638f52",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.3",
"OA-FLOW.NO.1",
"OA-FLOW.NO.2",
"OA-FLOW.NO.3",
"OA-FLOW.NO.4",
"OA-FLOW.NO.5",
"OA-FLOW.NO.6",
"OA-FLOW.NF.5",
"OA-FLOW.EW.5",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CV.3"
]
},
{
"id": "OA-FLOW.GF.1",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.gf1",
"start_line": 248,
"end_line": 304,
"range_sha256": "592a6d74e9c20f3778fa12e4fca58d9f28a2b12ef5c3374eb02e05e8b20cb3ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.4",
"OA-FLOW.MW.4",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.GF.2",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.gf2",
"start_line": 305,
"end_line": 380,
"range_sha256": "7ed809c0cf622542c04e7dd3bfb02c290429094ab54ac13da23c72319aae02fe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GF.1",
"OA-FLOW.GW.4",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF4",
"OA-FLOW.CF.1",
"OA-FLOW.KT.3",
"OA-FLOW.CP.6",
"OA-FLOW.FF.2",
"OA-FLOW.EW.3"
]
},
{
"id": "OA-FLOW.GF.3",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.gf3",
"start_line": 381,
"end_line": 452,
"range_sha256": "129c87fc643d09140604ad5bd94a3c236279ce96b4b3e4a0877da25334653754",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GF.1",
"OA-FLOW.GF.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.FF.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF4",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.GF.4",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.gf4",
"start_line": 453,
"end_line": 522,
"range_sha256": "85b57a40340a9d39492f187674cbd48e141eb30bc695df46a37c92128034a203",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GF.1",
"OA-FLOW.GF.2",
"OA-FLOW.GF.3",
"OA-FLOW.WS.2",
"OA-FLOW.WS.4",
"OA-FLOW.NO.2",
"OA-FLOW.NO.4",
"OA-FLOW.NO.6",
"OA-FLOW.CV.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF4",
"OA-FLOW.FF.2"
]
},
{
"id": "OA-FLOW.WS.5",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws5",
"start_line": 523,
"end_line": 530,
"range_sha256": "39326b34735d63163d7118d59261ee16c61903b0e77a97da1bd0456872f98555",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.1",
"OA-FLOW.WS.2",
"OA-FLOW.WS.3",
"OA-FLOW.WS.4",
"OA-FLOW.GF.1",
"OA-FLOW.GF.2",
"OA-FLOW.GF.3",
"OA-FLOW.GF.4"
]
},
{
"id": "OA-FLOW.WS.6",
"unit": "OA-FLOW-WS",
"source_file": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55",
"reader": "OA-FLOW-WS.html#oa-flow.weight-sum.ws6",
"start_line": 531,
"end_line": 564,
"range_sha256": "841d69cc11d4caaa3cc3114394f72d4ffe9cd09d518f3d416ec7b311ebed5722",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.1",
"OA-FLOW.WS.2",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.TD.1",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.1",
"start_line": 1,
"end_line": 72,
"range_sha256": "efe12fcf6e1ce222c14797ab8377e6da6e584c561e5f807be097dc0a81f4faf8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.1",
"OA-FLOW.ST.2",
"OA-FLOW.EP.4",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1"
]
},
{
"id": "OA-FLOW.TD.2",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.2",
"start_line": 73,
"end_line": 118,
"range_sha256": "3c045833a1efccad1fc34a9dd8b34d12a9f0fcaf113cc87ca5f89e78effc4e6d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5"
]
},
{
"id": "OA-FLOW.TD.3",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.3",
"start_line": 119,
"end_line": 204,
"range_sha256": "121919becf3b46329f5e33f4ee4ce3a8c83ac39eaeddc57144744f2dadc3b22d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.1",
"OA-FLOW.TD.2",
"OA-FLOW.HA-R.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.GW.4",
"OA-FLOW.EW.3",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.TD.4",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.4",
"start_line": 205,
"end_line": 230,
"range_sha256": "f4ca8b3daa12499cbf2462979e60782d02e6a9cb2eab72e95e7d16466dfdae5e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.2",
"OA-FLOW.TD.3",
"OA-FLOW.EP.1",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.WS.1",
"OA-FLOW.WS.2",
"OA-FLOW.WS.3",
"OA-FLOW.WS.4",
"OA-FLOW.WS.5",
"OA-FLOW.GF.1",
"OA-FLOW.GF.2",
"OA-FLOW.GF.3",
"OA-FLOW.GF.4"
]
},
{
"id": "OA-FLOW.TD.5",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.5",
"start_line": 231,
"end_line": 252,
"range_sha256": "9243f19d5f7971fdd46370d964c2332a772f9fe95f4cf7b84a73485f9604fb1f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.2",
"OA-FLOW.TD.3",
"OA-FLOW.TD.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.GW.1"
]
},
{
"id": "OA-FLOW.TD.6",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.6",
"start_line": 253,
"end_line": 309,
"range_sha256": "a6b7afc7192c61e2f94248c2a780ce716ec749f310658fde210d53fc20dcfa65",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.2",
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.TD.7",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.7",
"start_line": 310,
"end_line": 341,
"range_sha256": "515ac52ffeb0e4b5b175c4202a44dd9aa2e031e31ed5a424a5985030fe2cc61d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.2",
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.TD.6",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.FF.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.TD.8",
"unit": "OA-FLOW-TD",
"source_file": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59",
"reader": "OA-FLOW-TD.html#oa-flow.td.8",
"start_line": 342,
"end_line": 455,
"range_sha256": "afe9d89f05f6513818e5b04e73437a59c4772cae290e4b8c91e0c54c6a6584ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.1",
"OA-FLOW.TD.2",
"OA-FLOW.TD.3",
"OA-FLOW.TD.4",
"OA-FLOW.TD.6",
"OA-FLOW.TD.7",
"OA-FLOW.EP.2",
"OA-FLOW.EP.5",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.GW.4"
]
},
{
"id": "OA-FLOW.CZ.0",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.0",
"start_line": 1,
"end_line": 54,
"range_sha256": "53655d7145588076af77bf0aad4673a1d9ec3b1a8e54f7a1de55bad8a93f95fe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MA.2",
"OA-FLOW.MA.3",
"OA-FLOW.AS.2",
"OA-FLOW.AS.3",
"OA-FLOW.AS.4",
"OA-FLOW.SF.SF4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.MW.4"
]
},
{
"id": "OA-FLOW.CZ.1",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.1",
"start_line": 55,
"end_line": 136,
"range_sha256": "40f364dddeaa1be2e80c1875ef18fb00491485eb7829dcdb440728d477358b24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.EW.2",
"OA-FLOW.EW.3",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CZ.2",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.2",
"start_line": 137,
"end_line": 185,
"range_sha256": "7ca3569f9ceb5c3448c5f641504137276af5c0bdbc84764d743f08105bb2ed87",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CZ.3",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.3",
"start_line": 186,
"end_line": 278,
"range_sha256": "fe0c6d9ebaa9e5631649f131c17340f651b60e061f4f61db4626ab37ea5f7e67",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.CZ.4",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.4",
"start_line": 279,
"end_line": 328,
"range_sha256": "b8fa5ce7dcbaa5ccc2057131d91a5b33b92efb05636b7b513b5ea10c85c07924",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.MA.2",
"OA-FLOW.MA.3",
"OA-FLOW.AS.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4",
"OA-FLOW.MW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CZ.5",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.5",
"start_line": 329,
"end_line": 367,
"range_sha256": "8699eb05a6490560c93f3e730bd694e65a7fe57a544392f8d4a44233305634c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.KT.1",
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.KT.5",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CZ.6",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.6",
"start_line": 368,
"end_line": 414,
"range_sha256": "f7e9d39e6e43d21401243ea4e3e82575d72641d5724305ceb318135349717a4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.5",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CZ.7",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.7",
"start_line": 415,
"end_line": 466,
"range_sha256": "e904398422f0fe886fb5ca641e3b312fbf19ce2186f12a3871f62252d764a7fc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.BC.5",
"OA-FLOW.KT.5",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CZ.8",
"unit": "OA-FLOW-CZ",
"source_file": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1",
"reader": "OA-FLOW-CZ.html#oa-flow.cz.8",
"start_line": 467,
"end_line": 513,
"range_sha256": "67d8e69e2b8f416656b979b3520fdb5ae99faa8e2c539ab27737ee0eb346dc04",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.6",
"OA-FLOW.CZ.7",
"OA-FLOW.KT.5",
"OA-FLOW.CF.10",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.NC.1",
"unit": "OA-FLOW-NC",
"source_file": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d",
"reader": "OA-FLOW-NC.html#oa-flow.nc.1",
"start_line": 1,
"end_line": 40,
"range_sha256": "0be8aa3b46b8b28bd3f1b09627c8f6e97636dfb3a3c7dc036849f68e2b9f7d27",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.5",
"OA-FLOW.HA-R.4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.NC.2",
"unit": "OA-FLOW-NC",
"source_file": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d",
"reader": "OA-FLOW-NC.html#oa-flow.nc.2",
"start_line": 41,
"end_line": 122,
"range_sha256": "67d11d72e33b179df737bb76bbe47240ac51279209652782ac66c81185be45c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.1",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.WF.7",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.3",
"OA-FLOW.HA-R.4",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.NC.3",
"unit": "OA-FLOW-NC",
"source_file": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d",
"reader": "OA-FLOW-NC.html#oa-flow.nc.3",
"start_line": 123,
"end_line": 184,
"range_sha256": "c18910a3759d3d905d26d1765893b3b077db4bc79dc9369260ba168cbdde52e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.2",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.FF.2",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.NC.4",
"unit": "OA-FLOW-NC",
"source_file": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d",
"reader": "OA-FLOW-NC.html#oa-flow.nc.4",
"start_line": 185,
"end_line": 237,
"range_sha256": "c2ba6f32a99ec2c911b2f2ab69938554697820b98d745af6ed746552692a7fd4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.GF.1",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.NC.5",
"unit": "OA-FLOW-NC",
"source_file": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d",
"reader": "OA-FLOW-NC.html#oa-flow.nc.5",
"start_line": 238,
"end_line": 253,
"range_sha256": "93817ebaf1726d425bf21488476efb6820ef9d2c817e4760d648bc51e44266a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CR.1",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.1",
"start_line": 1,
"end_line": 81,
"range_sha256": "4903d6983ac7ff2535e5aa668d4587cf51a38cc14ac96dc206957bc6552ca89b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.NC.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SB0",
"OA-FLOW.PC.1"
]
},
{
"id": "OA-FLOW.CR.2",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.2",
"start_line": 82,
"end_line": 119,
"range_sha256": "e5da50d5d743dc536928b001d163956588ba424ec57ef07fa0e1ee884cf58eab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.PC.1",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.CR.3",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.3",
"start_line": 120,
"end_line": 161,
"range_sha256": "a022a7d461cc382d4756306b0cd042faa26eb1f6611ae9112e375aef713c0a00",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.CR.2",
"OA-FLOW.NC.4",
"OA-FLOW.ST.2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CR.4",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.4",
"start_line": 162,
"end_line": 192,
"range_sha256": "cea44a298fe06c04bf6997a5d369c966b601be76cc9d677928d909b5fb640391",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.3",
"OA-FLOW.NC.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.HA-R.7",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CR.5",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.5",
"start_line": 193,
"end_line": 261,
"range_sha256": "aacdb23047add1b291937609e8b36e8ebbc0b1e341f5c537301c03813d1baf22",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.4",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.5",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1"
]
},
{
"id": "OA-FLOW.CR.6",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.6",
"start_line": 262,
"end_line": 293,
"range_sha256": "6279593e6e8db44b21a13be66e71e491aa48278eaec5327147834f5656ee162e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.5",
"OA-FLOW.EP.2",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4"
]
},
{
"id": "OA-FLOW.CR.7",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.7",
"start_line": 294,
"end_line": 402,
"range_sha256": "38fe3cfce5bfe09e104de7bed9fc41e6c2017d8bb34e8eeea5d34ba1d5af9075",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.CF.8",
"OA-FLOW.CF.1",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.FF.1",
"OA-FLOW.EP.2"
]
},
{
"id": "OA-FLOW.CR.8",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.8",
"start_line": 403,
"end_line": 439,
"range_sha256": "165dbe487df1a5626618d5abdcffaa5904efb59a223573ea40da0d9adb63c5a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.6",
"OA-FLOW.CR.7",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.CR.9",
"unit": "OA-FLOW-CR",
"source_file": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380",
"reader": "OA-FLOW-CR.html#oa-flow.cr.9",
"start_line": 440,
"end_line": 450,
"range_sha256": "74d3265543c9286ea986392b0a9e95a3bdb63eb9402f36905b2cf261268bf464",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.EP.1",
"OA-FLOW.CF.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.MC.1",
"unit": "OA-FLOW-MC",
"source_file": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc",
"reader": "OA-FLOW-MC.html#oa-flow.mc.1",
"start_line": 1,
"end_line": 42,
"range_sha256": "4b7e93715f4217567f50404784c64f3b05ab756110e407201e0ef86bd0bf972c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.8",
"OA-FLOW.CR.9",
"OA-FLOW.CF.10",
"OA-FLOW.ST.2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.MC.2",
"unit": "OA-FLOW-MC",
"source_file": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc",
"reader": "OA-FLOW-MC.html#oa-flow.mc.2",
"start_line": 43,
"end_line": 98,
"range_sha256": "06fff9224aa2017fd3e7e562923e1a743e10f7bfd9be6ef2b74f14c8bc9510d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3"
]
},
{
"id": "OA-FLOW.MC.3",
"unit": "OA-FLOW-MC",
"source_file": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc",
"reader": "OA-FLOW-MC.html#oa-flow.mc.3",
"start_line": 99,
"end_line": 157,
"range_sha256": "051d425bca2f15f9c286600156b5736017178adeef16653c7eb72fa304a8828c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.2",
"OA-FLOW.CR.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.6",
"OA-FLOW.CR.8",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2"
]
},
{
"id": "OA-FLOW.MC.4",
"unit": "OA-FLOW-MC",
"source_file": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc",
"reader": "OA-FLOW-MC.html#oa-flow.mc.4",
"start_line": 158,
"end_line": 196,
"range_sha256": "af0d60872dc4bba50180a792856334d1120995f278c735ededaf2ef8f01ff725",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.2",
"OA-FLOW.MC.3",
"OA-FLOW.CR.6",
"OA-FLOW.CR.8",
"OA-FLOW.NC.4"
]
},
{
"id": "OA-FLOW.MC.5",
"unit": "OA-FLOW-MC",
"source_file": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc",
"reader": "OA-FLOW-MC.html#oa-flow.mc.5",
"start_line": 197,
"end_line": 290,
"range_sha256": "4d35691496be26b6b3ec8fdc8520817041f9502bf551d6ce5d9d6e08ab00b8cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.1",
"OA-FLOW.MC.2",
"OA-FLOW.MC.3",
"OA-FLOW.MC.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.8"
]
},
{
"id": "OA-FLOW.CC.0",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.0",
"start_line": 1,
"end_line": 112,
"range_sha256": "7a764f50eff2e785ebb1845bb794a526dcbe50d190d8844cc0e27b866d07b640",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.CC.1",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.1",
"start_line": 113,
"end_line": 182,
"range_sha256": "0abbe6827dbcdafaa1bf2040bff086fb7965808b3add9d0edc87385a35738b9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.5",
"OA-FLOW.MW.4"
]
},
{
"id": "OA-FLOW.CC.2",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.2",
"start_line": 183,
"end_line": 227,
"range_sha256": "4bf1c9aef5fdcffb33bfc2b560ef56e3e08f012022cd972128789aaac19ecd78",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.CC.1",
"OA-FLOW.CF.1",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CC.3",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.3",
"start_line": 228,
"end_line": 271,
"range_sha256": "23078b8b2132746abda1f49f6a189096f05e939151ca35b240101107762dcf71",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.CC.1",
"OA-FLOW.CC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CC.4",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.4",
"start_line": 272,
"end_line": 310,
"range_sha256": "a70432b05bbbd33bb71d1bda5265993eeee104ceb65a23d71d2bc34e2b60b6b0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.3",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.CC.5",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.5",
"start_line": 311,
"end_line": 343,
"range_sha256": "c2e1444b77293ba047979827eb0b950ae3e3b85ad0e4bcee80139a561d2bd38d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.4",
"OA-FLOW.EP.1",
"OA-FLOW.EP.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CC.6",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.6",
"start_line": 344,
"end_line": 413,
"range_sha256": "917e21f34e7028c6d22e96e82e35f7562abdd077376956e777296d49473dd61b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.2",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.EP.3",
"OA-FLOW.EP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.CC.7",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.7",
"start_line": 414,
"end_line": 441,
"range_sha256": "88f5be991118cc96ebc1969a55678ff5405ec89908b0bc044ead244210a17da7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4"
]
},
{
"id": "OA-FLOW.CC.8",
"unit": "OA-FLOW-CC",
"source_file": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec",
"reader": "OA-FLOW-CC.html#oa-flow.cc.8",
"start_line": 442,
"end_line": 521,
"range_sha256": "465586f590f1ea63ece1fdb5e77439ff7b88a4a067ed1e2bacbd979884124f0b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.CC.2",
"OA-FLOW.CC.3",
"OA-FLOW.CC.6",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.RF.1",
"unit": "OA-FLOW-RF",
"source_file": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b",
"reader": "OA-FLOW-RF.html#oa-flow.rf.1",
"start_line": 1,
"end_line": 45,
"range_sha256": "60dbc6ba571a17c4e6609cf653bd475ef1f4d9944e37e794a13aa7552348ba30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.RF.2",
"unit": "OA-FLOW-RF",
"source_file": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b",
"reader": "OA-FLOW-RF.html#oa-flow.rf.2",
"start_line": 46,
"end_line": 85,
"range_sha256": "7959bc2c4abd58d193a50012d7b8820461fa779f13329e5b2108f86e8cecf3b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.7",
"OA-FLOW.CF.10",
"OA-FLOW.FF.2"
]
},
{
"id": "OA-FLOW.RF.3",
"unit": "OA-FLOW-RF",
"source_file": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b",
"reader": "OA-FLOW-RF.html#oa-flow.rf.3",
"start_line": 86,
"end_line": 120,
"range_sha256": "38b166b8690529b99090ebda2cb08b8c31c14a5e6c00da84cbd640f6d477e26b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.SC.4",
"OA-FLOW.FF.2"
]
},
{
"id": "OA-FLOW.RF.4",
"unit": "OA-FLOW-RF",
"source_file": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b",
"reader": "OA-FLOW-RF.html#oa-flow.rf.4",
"start_line": 121,
"end_line": 143,
"range_sha256": "20a62c7598cc4e0f6cb0483890d1f70091ba565ef1d9180d312d20f19f757a0f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.FF.1",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.RF.5",
"unit": "OA-FLOW-RF",
"source_file": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b",
"reader": "OA-FLOW-RF.html#oa-flow.rf.5",
"start_line": 144,
"end_line": 220,
"range_sha256": "d571af08ed4d410c8e017bcc31923587ba545b6fead86e28cac4ad161bd6b75c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.CF.1",
"OA-FLOW.SC.5",
"OA-FLOW.FF.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.AL.1",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.1",
"start_line": 1,
"end_line": 69,
"range_sha256": "b04e67fe5e9ce0579aa33f8ebfe8539fafb4aacf3fe63edf26a74152917b289f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.RF.3",
"OA-FLOW.CF.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2"
]
},
{
"id": "OA-FLOW.AL.2",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.2",
"start_line": 70,
"end_line": 122,
"range_sha256": "b438feb709279cd10e35c435da1aa31ae1d5b52e8020292374e5167eef61711c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.RF.2",
"OA-FLOW.RF.3",
"OA-FLOW.AL.1",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.AL.3",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.3",
"start_line": 123,
"end_line": 198,
"range_sha256": "1bb09aca3824a597c14cfe8caf476f45195340fd326732749321b6d2983c5439",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.RF.3",
"OA-FLOW.AL.1",
"OA-FLOW.AL.2"
]
},
{
"id": "OA-FLOW.AL.4",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.4",
"start_line": 199,
"end_line": 300,
"range_sha256": "586fc0c4f0aef092089b24bbece28bf53b410c784604235488f234917bdf29fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.RF.3",
"OA-FLOW.AL.1",
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.SF.SF2",
"OA-FLOW.FF.1",
"OA-FLOW.SC.8",
"OA-FLOW.SF.SF4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.AL.5",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.5",
"start_line": 301,
"end_line": 365,
"range_sha256": "f2148a476b72cc8ae24fa279298627b5a21321ab26f9b246913db9cb0d003169",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.5",
"OA-FLOW.AL.1",
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4",
"OA-FLOW.GW.3",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.NC.5",
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.8",
"OA-FLOW.MC.1",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.AL.6",
"unit": "OA-FLOW-AL",
"source_file": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6",
"reader": "OA-FLOW-AL.html#oa-flow.al.6",
"start_line": 366,
"end_line": 470,
"range_sha256": "6e3778e74229bef16e7e2a51b54dc470206d0b5a8d0e9e26c30f900d93f510d1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4",
"OA-FLOW.AL.5",
"OA-FLOW.PC.1",
"OA-FLOW.SF.SF0"
]
},
{
"id": "OA-FLOW.CT.1",
"unit": "OA-FLOW-CT",
"source_file": "src/OA-FLOW-CT.md",
"source_sha256": "ffb26a3191feb5c5bdfa27475ef3e12e07134500cab4c6932a6d6072da760466",
"reader": "OA-FLOW-CT.html#oa-flow.ct.1",
"start_line": 1,
"end_line": 41,
"range_sha256": "7c95ad15e67e2f9b36010b48854d0a78854e3146b0a40c256ac1d5a8c97a1d5e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.CP.6",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.CT.2",
"unit": "OA-FLOW-CT",
"source_file": "src/OA-FLOW-CT.md",
"source_sha256": "ffb26a3191feb5c5bdfa27475ef3e12e07134500cab4c6932a6d6072da760466",
"reader": "OA-FLOW-CT.html#oa-flow.ct.2",
"start_line": 42,
"end_line": 53,
"range_sha256": "d080568ba67c3e17f357cd5e7a215f8427c95c3894124ac94b62e46649b42b06",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.PC.8",
"OA-FLOW.CP.4",
"OA-FLOW.PC.7"
]
},
{
"id": "OA-FLOW.CT.3",
"unit": "OA-FLOW-CT",
"source_file": "src/OA-FLOW-CT.md",
"source_sha256": "ffb26a3191feb5c5bdfa27475ef3e12e07134500cab4c6932a6d6072da760466",
"reader": "OA-FLOW-CT.html#oa-flow.ct.3",
"start_line": 54,
"end_line": 69,
"range_sha256": "222198bc2efb1a379554fd537ac8daf95a033155ece86625fc1b023a075aabc5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.TS.1",
"unit": "OA-FLOW-TS",
"source_file": "src/OA-FLOW-TS.md",
"source_sha256": "f3ccd40e8eaedf8d0eef3959f117b6854fc1a6b1e66a0d74a8ce1643199fd2fb",
"reader": "OA-FLOW-TS.html#oa-flow.ts.1",
"start_line": 1,
"end_line": 67,
"range_sha256": "69a5f01f995a3eca0249f161367d6fc8f364fbf8aa274cd5ae0f341c8f9cf391",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AL.3",
"OA-FLOW.AL.5",
"OA-FLOW.RF.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.TS.2",
"unit": "OA-FLOW-TS",
"source_file": "src/OA-FLOW-TS.md",
"source_sha256": "f3ccd40e8eaedf8d0eef3959f117b6854fc1a6b1e66a0d74a8ce1643199fd2fb",
"reader": "OA-FLOW-TS.html#oa-flow.ts.2",
"start_line": 68,
"end_line": 123,
"range_sha256": "cbaddb8c16397ac832692abae25f3933f7b37eda6d6c4698536c8d93ef61932c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TS.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.TS.3",
"unit": "OA-FLOW-TS",
"source_file": "src/OA-FLOW-TS.md",
"source_sha256": "f3ccd40e8eaedf8d0eef3959f117b6854fc1a6b1e66a0d74a8ce1643199fd2fb",
"reader": "OA-FLOW-TS.html#oa-flow.ts.3",
"start_line": 124,
"end_line": 175,
"range_sha256": "96c134639540febaed1d5c63685f0c746253f2a94c624c5009fad33398381891",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TS.1",
"OA-FLOW.TS.2",
"OA-FLOW.AL.5",
"OA-FLOW.AL.6",
"OA-FLOW.CT.3",
"OA-FLOW.BC.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.4",
"OA-FLOW.CR.8",
"OA-FLOW.MC.1"
]
},
{
"id": "OA-FLOW.CD.0",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.0",
"start_line": 1,
"end_line": 152,
"range_sha256": "3a065e4719335bdc6b7a14a6f7c91ef675ec1678d56580e5121adc1d66e95fab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.CC.1",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.CC.7"
],
"range_bytes": 7920
},
{
"id": "OA-FLOW.CD.1",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.1",
"start_line": 153,
"end_line": 228,
"range_sha256": "0b1a5def417832d5b3075db938d0acee96e817b7827354c7711ae9a6c8d71620",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CD.0",
"OA-FLOW.CC.1",
"OA-FLOW.CC.2",
"OA-FLOW.CC.3",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.7",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.MW.4"
],
"range_bytes": 2622
},
{
"id": "OA-FLOW.CD.2",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.2",
"start_line": 229,
"end_line": 273,
"range_sha256": "bc23e6caf3a5c40f34636ab3b2db12bad09770df00d9d88762c629e6d8c27143",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CD.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.CZ.0"
],
"range_bytes": 1974
},
{
"id": "OA-FLOW.CD.3",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.3",
"start_line": 274,
"end_line": 327,
"range_sha256": "e6337343bef5b9cb8fdd6aa5288a22371b0dcaf25f73e7dfade604f5a5f2de63",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CD.0",
"OA-FLOW.CD.2",
"OA-FLOW.CC.1",
"OA-FLOW.CC.3",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.CC.7",
"OA-FLOW.OT.1",
"OA-FLOW.OT.2",
"OA-FLOW.OT.3",
"OA-FLOW.OT.4",
"OA-FLOW.OT.5",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CZ.2"
],
"range_bytes": 2864
},
{
"id": "OA-FLOW.CD.4",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.4",
"start_line": 328,
"end_line": 393,
"range_sha256": "fb7a83b9a784babe6fcae46334c9de07797c7c31eba4b24ee48dd9eff0d4b965",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CD.0",
"OA-FLOW.CD.1",
"OA-FLOW.CD.2",
"OA-FLOW.CD.3",
"OA-FLOW.CC.2",
"OA-FLOW.CC.8",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.CZ.7",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3"
],
"range_bytes": 3810
},
{
"id": "OA-FLOW.CD.5",
"unit": "OA-FLOW-CD",
"source_file": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189",
"reader": "OA-FLOW-CD.html#oa-flow.cd.5",
"start_line": 394,
"end_line": 479,
"range_sha256": "87ef2b3ffb5d2cf5c2e3fd0a1bb598e499640f90328990f3f4276065e45a2329",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CD.3",
"OA-FLOW.CD.4",
"OA-FLOW.CC.8",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"range_bytes": 8083
},
{
"id": "OA-FLOW.FP.1",
"unit": "OA-FLOW-FP",
"source_file": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8",
"reader": "OA-FLOW-FP.html#oa-flow.fp.1",
"start_line": 1,
"end_line": 62,
"range_sha256": "ab5603ccbd135a99a661799ad73c71af44a54659ce6ebed7b5070ad53c10e92c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.4",
"OA-FLOW.CR.8",
"OA-FLOW.CF.8",
"OA-FLOW.PC.1"
]
},
{
"id": "OA-FLOW.FP.2",
"unit": "OA-FLOW-FP",
"source_file": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8",
"reader": "OA-FLOW-FP.html#oa-flow.fp.2",
"start_line": 63,
"end_line": 114,
"range_sha256": "dc81e316d57e9f26862fe3abd71cc4c1c158ec06e11240ed14da3d88df3b6213",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FP.1",
"OA-FLOW.SC.6",
"OA-FLOW.SF.SB4"
]
},
{
"id": "OA-FLOW.FP.3",
"unit": "OA-FLOW-FP",
"source_file": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8",
"reader": "OA-FLOW-FP.html#oa-flow.fp.3",
"start_line": 115,
"end_line": 173,
"range_sha256": "89d37445445f98c11361d0efd891e55d1fa31cfd91e8604c908653498531b706",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FP.2",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.FP.4",
"unit": "OA-FLOW-FP",
"source_file": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8",
"reader": "OA-FLOW-FP.html#oa-flow.fp.4",
"start_line": 174,
"end_line": 224,
"range_sha256": "cbd50585a5bb73350fa9fdbdbc37d3dadc0a9c0d0c0f9f040da3257920dae5a9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FP.3",
"OA-FLOW.PC.1",
"OA-FLOW.FF.1",
"OA-FLOW.MC.2",
"OA-FLOW.MC.3",
"OA-FLOW.MC.4"
]
},
{
"id": "OA-FLOW.FP.5",
"unit": "OA-FLOW-FP",
"source_file": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8",
"reader": "OA-FLOW-FP.html#oa-flow.fp.5",
"start_line": 225,
"end_line": 250,
"range_sha256": "06813ce443e3b791bd919cf5602b0cc7951825bb22bda7977710e83b8d906460",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FP.1",
"OA-FLOW.FP.4"
]
},
{
"id": "OA-FLOW.HC.2",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.2",
"start_line": 1,
"end_line": 115,
"range_sha256": "7414eeed2ecacddf174723f3ec8e9b32f8d8bed699eab958c4b1c13f2d9e6239",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FP.1",
"OA-FLOW.NC.4",
"OA-FLOW.CR.1",
"OA-FLOW.CR.3",
"OA-FLOW.CR.8",
"OA-FLOW.CR.9",
"OA-FLOW.MC.2",
"OA-FLOW.MC.3",
"OA-FLOW.MC.4",
"OA-FLOW.CZ.0",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CF.8",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
],
"range_bytes": 8160
},
{
"id": "OA-FLOW.HC.3",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.3",
"start_line": 116,
"end_line": 170,
"range_sha256": "5002b60b65614cd10c064c3cf4e860b26498d9f8c21e6cae7709cd2ea46ce5e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HC.2",
"OA-FLOW.FP.5",
"OA-FLOW.TS.1",
"OA-FLOW.TS.2",
"OA-FLOW.TS.3",
"OA-FLOW.AL.5",
"OA-FLOW.AL.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CT.3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.5",
"OA-FLOW.CR.4",
"OA-FLOW.MC.1"
],
"range_bytes": 3316
},
{
"id": "OA-FLOW.HC.4",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.4",
"start_line": 171,
"end_line": 254,
"range_sha256": "e3856293adfefd4da6120ee4c2efc76cbdee7e49b5f40a7cfd600e99add1ea40",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HC.2",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.SC.4",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.6"
],
"range_bytes": 5549
},
{
"id": "OA-FLOW.HC.5",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.5",
"start_line": 255,
"end_line": 359,
"range_sha256": "4519b68df0a9d78e0fbd793bcf45c4aa0c900a4316ade54382dccc21567d02ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HC.2",
"OA-FLOW.HC.3",
"OA-FLOW.HC.4",
"OA-FLOW.MC.2",
"OA-FLOW.MC.3",
"OA-FLOW.MC.4",
"OA-FLOW.CT.2",
"OA-FLOW.CT.3",
"OA-FLOW.CZ.0"
],
"range_bytes": 5368
},
{
"id": "OA-FLOW.HC.6",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.6",
"start_line": 360,
"end_line": 474,
"range_sha256": "2ad8f594e50ed2a9bec583b454d3289e4a7d7182af50172c5c8172aa7da9092a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HC.2",
"OA-FLOW.HC.4",
"OA-FLOW.HC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GNS.4.1"
],
"range_bytes": 4942
},
{
"id": "OA-FLOW.HC.7",
"unit": "OA-FLOW-HC",
"source_file": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5",
"reader": "OA-FLOW-HC.html#oa-flow.hc.7",
"start_line": 475,
"end_line": 561,
"range_sha256": "f647ed18cdb1e0186440e95c75afbc0ad7e9fb134f3047329b2ae88b6a50cc8d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HC.6",
"OA-FLOW.PC.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF1",
"OA-FLOW.GNS.4.1"
],
"range_bytes": 8161
},
{
"id": "OA-FLOW.FD.0",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.0",
"start_line": 1,
"end_line": 73,
"range_sha256": "d5ff0890301810a4c4b52c6121ac5e810b145101a4a90630e804d0933f7b3b3e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.1",
"OA-FLOW.CC.3",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.CD.4",
"OA-FLOW.TD.2",
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.TD.6",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.FD.1",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.1",
"start_line": 74,
"end_line": 99,
"range_sha256": "04d0f56289ae0bf368f23258e4863ebb2f3f1583d7b607d20283e9590d728ddf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.CC.3",
"OA-FLOW.CC.4"
]
},
{
"id": "OA-FLOW.FD.2",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.2",
"start_line": 100,
"end_line": 136,
"range_sha256": "20dca77a2a809b1aef26d447639506afec0c3a4293f75a18d18c5da210accc09",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.4",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6"
]
},
{
"id": "OA-FLOW.FD.3",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.3",
"start_line": 137,
"end_line": 200,
"range_sha256": "17beda1b8f2782e342fe0ae1ff56768294b71f6c488ce0cc07916f6769b63282",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.0",
"OA-FLOW.FD.1",
"OA-FLOW.FD.2",
"OA-FLOW.TD.1",
"OA-FLOW.TD.2",
"OA-FLOW.TD.5",
"OA-FLOW.TD.7",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.CC.4",
"OA-FLOW.SC.4",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.FD.4",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.4",
"start_line": 201,
"end_line": 236,
"range_sha256": "8b9e6d9c25021dad7a7e85dbb6c29d8ceb49c6e30577bd94c0174b30712fbed7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.0",
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.TD.1",
"OA-FLOW.TD.2",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CC.6",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.FD.5",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.5",
"start_line": 237,
"end_line": 280,
"range_sha256": "628a0bda61ef601663915ee46372338b624a36c8edbcb82e87f7f66e7626f248",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.3",
"OA-FLOW.FD.4",
"OA-FLOW.CC.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.FD.6",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.6",
"start_line": 281,
"end_line": 325,
"range_sha256": "b386f0ecade333cc67c547a281fa637c565899a78ec48e21f66535edfb1e8955",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.FD.5",
"OA-FLOW.CC.5",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3"
]
},
{
"id": "OA-FLOW.FD.7",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.7",
"start_line": 326,
"end_line": 343,
"range_sha256": "0199618855cf7a84ea13d860bcd1da3ca7ea361c88eccffc76cc3e6f8c3c4190",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.0",
"OA-FLOW.FD.1",
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.FD.4",
"OA-FLOW.FD.5",
"OA-FLOW.FD.6",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.FD.8",
"unit": "OA-FLOW-FD",
"source_file": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c",
"reader": "OA-FLOW-FD.html#oa-flow.fd.8",
"start_line": 344,
"end_line": 420,
"range_sha256": "ea7bc5d75403580e101daca5e649ff3df5422dc65b61ee8c747bdf403a1df3fc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.FD.6",
"OA-FLOW.FD.7",
"OA-FLOW.CD.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.VD.0",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.0",
"start_line": 1,
"end_line": 106,
"range_sha256": "e5b4db7a039293342125f8ede904ac534f132c3e299bb50b53a54667095bac36",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.VD.1",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.1",
"start_line": 107,
"end_line": 194,
"range_sha256": "b44da622cbb7140995c67591a28a7d92225d0a326efb8f3fdd18ad4ee83d1c29",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0"
]
},
{
"id": "OA-FLOW.VD.2",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.2",
"start_line": 195,
"end_line": 264,
"range_sha256": "1d5477cb2ed83d8d4881c56b5d1105e1f3850ab26fbf58f1ce9840c2b58fd4a3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1"
]
},
{
"id": "OA-FLOW.VD.3",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.3",
"start_line": 265,
"end_line": 334,
"range_sha256": "08664ba67c68826a8826ae22e80acd36112a928f1c05883bce3ac30230d1ee69",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2"
]
},
{
"id": "OA-FLOW.VD.4",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.4",
"start_line": 335,
"end_line": 375,
"range_sha256": "83c3ced930017f5aa01ed22851cbe30b654abf87365bc655c456ce633c695d77",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3"
]
},
{
"id": "OA-FLOW.VD.5",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.5",
"start_line": 376,
"end_line": 469,
"range_sha256": "bf9be4c938fcf15e216fc0d41243091b09945545f26e49cf96b0e7a3f5cc96c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4"
]
},
{
"id": "OA-FLOW.VD.6",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.6",
"start_line": 470,
"end_line": 510,
"range_sha256": "8f9e1ed59029806d5bdc70277448ff0fc535cccf00dfde350199f7163d49f2f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5"
]
},
{
"id": "OA-FLOW.VD.7",
"unit": "OA-FLOW-VD",
"source_file": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5",
"reader": "OA-FLOW-VD.html#oa-flow.vd.7",
"start_line": 511,
"end_line": 566,
"range_sha256": "3b7ff57c16237202c326f62357361f94a311f95b7bf28bb61e3af8fca729f777",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SF2",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.2",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6"
]
},
{
"id": "OA-FLOW.PW.1",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.1",
"start_line": 1,
"end_line": 48,
"range_sha256": "674b74e5eb986dfaf9dbfc574d63eaf377f112360325b1bbfe99fe6fa4fbaa66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.PW.2",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.2",
"start_line": 49,
"end_line": 135,
"range_sha256": "807b304a861a781439bb39396dfe10bc55b100952080e2648f0e329c275d1302",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.PW.1"
]
},
{
"id": "OA-FLOW.PW.3",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.3",
"start_line": 136,
"end_line": 233,
"range_sha256": "af16c139c1bd4fd0a8d65766cd185e7b9ad81f988b20a3a3612111cc457ddc30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2"
]
},
{
"id": "OA-FLOW.PW.4",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.4",
"start_line": 234,
"end_line": 288,
"range_sha256": "060734aa8ede471d94a51608d0b5e6ad6ea2e297214318f6957bc41f267125d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3"
]
},
{
"id": "OA-FLOW.PW.5",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.5",
"start_line": 289,
"end_line": 358,
"range_sha256": "687785fb78bfced3adc1ab13020edc58d23f596c652d6d02a0a446b8edc7738a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3",
"OA-FLOW.PW.4"
]
},
{
"id": "OA-FLOW.PW.6",
"unit": "OA-FLOW-PW",
"source_file": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245",
"reader": "OA-FLOW-PW.html#oa-flow.pw.6",
"start_line": 359,
"end_line": 433,
"range_sha256": "76222e8c8474a48e0b01b9196c5964078b4ac1ba4ed9aee7452a554698e5da1e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.WS.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.KT.2",
"OA-FLOW.KT.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5"
]
},
{
"id": "OA-FLOW.PF.1",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.1",
"start_line": 1,
"end_line": 115,
"range_sha256": "21458e50ba887bdfd450fa6fe2c09da7e8ea94bc95c41e3193745e9aa0ae4c4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6"
]
},
{
"id": "OA-FLOW.PF.2",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.2",
"start_line": 116,
"end_line": 180,
"range_sha256": "72dafa0b7d8694ce7f75a01865a2ebd47f6c10b56010abf8ff03305a98be3524",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1"
]
},
{
"id": "OA-FLOW.PF.3",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.3",
"start_line": 181,
"end_line": 221,
"range_sha256": "f3e1a260b6e3bc89872a41b506b1ada63d42271d7a7b89dec10770b704973434",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2"
]
},
{
"id": "OA-FLOW.PF.4",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.4",
"start_line": 222,
"end_line": 260,
"range_sha256": "0138ae02a3087bcb79fe7ce81c6229f02129b2946509d6d16d0ecfe14109e305",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3"
]
},
{
"id": "OA-FLOW.PF.5",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.5",
"start_line": 261,
"end_line": 311,
"range_sha256": "be4db4baf7462d553d5426e92a770eb47830697aac8aaf6ed84f279fbae41e07",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4"
]
},
{
"id": "OA-FLOW.PF.6",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.6",
"start_line": 312,
"end_line": 330,
"range_sha256": "644398121e2ccbd5bf40a938a808bdf88fe63c1bcb40eac9d2632badbfec71d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5"
]
},
{
"id": "OA-FLOW.PF.7",
"unit": "OA-FLOW-PF",
"source_file": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655",
"reader": "OA-FLOW-PF.html#oa-flow.pf.7",
"start_line": 331,
"end_line": 408,
"range_sha256": "f9cc16ae298c42a908d4510bb41ab85f9c21a30cda146733793917d9160e62ca",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CZ.5",
"OA-FLOW.TS.1",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.CC.0",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PW.6",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6"
]
},
{
"id": "OA-FLOW.GT.1",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.1",
"start_line": 1,
"end_line": 103,
"range_sha256": "361a79e22aed69771db63de429efd1cde1ef0f7f56b872295a1a0725f341a09a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6"
]
},
{
"id": "OA-FLOW.GT.2",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.2",
"start_line": 104,
"end_line": 200,
"range_sha256": "dbc1f511e031df1795b9d8241e85b28074b4a5a1c196e0bb7656671b8491adc6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6",
"OA-FLOW.GT.1"
]
},
{
"id": "OA-FLOW.GT.3",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.3",
"start_line": 201,
"end_line": 246,
"range_sha256": "125f09b51e83627950065c01f8b3fc7e06908e1fb2c086a6fd700d3f664bcc41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6",
"OA-FLOW.GT.1",
"OA-FLOW.GT.2"
]
},
{
"id": "OA-FLOW.GT.4",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.4",
"start_line": 247,
"end_line": 321,
"range_sha256": "189e7bfb8b8d737743ea21ff51bfa4fd6842cab487f353391a95a465677d3c43",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6",
"OA-FLOW.GT.1",
"OA-FLOW.GT.2",
"OA-FLOW.GT.3"
]
},
{
"id": "OA-FLOW.GT.5",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.5",
"start_line": 322,
"end_line": 385,
"range_sha256": "453e3ae12c710633370b9da74430ef56005e5c17474e403ca7bacc1caf3f6812",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6",
"OA-FLOW.GT.1",
"OA-FLOW.GT.2",
"OA-FLOW.GT.3",
"OA-FLOW.GT.4"
]
},
{
"id": "OA-FLOW.GT.6",
"unit": "OA-FLOW-GT",
"source_file": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7",
"reader": "OA-FLOW-GT.html#oa-flow.gt.6",
"start_line": 386,
"end_line": 566,
"range_sha256": "9b855325d0be72e623662ae6c0bffbdfc936366ded2982994351cd83a673d8d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.6",
"OA-FLOW.CT.1",
"OA-FLOW.CT.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.4",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SF3",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6",
"OA-FLOW.GT.1",
"OA-FLOW.GT.2",
"OA-FLOW.GT.3",
"OA-FLOW.GT.4",
"OA-FLOW.GT.5"
]
},
{
"id": "OA-FLOW.CA.0",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.0",
"start_line": 1,
"end_line": 65,
"range_sha256": "86f6418a2c1f2e0711b887da6965d8327e09f8b3e70c7781571dd012d4285c20",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.CA.1",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.1",
"start_line": 66,
"end_line": 151,
"range_sha256": "b353e64fd3bcf48bd371f31195ee884cc5267a27c26b38b5f7e9ba74e3e9fbb5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0"
]
},
{
"id": "OA-FLOW.CA.2",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.2",
"start_line": 152,
"end_line": 193,
"range_sha256": "d541c4331967c74991a9cf433bf46352874d073a4ebc253ff52c642263e47f28",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1"
]
},
{
"id": "OA-FLOW.CA.3",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.3",
"start_line": 194,
"end_line": 244,
"range_sha256": "e347f463caefd76c06e9a696f75e8e7ba6af904d39a927d188112f9435688053",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2"
]
},
{
"id": "OA-FLOW.CA.4",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.4",
"start_line": 245,
"end_line": 343,
"range_sha256": "51a0e9b14c175b9031dc8a08178c660cf9888a928e0cdf61daef208b71dd0b2e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2",
"OA-FLOW.CA.3"
]
},
{
"id": "OA-FLOW.CA.5",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.5",
"start_line": 344,
"end_line": 404,
"range_sha256": "d3a40a42fc2803ec433937693d3fae7c439e51ab51f419553db301ed9054330c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2",
"OA-FLOW.CA.3",
"OA-FLOW.CA.4"
]
},
{
"id": "OA-FLOW.CA.6",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.6",
"start_line": 405,
"end_line": 448,
"range_sha256": "7b85a4217435ffaa1c5ba819efa3ada925ce534a6ddb64fb240ad47a09a2567a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2",
"OA-FLOW.CA.3",
"OA-FLOW.CA.4",
"OA-FLOW.CA.5"
]
},
{
"id": "OA-FLOW.CA.7",
"unit": "OA-FLOW-CA",
"source_file": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9",
"reader": "OA-FLOW-CA.html#oa-flow.ca.7",
"start_line": 449,
"end_line": 562,
"range_sha256": "6455f635a837dff043ddcfdf26bbe345596dd1be532ca16377390c88c9e06c44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.VD.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.WR.3",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.7.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CT.1",
"OA-FLOW.SC.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2",
"OA-FLOW.CA.3",
"OA-FLOW.CA.4",
"OA-FLOW.CA.5",
"OA-FLOW.CA.6"
]
},
{
"id": "OA-FLOW.CS.0",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.0",
"start_line": 1,
"end_line": 48,
"range_sha256": "f6e40b34ee830399c44f018931cc0dcfff7a97a7ff6a47fb0b8cf6f3ec7df573",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF0",
"OA-FLOW.AL.1",
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
]
},
{
"id": "OA-FLOW.CS.1",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.1",
"start_line": 49,
"end_line": 93,
"range_sha256": "ea62c61630526b999916556a4c2c2ccaf1898aa39572f6120c3c641e4e727f03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4",
"OA-FLOW.SF.SF2"
]
},
{
"id": "OA-FLOW.CS.2",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.2",
"start_line": 94,
"end_line": 100,
"range_sha256": "fe74c5cabe8d10d3673cd4b46389affc083833707b9ac1837522ffaf53c6280e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.1",
"OA-FLOW.PC.2",
"OA-FLOW.RF.1",
"OA-FLOW.RF.2",
"OA-FLOW.RF.3",
"OA-FLOW.AL.3"
]
},
{
"id": "OA-FLOW.CS.3",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.3",
"start_line": 101,
"end_line": 127,
"range_sha256": "48f346a093556af2dba39a41cdda5253efd14948133b46ff5ac572051279a8dd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4"
]
},
{
"id": "OA-FLOW.CS.4",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.4",
"start_line": 128,
"end_line": 143,
"range_sha256": "e808e753c9da907519a5b883db686153f614e9e831c6e49a3651f06058fa996f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.3",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4"
]
},
{
"id": "OA-FLOW.CS.5",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.5",
"start_line": 144,
"end_line": 184,
"range_sha256": "82942140b6de30afce349301bf3804b3d022c556ddbb2e52f6e294573be8f3bf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.SF.SF2",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.CS.6",
"unit": "OA-FLOW-CS",
"source_file": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5",
"reader": "OA-FLOW-CS.html#oa-flow.cs.6",
"start_line": 185,
"end_line": 203,
"range_sha256": "ba7ba3de6ad1c4678d59d526d5be47f344db00043a5c25f9b3307dffd7cba7ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.0",
"OA-FLOW.CS.3",
"OA-FLOW.CS.5"
]
},
{
"id": "OA-FLOW.MG.0",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.0",
"start_line": 1,
"end_line": 61,
"range_sha256": "2020eafad67911055ce1ff491f2e082acdeb3db6a086a4324acded9ffc63ee26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.WR.4",
"OA-FLOW.WR.5",
"OA-FLOW.MW.4",
"OA-FLOW.EW.3",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF3",
"OA-FLOW.RF.5",
"OA-FLOW.AL.1",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.MG.1",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.1",
"start_line": 62,
"end_line": 96,
"range_sha256": "f3e81507cd8597b90e718bd40feb9fa02d0ec500e317dce9da57b34b58eb8481",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.0",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF1",
"OA-FLOW.RF.1",
"OA-FLOW.AL.2"
]
},
{
"id": "OA-FLOW.MG.2",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.2",
"start_line": 97,
"end_line": 136,
"range_sha256": "c4090c122370b77eba809eb86f0b3c226f076c5366aca9f102ef9e5c772ea604",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.0",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.4",
"OA-FLOW.CZ.0",
"OA-FLOW.KT.2",
"OA-FLOW.KU.3"
]
},
{
"id": "OA-FLOW.MG.3",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.3",
"start_line": 137,
"end_line": 171,
"range_sha256": "5f4072ba439121d523b03e640b676fb0df87d16408385edc00138c0ab48bca57",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.1",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.CS.5",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3"
]
},
{
"id": "OA-FLOW.MG.4",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.4",
"start_line": 172,
"end_line": 197,
"range_sha256": "7fba75385e94d670b77ff7266b64988ad12d208bbefee4510511c7880f7f411d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2"
]
},
{
"id": "OA-FLOW.MG.5",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.5",
"start_line": 198,
"end_line": 210,
"range_sha256": "ccca97271b11c72cd6255fc4c72bb298c041f5b7325273650ef8b3eb69e32074",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.1",
"OA-FLOW.MG.3",
"OA-FLOW.TD.1",
"OA-FLOW.MW.4",
"OA-FLOW.CS.0"
]
},
{
"id": "OA-FLOW.MG.6",
"unit": "OA-FLOW-MG",
"source_file": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d",
"reader": "OA-FLOW-MG.html#oa-flow.mg.6",
"start_line": 211,
"end_line": 281,
"range_sha256": "c087f697f68187648d10bb6100a5cb0bfda3343fbe1afc8ee0b57e0dca18bd6b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.1",
"OA-FLOW.MG.5",
"OA-FLOW.CS.6",
"OA-FLOW.MG.0",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.CS.5"
]
},
{
"id": "OA-FLOW.IP.0",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.0",
"start_line": 1,
"end_line": 45,
"range_sha256": "6727151a2a754dc600034b51cc822ca79f5c06c3cca3a56e0d319f169df6800b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.4",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MW.4",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.IP.1",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.1",
"start_line": 46,
"end_line": 61,
"range_sha256": "0ae94d2d2b623751ae39abfc8d08381f47580f10e445d8e84d4c75f34f8129c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IP.0",
"OA-FLOW.CF.1",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4"
]
},
{
"id": "OA-FLOW.IP.2",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.2",
"start_line": 62,
"end_line": 98,
"range_sha256": "b9f7526ab2fd7df0fbb3d9b743ae184d21631e3943d19b45bf188e25fa1ad8f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AL.1",
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.RF.1",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.IP.3",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.3",
"start_line": 99,
"end_line": 169,
"range_sha256": "6f57500fe2b346ae7c6f24bdb6daa11c1e98b55d2de4f534585f18e4e8e616b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.IP.4",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.4",
"start_line": 170,
"end_line": 194,
"range_sha256": "2083b6ecd7c59ce00657f58452bea91695bb1d1c4427664d3cda2e0ef50d8585",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IP.1",
"OA-FLOW.IP.2",
"OA-FLOW.IP.3"
]
},
{
"id": "OA-FLOW.IP.5",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.5",
"start_line": 195,
"end_line": 273,
"range_sha256": "7ede7451d2aa7bf5597362829886c4f2408fea9553bd217658d1ee13b1fdb9a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IP.4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER"
]
},
{
"id": "OA-FLOW.IP.6",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.6",
"start_line": 274,
"end_line": 307,
"range_sha256": "6263f9d021383aa70d5602f8c536370c4a7e462fb95474c05622058043ceea0c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IP.0",
"OA-FLOW.IP.5",
"OA-FLOW.CT.2",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.IP.7",
"unit": "OA-FLOW-IP",
"source_file": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012",
"reader": "OA-FLOW-IP.html#oa-flow.ip.7",
"start_line": 308,
"end_line": 410,
"range_sha256": "c69f7ad3d371ad8aa1132360868f2d77ed94836da91084474603106e9540e4e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IP.5",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.IP.0",
"OA-FLOW.IP.1",
"OA-FLOW.IP.2",
"OA-FLOW.IP.3",
"OA-FLOW.IP.4",
"OA-FLOW.IP.6",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.MT.1",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.1",
"start_line": 1,
"end_line": 90,
"range_sha256": "ad69c49ea406a68431182901f0ae946c04575ebdfc93871e549b3d194ea10af1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER"
],
"range_bytes": 5206
},
{
"id": "OA-FLOW.MT.2",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.2",
"start_line": 91,
"end_line": 136,
"range_sha256": "218746e4a5d408a01a63877c59c72a7a29edf50dc863dd51b062f5f1ced01261",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.MT.1"
],
"range_bytes": 2083
},
{
"id": "OA-FLOW.MT.3",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.3",
"start_line": 137,
"end_line": 188,
"range_sha256": "141890c5f3989bd673a5d45d5f98d067b78fb8124f9b0434e7d85aa3d15923af",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.MT.1",
"OA-FLOW.MT.2"
],
"range_bytes": 1714
},
{
"id": "OA-FLOW.MT.4",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.4",
"start_line": 189,
"end_line": 222,
"range_sha256": "5984f10e3e3120858706ad4bdba6d2ebbee36453337971fbbd2b2d68872285b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.MT.1",
"OA-FLOW.MT.2",
"OA-FLOW.MT.3"
],
"range_bytes": 1847
},
{
"id": "OA-FLOW.MT.5",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.5",
"start_line": 223,
"end_line": 270,
"range_sha256": "fcb8c02cb20a800c6178e1bc396e75229c1cf5c8f0ead29bfb5a3e83bde835ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.MT.1",
"OA-FLOW.MT.2",
"OA-FLOW.MT.3",
"OA-FLOW.MT.4"
],
"range_bytes": 2531
},
{
"id": "OA-FLOW.MT.6",
"unit": "OA-FLOW-MT",
"source_file": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081",
"reader": "OA-FLOW-MT.html#oa-flow.mt.6",
"start_line": 271,
"end_line": 334,
"range_sha256": "ec548259d4b1004467707b1c96394289e9d2b34b5873ba854e5b8a97aff06b2c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.SF.SF3",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.KT.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.2",
"OA-FLOW.MG.3",
"OA-FLOW.MG.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.GI.TAILS",
"OA-FLOW.GI.TRANSFER",
"OA-FLOW.MT.1",
"OA-FLOW.MT.2",
"OA-FLOW.MT.3",
"OA-FLOW.MT.4",
"OA-FLOW.MT.5"
],
"range_bytes": 7642
},
{
"id": "OA-FLOW.DD.1",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.1",
"start_line": 1,
"end_line": 115,
"range_sha256": "4e64a4af392d36d19495885725665e4ae5868f08274862e4e39d5ee6bbd68924",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.PC.5"
],
"range_bytes": 6692
},
{
"id": "OA-FLOW.DD.2",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.2",
"start_line": 116,
"end_line": 163,
"range_sha256": "97a634cedbdbcf8ee35977f3a8b6740c400a0e9c873c8aebcfcbc92b3fbaa610",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.1",
"OA-FLOW.IP.0",
"OA-FLOW.IP.1",
"OA-FLOW.IP.2",
"OA-FLOW.IP.3",
"OA-FLOW.IP.4",
"OA-FLOW.IP.5",
"OA-FLOW.IP.6",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3"
],
"range_bytes": 1687
},
{
"id": "OA-FLOW.DD.3",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.3",
"start_line": 164,
"end_line": 243,
"range_sha256": "af96fafa28eba5461e8c7ec8b59dd66674e0dfd9e432ca59f86c735061e134e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.2",
"OA-FLOW.PC.8",
"OA-FLOW.PC.5",
"OA-FLOW.CA.0",
"OA-FLOW.CA.1",
"OA-FLOW.CA.2",
"OA-FLOW.CA.3",
"OA-FLOW.CA.4",
"OA-FLOW.CA.5",
"OA-FLOW.CA.6",
"OA-FLOW.CA.7",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4"
],
"range_bytes": 3113
},
{
"id": "OA-FLOW.DD.4",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.4",
"start_line": 244,
"end_line": 268,
"range_sha256": "07e409ac311bebeccf53870b63bb63515e4b47c8666161674f7d174059da1ee1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PF.1",
"OA-FLOW.PF.2",
"OA-FLOW.PF.3",
"OA-FLOW.PF.4",
"OA-FLOW.PF.5",
"OA-FLOW.PF.6",
"OA-FLOW.PF.7"
],
"range_bytes": 2146
},
{
"id": "OA-FLOW.DD.5",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.5",
"start_line": 269,
"end_line": 319,
"range_sha256": "6d37902ee9ef3c33cc6154d3c8c0cfde443ad58f4ff4676d5f489e062abbe010",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.4",
"OA-FLOW.GT.1",
"OA-FLOW.GT.2",
"OA-FLOW.GT.3",
"OA-FLOW.GT.4",
"OA-FLOW.GT.5",
"OA-FLOW.CZ.0"
],
"range_bytes": 3064
},
{
"id": "OA-FLOW.DD.6",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.6",
"start_line": 320,
"end_line": 405,
"range_sha256": "6b6a41cfb6c6f8e6defa443509ef040ea2ddaa450ee621edeb3f3d5b7071b414",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.5",
"OA-FLOW.GT.4",
"OA-FLOW.MW.4",
"OA-FLOW.RF.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CF.1"
],
"range_bytes": 4865
},
{
"id": "OA-FLOW.DD.7",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.7",
"start_line": 406,
"end_line": 438,
"range_sha256": "c2746b7ec9a53611acdce57deb31bdf42c7607dcb1b2eceaa99bbe42ab99cf96",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.6",
"OA-FLOW.PF.2",
"OA-FLOW.SF.SB4"
],
"range_bytes": 2247
},
{
"id": "OA-FLOW.DD.8",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.8",
"start_line": 439,
"end_line": 524,
"range_sha256": "cb49dc48b2dedb2ad31a80aad3697610d37fdaaafc22c1ad1a0ab095fda03319",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.5",
"OA-FLOW.GT.6",
"OA-FLOW.VD.0",
"OA-FLOW.VD.1",
"OA-FLOW.VD.2",
"OA-FLOW.VD.3",
"OA-FLOW.VD.4",
"OA-FLOW.VD.5",
"OA-FLOW.VD.6"
],
"range_bytes": 3301
},
{
"id": "OA-FLOW.DD.9",
"unit": "OA-FLOW-DD",
"source_file": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb",
"reader": "OA-FLOW-DD.html#oa-flow.dd.9",
"start_line": 525,
"end_line": 611,
"range_sha256": "f0d8ad8157651f51c749f1fe98ad222325716af8a68559bb5888a16dfaa0a47c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.6",
"OA-FLOW.DD.7",
"OA-FLOW.DD.8",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.DD.1",
"OA-FLOW.DD.2",
"OA-FLOW.DD.3",
"OA-FLOW.DD.4",
"OA-FLOW.DD.5"
],
"range_bytes": 8425
},
{
"id": "OA-FLOW.GM.1",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.1",
"start_line": 1,
"end_line": 121,
"range_sha256": "7bcc758a3edbaefc8989c3e02cf98c670813c741cf7f048afe8c068a54ce34ab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.MW.4",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9"
]
},
{
"id": "OA-FLOW.GM.2",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.2",
"start_line": 122,
"end_line": 145,
"range_sha256": "1cafff150d4a661f3b71099ce5cd0f99b0efc00393313168990f7ad0e636a9de",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.GM.1"
]
},
{
"id": "OA-FLOW.GM.3",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.3",
"start_line": 146,
"end_line": 199,
"range_sha256": "26dbca21fe06fa753d4de6bb7bfbd0774569173793f62fd184fb88ee1816c655",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3",
"OA-FLOW.PW.4",
"OA-FLOW.PW.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SC.9",
"OA-FLOW.CF.8",
"OA-FLOW.MW.4",
"OA-FLOW.GM.1",
"OA-FLOW.GM.2"
]
},
{
"id": "OA-FLOW.GM.4",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.4",
"start_line": 200,
"end_line": 217,
"range_sha256": "c8b95ed3c70d00794dad3af54dde7bd81b5f1b082487e9d7f667bf3b6856420f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KT.2",
"OA-FLOW.KT.3",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.RF.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.CF.6",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GM.1",
"OA-FLOW.GM.2",
"OA-FLOW.GM.3"
]
},
{
"id": "OA-FLOW.GM.5",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.5",
"start_line": 218,
"end_line": 277,
"range_sha256": "b62889178dda25e2c5df5d912a47d3c37dddce9b50a78aeaa69f2c50ec1b6f2e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.7.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.CF.8",
"OA-FLOW.GM.1",
"OA-FLOW.GM.2",
"OA-FLOW.GM.3",
"OA-FLOW.GM.4"
]
},
{
"id": "OA-FLOW.GM.6",
"unit": "OA-FLOW-GM",
"source_file": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398",
"reader": "OA-FLOW-GM.html#oa-flow.gm.6",
"start_line": 278,
"end_line": 319,
"range_sha256": "297137bc5ff24142a75f3d6967a8134f4fe7a3eedb99b5686355b75f07198ae8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.GW.3",
"OA-FLOW.MW.4",
"OA-FLOW.GM.1",
"OA-FLOW.GM.2",
"OA-FLOW.GM.3",
"OA-FLOW.GM.4",
"OA-FLOW.GM.5"
]
},
{
"id": "OA-FLOW.Z.1",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.1",
"start_line": 1,
"end_line": 59,
"range_sha256": "9ead1de5da4b2cfe5daaa71dd8e179af3b2b2eeb85bbb1061ada978c79ae124c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.4.1",
"OA-FLOW.MG.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
]
},
{
"id": "OA-FLOW.Z.2",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.2",
"start_line": 60,
"end_line": 95,
"range_sha256": "77232975df7041ffadea5dfa1edeb6afd2d680b6b96e86aa74673cabf2d8bca4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.3",
"OA-FLOW.CF.1",
"OA-FLOW.MG.2",
"OA-FLOW.Z.1"
]
},
{
"id": "OA-FLOW.Z.3",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.3",
"start_line": 96,
"end_line": 190,
"range_sha256": "cd7ecd31685f2ad08b4f3e579a30daede94c07d10a429f48814708c30bbeb78d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.1",
"OA-FLOW.RF.2",
"OA-FLOW.RF.3",
"OA-FLOW.AL.1",
"OA-FLOW.AL.2",
"OA-FLOW.AL.3",
"OA-FLOW.AL.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.GI.FILTERS",
"OA-FLOW.GI.DIAGONAL",
"OA-FLOW.Z.1",
"OA-FLOW.Z.2"
]
},
{
"id": "OA-FLOW.Z.4",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.4",
"start_line": 191,
"end_line": 296,
"range_sha256": "c8ce2c9b5a8fe9f02c7ec57d0478368d7ecfcff7d836f0afcc7c7feb5e31a704",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.Z.1",
"OA-FLOW.Z.2",
"OA-FLOW.Z.3"
]
},
{
"id": "OA-FLOW.Z.5",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.5",
"start_line": 297,
"end_line": 312,
"range_sha256": "c5daa4b8b6313d6a12144efdc2139d76fb23659cc646a8013bcddc2ce22850a9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.CT.2",
"OA-FLOW.Z.1",
"OA-FLOW.Z.2",
"OA-FLOW.Z.3",
"OA-FLOW.Z.4"
]
},
{
"id": "OA-FLOW.Z.6",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.6",
"start_line": 313,
"end_line": 340,
"range_sha256": "59d76052f688f27be30038c1fc67be13dd6e43b3d897e251c44c2bc0fe19b17e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CS.3",
"OA-FLOW.MG.2",
"OA-FLOW.MG.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.KT.5",
"OA-FLOW.CF.1",
"OA-FLOW.Z.1",
"OA-FLOW.Z.2",
"OA-FLOW.Z.3",
"OA-FLOW.Z.4",
"OA-FLOW.Z.5"
]
},
{
"id": "OA-FLOW.Z.7",
"unit": "OA-FLOW-Z",
"source_file": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4",
"reader": "OA-FLOW-Z.html#oa-flow.z.7",
"start_line": 341,
"end_line": 446,
"range_sha256": "67052e1a6c1a6a8360f687e9ac44f8fccd8c1147b00ab350df5b9a03c294cb43",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.4.1",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.4",
"OA-FLOW.Z.1",
"OA-FLOW.Z.2",
"OA-FLOW.Z.3",
"OA-FLOW.Z.4",
"OA-FLOW.Z.5",
"OA-FLOW.Z.6",
"OA-FLOW.GM.6"
]
},
{
"id": "OA-FLOW.DP.1",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.1",
"start_line": 34,
"end_line": 47,
"range_sha256": "87af5259452b5f5f72b2bbaa48e4ef05d6d667ad82ab20621eb67b2a42a368f6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.GNS.4.1"
]
},
{
"id": "OA-FLOW.DP.2",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.2",
"start_line": 48,
"end_line": 127,
"range_sha256": "027dbf81317d1b5a104c5704c0272b3d0c962f20cdaf326709e554d21933d778",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DD.1",
"OA-FLOW.DD.2",
"OA-FLOW.DD.3",
"OA-FLOW.DD.4",
"OA-FLOW.DD.5",
"OA-FLOW.DD.6",
"OA-FLOW.DD.7",
"OA-FLOW.DD.8",
"OA-FLOW.CC.0",
"OA-FLOW.CC.1",
"OA-FLOW.CC.2",
"OA-FLOW.CC.3",
"OA-FLOW.CC.4",
"OA-FLOW.CC.5",
"OA-FLOW.CC.6",
"OA-FLOW.CC.7",
"OA-FLOW.CC.8",
"OA-FLOW.CD.0",
"OA-FLOW.CD.1",
"OA-FLOW.CD.2",
"OA-FLOW.CD.3",
"OA-FLOW.CD.4",
"OA-FLOW.CZ.0",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.CZ.7",
"OA-FLOW.TD.1",
"OA-FLOW.TD.2",
"OA-FLOW.TD.3",
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.TD.6",
"OA-FLOW.TD.7",
"OA-FLOW.TD.8",
"OA-FLOW.FD.0",
"OA-FLOW.FD.1",
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.FD.4",
"OA-FLOW.FD.5",
"OA-FLOW.FD.6",
"OA-FLOW.NF.1",
"OA-FLOW.SF.SF2",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8"
]
},
{
"id": "OA-FLOW.DP.SI",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.si",
"start_line": 128,
"end_line": 163,
"range_sha256": "caa6735d23c261bd9cbf7df034fac73055f463c75ba135d4d50af136fc03329b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.DP.3",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.3",
"start_line": 164,
"end_line": 269,
"range_sha256": "e9b281dbc4b1067c376926174e63436452f056f0926e34919f8e8f4112f82521",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.2",
"OA-FLOW.DP.SI",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF1",
"OA-FLOW.TD.8",
"OA-FLOW.CC.6",
"OA-FLOW.EP.5"
]
},
{
"id": "OA-FLOW.DP.4",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.4",
"start_line": 270,
"end_line": 351,
"range_sha256": "a1b1f3145765cd0dc51bbf29903a30b2d03ff2b7ec40c06d989a2025d118b82e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.3",
"OA-FLOW.ST.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CF.3",
"OA-FLOW.CF.6",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.GNS.4.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.PC.1",
"OA-FLOW.TD.5"
]
},
{
"id": "OA-FLOW.DP.5",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.5",
"start_line": 352,
"end_line": 457,
"range_sha256": "6220e112c3982a51d01763ade6039a23692d5f8f9187136f21c6f69173a4fd00",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.1",
"OA-FLOW.DP.3",
"OA-FLOW.DP.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.PC.8",
"OA-FLOW.PF.6",
"OA-FLOW.TD.5",
"OA-FLOW.TD.7",
"OA-FLOW.TD.8",
"OA-FLOW.CC.3",
"OA-FLOW.CC.6",
"OA-FLOW.DD.1",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.4.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF1"
]
},
{
"id": "OA-FLOW.DP.6",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.6",
"start_line": 458,
"end_line": 646,
"range_sha256": "eb3ce77c60e53dbca6dfb9b9c7cff7c22369f6e9b07de6bbf86e3975cb6058fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.1",
"OA-FLOW.DP.2",
"OA-FLOW.DP.3",
"OA-FLOW.DP.5",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.TD.2",
"OA-FLOW.TD.6",
"OA-FLOW.TD.8",
"OA-FLOW.FD.0",
"OA-FLOW.FD.1",
"OA-FLOW.FD.2",
"OA-FLOW.FD.3",
"OA-FLOW.FD.4",
"OA-FLOW.FD.5",
"OA-FLOW.FD.6",
"OA-FLOW.DD.1"
]
},
{
"id": "OA-FLOW.DP.7",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.7",
"start_line": 647,
"end_line": 769,
"range_sha256": "fbf0918fe7a6f3b66f31eb13be6008fc43909a4960a3dec5d87e80f563e9512f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.SI",
"OA-FLOW.DP.3",
"OA-FLOW.DP.6",
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.DP.8",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.8",
"start_line": 770,
"end_line": 807,
"range_sha256": "40d4831bf9c047f7fc29625ccb513e0a63432e48c6aed80e15350fef812e47c8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.1",
"OA-FLOW.DP.4",
"OA-FLOW.DP.6",
"OA-FLOW.DP.7",
"OA-FLOW.SC.0",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.DP.9",
"unit": "OA-FLOW-DP",
"source_file": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1",
"reader": "OA-FLOW-DP.html#oa-flow.dp.9",
"start_line": 808,
"end_line": 835,
"range_sha256": "2407c22ee28be3b3cdd20b33a949c91333726a14dc7d501af5e875b2f728af7c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DP.8",
"OA-FLOW.Z.7",
"OA-FLOW.GM.6",
"OA-FLOW.HC.6",
"OA-FLOW.HC.7"
],
"proof_available_after_line": 835
},
{
"id": "OA-FLOW.UC.0",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.0",
"start_line": 1,
"end_line": 42,
"range_sha256": "6d65fa20805798ab1df0c658bc7149a812c6f2361ae8522a50a208cace522d75",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.UC.1",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.1",
"start_line": 43,
"end_line": 88,
"range_sha256": "17fd49699a8b51962920eeaf763141c31637b88ce49fd8339c119d06a0a5862f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0"
]
},
{
"id": "OA-FLOW.UC.2",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.2",
"start_line": 89,
"end_line": 118,
"range_sha256": "0747dd72fd1bef4215684b0d9cd35c325477d74412a54ee7e2cf1880361f0ac8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.1"
]
},
{
"id": "OA-FLOW.UC.3",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.3",
"start_line": 119,
"end_line": 161,
"range_sha256": "46c5bb1152edbe7b1a00e211b2d94715e661d70c8e7a545313570e859296a31b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.UC.1"
]
},
{
"id": "OA-FLOW.UC.4",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.4",
"start_line": 162,
"end_line": 205,
"range_sha256": "dd9b74f1a73dd5ab81318bb3604da0baaa525c036c3884004f67f2bd30dbfb05",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.UC.2"
]
},
{
"id": "OA-FLOW.UC.5",
"unit": "OA-FLOW-UC",
"source_file": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c",
"reader": "OA-FLOW-UC.html#oa-flow.uc.5",
"start_line": 206,
"end_line": 224,
"range_sha256": "03fe77c74e871f5cacfc5c2112389d0dea83bd3a9bb31619026252927c7600a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.1",
"OA-FLOW.UC.2",
"OA-FLOW.UC.3",
"OA-FLOW.UC.4"
]
},
{
"id": "OA-FLOW.AC.0",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.0",
"start_line": 1,
"end_line": 16,
"range_sha256": "c9d83dc0d81b1fbf988d8f844525b4adb247fba5840df800a9e870e6656ff1a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.1"
]
},
{
"id": "OA-FLOW.AC.1",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.1",
"start_line": 17,
"end_line": 41,
"range_sha256": "ef38d213783cc25a3fd58d5f66e9d51aa07ff99f655c49458c39288705fbe8fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AC.0",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.AC.2",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.2",
"start_line": 42,
"end_line": 78,
"range_sha256": "836f23877c6cdaa0a914a752b6f32c0af4e54586aa4cad611694027aeaa86bd7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.AC.0",
"OA-FLOW.AC.1",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.AC.3",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.3",
"start_line": 79,
"end_line": 111,
"range_sha256": "d8ed9eb07adb235ac3597c78431fea7e013b679309ebaaa2c12a29e5c0144375",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.AC.0",
"OA-FLOW.AC.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.AC.4",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.4",
"start_line": 112,
"end_line": 174,
"range_sha256": "9db4b40d82f3ec884d85ab217d2d7334e37bd87942b4f566253a739a61ea8fe1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.AC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
]
},
{
"id": "OA-FLOW.AC.5",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.5",
"start_line": 175,
"end_line": 184,
"range_sha256": "bc1973fb9a0372795800f2a31b84a808918ff805bd1d5da083bb935d3aef99b6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AC.1",
"OA-FLOW.AC.2",
"OA-FLOW.AC.3",
"OA-FLOW.AC.4",
"OA-FLOW.UC.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.6"
]
},
{
"id": "OA-FLOW.AC.6",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.6",
"start_line": 185,
"end_line": 198,
"range_sha256": "d35d6c541cf1858044a60fdb1a4f27a76ce4e217f98f39e8d3c867c38344ef81",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AC.2",
"OA-FLOW.AC.3",
"OA-FLOW.AC.4",
"OA-FLOW.UC.1"
]
},
{
"id": "OA-FLOW.AC.FIGURE",
"unit": "OA-FLOW-AC",
"source_file": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a",
"reader": "OA-FLOW-AC.html#oa-flow.ac.figure",
"start_line": 199,
"end_line": 217,
"range_sha256": "31ec1ecad9b55b53f6a4e9cccf7cca16179267d857a24e722e0dc87867e030ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.1",
"OA-FLOW.UC.2",
"OA-FLOW.AC.1",
"OA-FLOW.AC.2",
"OA-FLOW.AC.3",
"OA-FLOW.AC.4",
"OA-FLOW.AC.5"
]
},
{
"id": "OA-FLOW.LF.0",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-0",
"start_line": 1,
"end_line": 74,
"range_sha256": "d889369baaa0cba456e226f3e172d6a846628c669c414befa79b716dedf0a884",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.SC.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.SP.3",
"OA-FLOW.XGAPS.CSTAR.H2"
]
},
{
"id": "OA-FLOW.LF.1",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-1",
"start_line": 75,
"end_line": 133,
"range_sha256": "7e37753391a185b0755db15bee2ed2bd9c94901484b32affa6044d04d8202365",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.0",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6"
]
},
{
"id": "OA-FLOW.LF.2",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-2",
"start_line": 134,
"end_line": 178,
"range_sha256": "5b5d0887bbf203c35c4af217dcdfcd15020996203a657ac462441301b3166cb4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3"
]
},
{
"id": "OA-FLOW.LF.3",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-3",
"start_line": 179,
"end_line": 230,
"range_sha256": "ecb4999185abe1042fff6fb590e6b23132d43fb772f6d32e390ec8cd137b6157",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.2",
"OA-FLOW.LF.0"
]
},
{
"id": "OA-FLOW.LF.4",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-4",
"start_line": 231,
"end_line": 247,
"range_sha256": "7feb12642205783d04894a6e829134d8e395770f2f01c335a334e21bd90157fa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.1",
"OA-FLOW.LF.0"
]
},
{
"id": "OA-FLOW.LF.5",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-5",
"start_line": 248,
"end_line": 311,
"range_sha256": "3f1dcc5b4747d9c420037a6eeb2ee80222c58a0cbacba88a801b5eb5178d1c03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.0",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.HR.3",
"OA-FLOW.SP.3"
]
},
{
"id": "OA-FLOW.LF.6",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-6",
"start_line": 312,
"end_line": 361,
"range_sha256": "795e3912acdd8b4de3f9a09317fec2c1ba833131ce653c52540bbda8466893d7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.0",
"OA-FLOW.LF.3",
"OA-FLOW.LF.4",
"OA-FLOW.LF.5"
]
},
{
"id": "OA-FLOW.LF.7",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-7",
"start_line": 362,
"end_line": 398,
"range_sha256": "6656f51441d1f29de4c880c487dfc85f75d8eee03ff0c2f73b9678fbee89cee4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.6"
]
},
{
"id": "OA-FLOW.LF.8",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#lf-8",
"start_line": 399,
"end_line": 444,
"range_sha256": "04d07bc42a54a65250ff68fef7f863103780c0b0f2c14cca2a3a0281de7dcbb9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.0",
"OA-FLOW.LF.1",
"OA-FLOW.LF.3",
"OA-FLOW.SP.4"
]
},
{
"id": "OA-FLOW.LF.FIGURE",
"unit": "OA-FLOW-LF",
"source_file": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c",
"reader": "OA-FLOW-LF.html#oa-flow.lf.figure",
"start_line": 445,
"end_line": 491,
"range_sha256": "0da3753f97feab883aff171e77039433a754faf6ba9b505e9595893284f54953",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.6",
"OA-FLOW.LF.8"
]
},
{
"id": "OA-FLOW.GBAC.SETTING",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-setting",
"start_line": 3,
"end_line": 19,
"range_sha256": "f06fc991f76e9826806ad90805cdde346ccaac4faae8e2162c49ba80e1d4588e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 3,
"through_line": 19,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
}
]
},
{
"id": "OA-FLOW.GBAC.INPUTS",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-inputs",
"start_line": 20,
"end_line": 30,
"range_sha256": "613e294f1bae96832c45de75fe53a9e5bea27b8b27701ea1e650ce09a756a828",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 20,
"through_line": 30,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
}
]
},
{
"id": "OA-FLOW.GBAC.BOUNDS",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-local-bound",
"start_line": 31,
"end_line": 48,
"range_sha256": "425ceb6e416fc3f7513cbc7179081286cc9489d27938acc3bd6b3b2321aede98",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 47,
"definition_context": [
"OA-FLOW.GBAC.SETTING"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 31,
"through_line": 48,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
},
{
"source": "src/OA-FLOW-GBAC.md",
"line": 3,
"through_line": 19,
"description": "Earlier same-lesson group, Banach space, action and scalar-continuity hypotheses; the opening conclusion is a forward statement, not a proof premise."
}
]
},
{
"id": "OA-FLOW.GBAC.BARYCENTRES",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-metric-barycentre",
"start_line": 49,
"end_line": 101,
"range_sha256": "33741300655bb19a32357bff6b900b2665f1abe9f30442f6889eaec958a61ee5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.BOUNDS",
"OA-FLOW.CF.1",
"OA-FLOW.HR.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 100,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 49,
"through_line": 101,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
}
]
},
{
"id": "OA-FLOW.GBAC.CONVEXHULL",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-closed-convex-hull",
"start_line": 102,
"end_line": 132,
"range_sha256": "8c3a2cc2ae67d20ed5d676fb045e48a09fa6c7a5366e101921687e2e665f1d9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.BARYCENTRES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 131,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 102,
"through_line": 132,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
}
]
},
{
"id": "OA-FLOW.GBAC.AVERAGES",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-scalar-averages",
"start_line": 133,
"end_line": 173,
"range_sha256": "af9662f2679b95abe7b7866426b2f4be4c124f78c900ddd19e13244357c88127",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.BOUNDS",
"OA-FLOW.GBAC.CONVEXHULL",
"OA-FLOW.HR.6",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 172,
"definition_context": [
"OA-FLOW.GBAC.SETTING"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 133,
"through_line": 173,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
},
{
"source": "src/OA-FLOW-GBAC.md",
"line": 3,
"through_line": 19,
"description": "Earlier same-lesson group, Banach space, action and scalar-continuity hypotheses; the opening conclusion is a forward statement, not a proof premise."
}
]
},
{
"id": "OA-FLOW.GBAC.SMOOTHING",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-smoothing",
"start_line": 174,
"end_line": 212,
"range_sha256": "3ddd6b2422e1250bbef0148e941dbf5569ef9e3f305477483295d2f54b7530e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.BOUNDS",
"OA-FLOW.GBAC.AVERAGES",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 211,
"definition_context": [
"OA-FLOW.GBAC.SETTING"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 174,
"through_line": 212,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
},
{
"source": "src/OA-FLOW-GBAC.md",
"line": 3,
"through_line": 19,
"description": "Earlier same-lesson group, Banach space, action and scalar-continuity hypotheses; the opening conclusion is a forward statement, not a proof premise."
}
]
},
{
"id": "OA-FLOW.GBAC.CONTINUITY",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-density",
"start_line": 213,
"end_line": 254,
"range_sha256": "e5d905efbce7d7f97462632762e3eccf544f836e0108b7d54a711e211af7e2d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.BOUNDS",
"OA-FLOW.GBAC.AVERAGES",
"OA-FLOW.GBAC.SMOOTHING",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 252,
"definition_context": [
"OA-FLOW.GBAC.SETTING"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 213,
"through_line": 254,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
},
{
"source": "src/OA-FLOW-GBAC.md",
"line": 3,
"through_line": 19,
"description": "Earlier same-lesson group, Banach space, action and scalar-continuity hypotheses; the opening conclusion is a forward statement, not a proof premise."
}
]
},
{
"id": "OA-FLOW.GBAC.CONTEXT",
"unit": "OA-FLOW-GBAC",
"source_file": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b",
"reader": "OA-FLOW-GBAC.html#gbac-context",
"start_line": 255,
"end_line": 281,
"range_sha256": "47d9b12c80f1b6a28723570bd91d4c59130394fd73ccacef67879278859ed5a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GBAC.md",
"line": 255,
"through_line": 281,
"description": "Exact stated hypotheses, definitions, theorem or proof scope of this item."
}
]
},
{
"id": "OA-FLOW.AT.1",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.1",
"start_line": 1,
"end_line": 49,
"range_sha256": "026e0588261cbf67810c451036231eb87b9255e99600a98ea29a7f7441895255",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.AT.2",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.2",
"start_line": 50,
"end_line": 113,
"range_sha256": "5c060193b83fd114d24befccc604f5df07036b1b6b64e4d1e1873936dab11ab0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.CP.6",
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.NF.4",
"OA-FLOW.HR.3",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS"
]
},
{
"id": "OA-FLOW.AT.3",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.3",
"start_line": 114,
"end_line": 171,
"range_sha256": "e6dc5e7b56d90c2c51bd998d950c3a2dbb84f40807da84199ba60dee94376fcd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.2",
"OA-FLOW.CF.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.AT.4",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.4",
"start_line": 172,
"end_line": 204,
"range_sha256": "d5d579c040fc531ba165957360e522ba03be74034d93468f1345a0ff28fe16e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.AT.3",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.AT.5",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.5",
"start_line": 205,
"end_line": 249,
"range_sha256": "b5a203d5e28abeb254554a0afe3521ab1f61ce536576b96ff89bfde6e55076a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.AT.6",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.6",
"start_line": 250,
"end_line": 288,
"range_sha256": "aa717687ea4fe84368ecef95a3252d6f444d857fd101e6efd472d78aed97380a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.AT.5",
"OA-FLOW.HR.3",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.AT.7",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.7",
"start_line": 289,
"end_line": 300,
"range_sha256": "d93f169a1b24b60030a5755fccf7d499d1f67e9e636ca36ef10aa4648fe11620",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6"
]
},
{
"id": "OA-FLOW.AT.FIGURE",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#oa-flow.at.figure",
"start_line": 301,
"end_line": 353,
"range_sha256": "01e746722d5aa058016ca8ef9a62f8bf2bda151b0b466f969c851b12cb9791b2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.8",
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.NF.4",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.AT.U-GROUP",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-u-group",
"start_line": 355,
"end_line": 417,
"range_sha256": "133930e5b80268e9c72503fc2ed8139451d6f1f66be3cf1b963ac22c41d46265",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CP.6",
"OA-FLOW.AT.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 417,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 355,
"through_line": 417,
"description": "Arbitrary von Neumann algebra and arbitrary directed automorphism nets. Joint ultraweak and intrinsic sigma-strong-star evaluation requires a fixed operator norm ball. The counterexample uses B(ell2), a net indexed by finite sets of vectors and positive normal functionals and an integer, positive rank-one operators of norm n squared, and unitary plane rotations of angle 1/n. The final sequence contrast uses the earlier Banach-Steinhaus proof."
}
]
},
{
"id": "OA-FLOW.AT.U-EVALUATION",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-u-evaluation",
"start_line": 418,
"end_line": 468,
"range_sha256": "e092d31423fb27e9b377a325563a98d6f51372b022ec07c6e280ff55f9dc89ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.U-GROUP",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 468,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 418,
"through_line": 468,
"description": "Arbitrary von Neumann algebra and arbitrary directed automorphism nets. Joint ultraweak and intrinsic sigma-strong-star evaluation requires a fixed operator norm ball. The counterexample uses B(ell2), a net indexed by finite sets of vectors and positive normal functionals and an integer, positive rank-one operators of norm n squared, and unitary plane rotations of angle 1/n. The final sequence contrast uses the earlier Banach-Steinhaus proof."
}
]
},
{
"id": "OA-FLOW.AT.UNBOUNDED-NET",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-unbounded-net",
"start_line": 469,
"end_line": 545,
"range_sha256": "8e33f4a6876e959b37944bdbcda38cb9cb502f4b7ee4d26422fd40ffb463d48e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.U-GROUP",
"OA-FLOW.AT.U-EVALUATION",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.6",
"OA-FLOW.AT.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 545,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 469,
"through_line": 545,
"description": "Arbitrary von Neumann algebra and arbitrary directed automorphism nets. Joint ultraweak and intrinsic sigma-strong-star evaluation requires a fixed operator norm ball. The counterexample uses B(ell2), a net indexed by finite sets of vectors and positive normal functionals and an integer, positive rank-one operators of norm n squared, and unitary plane rotations of angle 1/n. The final sequence contrast uses the earlier Banach-Steinhaus proof."
}
]
},
{
"id": "OA-FLOW.AT.LEBESGUE-PREDUAL",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-lebesgue-predual",
"start_line": 547,
"end_line": 603,
"range_sha256": "88dca1b898c02d00c11e82a3b90c46e225caa03957d1f13bdd47c43fc8c77586",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BD.1",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.IMPORT.FOUNDATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 603,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 547,
"through_line": 603,
"description": "The multiplication model uses Lebesgue measure on the whole real line. Translation continuity retains arbitrary locally compact groups. Banach coefficient tests quantify over every vector and bounded functional with local compactness; unitary and implemented-action arguments retain arbitrary topological groups and arbitrary Hilbert spaces. Joint strong-star evaluation is asserted only on bounded operator balls."
}
]
},
{
"id": "OA-FLOW.AT.TRANSLATION-DISTANCES",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-translation-distances",
"start_line": 604,
"end_line": 668,
"range_sha256": "d736f9ae72efa23e7f533caf2093488c2b39b188b419427275631873d64f4d39",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.LEBESGUE-PREDUAL",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.AT.1",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 668,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 604,
"through_line": 668,
"description": "The multiplication model uses Lebesgue measure on the whole real line. Translation continuity retains arbitrary locally compact groups. Banach coefficient tests quantify over every vector and bounded functional with local compactness; unitary and implemented-action arguments retain arbitrary topological groups and arbitrary Hilbert spaces. Joint strong-star evaluation is asserted only on bounded operator balls."
}
]
},
{
"id": "OA-FLOW.AT.BANACH-SOLVED-TESTS",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-banach-solved-tests",
"start_line": 669,
"end_line": 712,
"range_sha256": "e5dd8e0818ef5928f786aa39793c99d7bd88f0814dee7c3200e074a277645de9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GBAC.CONTINUITY",
"OA-FLOW.GBAC.AVERAGES",
"OA-FLOW.CF.1",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 712,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 669,
"through_line": 712,
"description": "The multiplication model uses Lebesgue measure on the whole real line. Translation continuity retains arbitrary locally compact groups. Banach coefficient tests quantify over every vector and bounded functional with local compactness; unitary and implemented-action arguments retain arbitrary topological groups and arbitrary Hilbert spaces. Joint strong-star evaluation is asserted only on bounded operator balls."
}
]
},
{
"id": "OA-FLOW.AT.UNITARY-IMPLEMENTED-ACTIONS",
"unit": "OA-FLOW-AT",
"source_file": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014",
"reader": "OA-FLOW-AT.html#at-unitary-implemented-actions",
"start_line": 713,
"end_line": 774,
"range_sha256": "09113f508b77989679fb4f254f3999d7e80c9f12cee65093f87b7290edfa6c26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.AT.U-EVALUATION",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 774,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-AT.md",
"line": 713,
"through_line": 774,
"description": "The multiplication model uses Lebesgue measure on the whole real line. Translation continuity retains arbitrary locally compact groups. Banach coefficient tests quantify over every vector and bounded functional with local compactness; unitary and implemented-action arguments retain arbitrary topological groups and arbitrary Hilbert spaces. Joint strong-star evaluation is asserted only on bounded operator balls."
}
]
},
{
"id": "OA-FLOW.NR.1",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.1",
"start_line": 1,
"end_line": 47,
"range_sha256": "8ab599396160000d33250c4a6ff7f80387a75f585401b1eefaac7de403ce2e5d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.ST.2",
"OA-FLOW.NC.1",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.CR.1",
"OA-FLOW.CR.6",
"OA-FLOW.CR.8",
"OA-FLOW.CR.9",
"OA-FLOW.MC.1"
],
"range_bytes": 4456
},
{
"id": "OA-FLOW.NR.2",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.2",
"start_line": 48,
"end_line": 90,
"range_sha256": "572768d65e993104b252f440a48892d1dc4ad4303533d116141053e85e4b4c45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6"
],
"range_bytes": 1940
},
{
"id": "OA-FLOW.NR.3",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.3",
"start_line": 91,
"end_line": 136,
"range_sha256": "7100fccbd681fd773bf42d861a750cb080a8ac5ef4b432ce2f411b634266a7cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.CCOV.INTEGRATION",
"OA-FLOW.CCOV.COEFFICIENTRECOVERY",
"OA-FLOW.CCOV.GROUPRECOVERY"
],
"range_bytes": 3505
},
{
"id": "OA-FLOW.NR.4",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.4",
"start_line": 137,
"end_line": 189,
"range_sha256": "919b9af236a6f18f59713f289195a557f238a1092a523f7ae7a4c360ffd79847",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.CR.8",
"OA-FLOW.ST.2",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4"
],
"range_bytes": 4742
},
{
"id": "OA-FLOW.NR.5",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.5",
"start_line": 190,
"end_line": 221,
"range_sha256": "697f6a76a1b462e2d47fb2be23f6927296c78cd83d1a264ca8a5bb9ac05acd7a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.CP.4",
"OA-FLOW.ST.2",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"range_bytes": 2822
},
{
"id": "OA-FLOW.NR.FIGURE",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.figure",
"start_line": 222,
"end_line": 247,
"range_sha256": "88aac4041058aca7bedfa86d3278a047ab0d94bbdb280132f11c4db58bba8b48",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.4",
"OA-FLOW.CF.8"
],
"range_bytes": 3440
},
{
"id": "OA-FLOW.GL.0",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-0",
"start_line": 1,
"end_line": 94,
"range_sha256": "dd0958d37cf299f75fbf973e8a846cc035e9c5d4319d822434e9db85e90ee9bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7"
]
},
{
"id": "OA-FLOW.GL.1",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-1",
"start_line": 95,
"end_line": 121,
"range_sha256": "46d9e643bed7b393f7b356deec741a8ca366acc826040f79d5ef1ae7c638e9f6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.LF.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.GL.2",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-2",
"start_line": 122,
"end_line": 176,
"range_sha256": "965caf8f70e7b97d992b093f24b2e90729d5ac64fcf776047fa9d6b124e42e16",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.GL.1",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7"
]
},
{
"id": "OA-FLOW.GL.3",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-3",
"start_line": 177,
"end_line": 219,
"range_sha256": "63b51b651fbf4f809d662e59b292fbb54f255e5f8cde55ffbafd18a26869017f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.LF.1",
"OA-FLOW.LF.7",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.GL.4",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-4",
"start_line": 220,
"end_line": 263,
"range_sha256": "b77f846c93f8cca41cbfbacd66e6bedb4200d45f61fb4a01a9add3038ff57a86",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.LF.4",
"OA-FLOW.LF.7",
"OA-FLOW.XGAPS.CSTAR.H0"
]
},
{
"id": "OA-FLOW.GL.5",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-5",
"start_line": 264,
"end_line": 323,
"range_sha256": "ea9b306f648f48e114735ea1ca7aa354d9075c1c33188ca1251d6130fbc30a3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.GL.3",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.GL.6",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-6",
"start_line": 324,
"end_line": 366,
"range_sha256": "058d15ccd3bedaed104949af00237a0602d35d0074350cdfc5fba6e72f30607b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.2",
"OA-FLOW.GL.4",
"OA-FLOW.GL.5",
"OA-FLOW.LF.1",
"OA-FLOW.XGAPS.CSTAR.H0"
]
},
{
"id": "OA-FLOW.GL.7",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-7",
"start_line": 367,
"end_line": 385,
"range_sha256": "91d0626f0cc66bd247aabe5b098928e462c54a11995c76d2f71c2a1a113c7edb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.6",
"OA-FLOW.LF.1"
]
},
{
"id": "OA-FLOW.GL.8",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#gl-8",
"start_line": 386,
"end_line": 432,
"range_sha256": "7bbb51d261e2664222d09b87d25912f20f83d9c5183801feaa1ba53ab2ba8bb2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.7",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.GL.FIGURE",
"unit": "OA-FLOW-GL",
"source_file": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f",
"reader": "OA-FLOW-GL.html#oa-flow.gl.figure",
"start_line": 433,
"end_line": 470,
"range_sha256": "3a1e41f404f1059bc1acbf2dfb7b2ba11604e32a3981a8a9593a3a9905c69b21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.GL.4",
"OA-FLOW.GL.5",
"OA-FLOW.GL.6",
"OA-FLOW.GL.8"
]
},
{
"id": "OA-FLOW.SS.0",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#ss-0",
"start_line": 1,
"end_line": 46,
"range_sha256": "b0b8ab26fc3d089b9ad8529f7614428230db06175d7da0c11fe03b19cd7a71ad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.0",
"OA-FLOW.LF.6",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.SS.1",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#ss-1",
"start_line": 47,
"end_line": 109,
"range_sha256": "b0c47bdbcd50c8f647cc61a019845c1879d87bdff7a7b0372fdc01f92990808a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SS.0",
"OA-FLOW.LF.1",
"OA-FLOW.LF.2",
"OA-FLOW.LF.5"
]
},
{
"id": "OA-FLOW.SS.2",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#ss-2",
"start_line": 110,
"end_line": 126,
"range_sha256": "89d012c57334605bb5cf324ff48cc213cb6d12268c9ef05e4edb0f87113d2f4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SS.1",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6"
]
},
{
"id": "OA-FLOW.SS.3",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#ss-3",
"start_line": 127,
"end_line": 189,
"range_sha256": "ebe283602d85af01ce24b1a49d1c02429a6134f500216497fb54fb14b00a614b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SS.2",
"OA-FLOW.GL.0",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6"
]
},
{
"id": "OA-FLOW.SS.4",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#ss-4",
"start_line": 190,
"end_line": 233,
"range_sha256": "d10d7bc6149b0378e838004ed0cd54ced9ca1d4f4364dcc4c66f4562469deb0b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SS.3",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.SS.FIGURE",
"unit": "OA-FLOW-SS",
"source_file": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14",
"reader": "OA-FLOW-SS.html#oa-flow.ss.figure",
"start_line": 234,
"end_line": 264,
"range_sha256": "b1cde269c863a39fe1031f87c58f508f8d8c8bf4ce968e86acec0ed20bb1ec38",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SS.1",
"OA-FLOW.SS.3",
"OA-FLOW.SS.4"
]
},
{
"id": "OA-FLOW.FS.1",
"unit": "OA-FLOW-FS",
"source_file": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c",
"reader": "OA-FLOW-FS.html#fs-1",
"start_line": 1,
"end_line": 51,
"range_sha256": "5d55754da9cb4ab3ae5a3b8806642ba43103f03e5e38ee4e82d80a96d21850d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.0",
"OA-FLOW.GL.2",
"OA-FLOW.GL.6",
"OA-FLOW.GL.7",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.PC.1"
]
},
{
"id": "OA-FLOW.FS.2",
"unit": "OA-FLOW-FS",
"source_file": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c",
"reader": "OA-FLOW-FS.html#fs-2",
"start_line": 52,
"end_line": 124,
"range_sha256": "16bd0b97bce457b11dc5a43e09ee3dc580afb8d2447b56ae6fc194594645b673",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FS.1",
"OA-FLOW.PC.1",
"OA-FLOW.GL.2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.FS.3",
"unit": "OA-FLOW-FS",
"source_file": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c",
"reader": "OA-FLOW-FS.html#fs-3",
"start_line": 125,
"end_line": 159,
"range_sha256": "73cf333921da813d620e79d399d77beae77eb423196f058c2886aac16a236d90",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FS.1",
"OA-FLOW.PC.1",
"OA-FLOW.CP.4",
"OA-FLOW.GL.0",
"OA-FLOW.GL.2"
]
},
{
"id": "OA-FLOW.FS.4",
"unit": "OA-FLOW-FS",
"source_file": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c",
"reader": "OA-FLOW-FS.html#fs-4",
"start_line": 160,
"end_line": 206,
"range_sha256": "121c84f943b11c7eb540c63433ee4ba07f9acfceb04288dc3c581c3ac636014f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FS.1",
"OA-FLOW.FS.2",
"OA-FLOW.GL.0",
"OA-FLOW.GL.2",
"OA-FLOW.CF.3"
]
},
{
"id": "OA-FLOW.FS.FIGURE",
"unit": "OA-FLOW-FS",
"source_file": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c",
"reader": "OA-FLOW-FS.html#oa-flow.ss.figure",
"start_line": 207,
"end_line": 217,
"range_sha256": "2558705d4d7eda67617d452218f2e160bd3fbc80b2258769cd0d6c4c17e2e317",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FS.2",
"OA-FLOW.FS.4"
]
},
{
"id": "OA-FLOW.CCM.4",
"unit": "OA-FLOW-CAI",
"source_file": "src/OA-FLOW-CAI.md",
"source_sha256": "b3aedb35dcd9852b967d00912696375d089a810b7196a21448c6e33f21264f30",
"reader": "OA-FLOW-CAI.html#ccm-4",
"start_line": 1,
"end_line": 202,
"range_sha256": "b3aedb35dcd9852b967d00912696375d089a810b7196a21448c6e33f21264f30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.AT.4",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.NR.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.ST.2",
"OA-FLOW.AT.3",
"OA-FND-HB-03",
"OA-FND-HB-04"
]
},
{
"id": "OA-FLOW.WY.0",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-0",
"start_line": 1,
"end_line": 56,
"range_sha256": "9a2a8de35983e55b0c80587bbd52cd4c1b36700140d4ad076ff3832e21adacba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.XGAPS.CSTAR.H0"
]
},
{
"id": "OA-FLOW.WY.1",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-1",
"start_line": 57,
"end_line": 108,
"range_sha256": "3c0799e99350bf9a3a8bb39abb439de746bb940c9a303cf074e95eb63a453a7b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.0",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.CF.6"
]
},
{
"id": "OA-FLOW.WY.2",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-2",
"start_line": 109,
"end_line": 192,
"range_sha256": "4c6f9c9277f9c3ec9a812c222b26687fe5133b1af215dfe7323e0658a6924ad0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.1",
"OA-FLOW.CCOV.ALGEBRA",
"OA-FLOW.CCOV.INTEGRATION",
"OA-FLOW.CCOV.TRANSFORMATIONS",
"OA-FLOW.GRP.CONTRACTIVITY",
"OA-FLOW.XGAPS.CSTAR.KERNELCLOSURE"
]
},
{
"id": "OA-FLOW.WY.3",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-3",
"start_line": 193,
"end_line": 238,
"range_sha256": "309076c18cf48b25e222b5b1bb48059a691b9e65eec5214ea2e6b408e355e39d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.2",
"OA-FLOW.CF.8",
"OA-FLOW.XGAPS.CSTAR.H0"
]
},
{
"id": "OA-FLOW.WY.4",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-4",
"start_line": 239,
"end_line": 268,
"range_sha256": "7acc6bc8c49bf66bd8d4a7e3b119c133a4d333c710694f66ece69024aa359898",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.3",
"OA-FLOW.CCOV.COEFFICIENTRECOVERY",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.GRP.INTERTWINERS"
]
},
{
"id": "OA-FLOW.WY.5",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-5",
"start_line": 269,
"end_line": 302,
"range_sha256": "c7d7f6f293bc6d56699be46ef16034775b467c665452aa732bf03f99e4f960d9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.WY.6",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-6",
"start_line": 303,
"end_line": 346,
"range_sha256": "d7af8687802e4b298bd050c102b048e480989500a91e6e14c6011e83d26fba25",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.5",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.WY.FIGURE",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#oa-flow.wy.figure",
"start_line": 347,
"end_line": 391,
"range_sha256": "2f17fca045e5ecf2778b75b00607493742e31b56fb990c75c912caa3bcee041d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.0",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WY.6",
"OA-FLOW.CF.3"
]
},
{
"id": "OA-FLOW.WY.LEFT-CONVENTION",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-left-convention",
"start_line": 393,
"end_line": 475,
"range_sha256": "ac87d8000c0100e549bce82a0f436f0b64d8b2a65d39eaaa98ec7f1acb4e9268",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.0",
"OA-FLOW.WY.1",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.CCOV.ALGEBRA"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 475,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-WY.md",
"line": 393,
"through_line": 475,
"description": "The Weyl results retain every locally compact Hausdorff abelian group and arbitrary Hilbert spaces and multiplicities, without countability or sigma-compactness assumptions. Compact kernel integrals use the qualified Radon-product contract on compact carriers. The corner and type-I assertions require a nonzero representation and nonzero rank-one inputs; the zero system is explicitly retained. The final example uses Z with counting Haar and its probability-Haar circle dual."
}
]
},
{
"id": "OA-FLOW.WY.COMPACT-KERNELS",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-compact-kernels",
"start_line": 476,
"end_line": 582,
"range_sha256": "1483ec55bce5cd276a4fa29aeded64e754e79a6abc0c775dba1c51f19af23d40",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.LEFT-CONVENTION",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.CCOV.TRANSFORMATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 582,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-WY.md",
"line": 476,
"through_line": 582,
"description": "The Weyl results retain every locally compact Hausdorff abelian group and arbitrary Hilbert spaces and multiplicities, without countability or sigma-compactness assumptions. Compact kernel integrals use the qualified Radon-product contract on compact carriers. The corner and type-I assertions require a nonzero representation and nonzero rank-one inputs; the zero system is explicitly retained. The final example uses Z with counting Haar and its probability-Haar circle dual."
}
]
},
{
"id": "OA-FLOW.WY.GENERATED-CORNERS",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-generated-corners",
"start_line": 583,
"end_line": 639,
"range_sha256": "49d90e2f94724104c09be5b7557fc104ed4d558e1872e6c3085a6641c1f0cf21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.COMPACT-KERNELS",
"OA-FLOW.WY.5",
"OA-FLOW.CF.8",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 639,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-WY.md",
"line": 583,
"through_line": 639,
"description": "The Weyl results retain every locally compact Hausdorff abelian group and arbitrary Hilbert spaces and multiplicities, without countability or sigma-compactness assumptions. Compact kernel integrals use the qualified Radon-product contract on compact carriers. The corner and type-I assertions require a nonzero representation and nonzero rank-one inputs; the zero system is explicitly retained. The final example uses Z with counting Haar and its probability-Haar circle dual."
}
]
},
{
"id": "OA-FLOW.WY.INFINITE-MULTIPLICITIES",
"unit": "OA-FLOW-WY",
"source_file": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15",
"reader": "OA-FLOW-WY.html#wy-infinite-multiplicities",
"start_line": 640,
"end_line": 691,
"range_sha256": "71747f7056d707a5ab3845831b913886a4d0a41556406db4fdcf01081857569f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WY.LEFT-CONVENTION",
"OA-FLOW.WY.GENERATED-CORNERS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 691,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-WY.md",
"line": 640,
"through_line": 691,
"description": "The Weyl results retain every locally compact Hausdorff abelian group and arbitrary Hilbert spaces and multiplicities, without countability or sigma-compactness assumptions. Compact kernel integrals use the qualified Radon-product contract on compact carriers. The corner and type-I assertions require a nonzero representation and nonzero rank-one inputs; the zero system is explicitly retained. The final example uses Z with counting Haar and its probability-Haar circle dual."
}
]
},
{
"id": "OA-FLOW.BS.0",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-0",
"start_line": 1,
"end_line": 72,
"range_sha256": "8e6aa8fce8683d4c99e26ee05b97f2637bfa7662548625ab95ba2ee31ec037bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.BS.1",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-1",
"start_line": 73,
"end_line": 133,
"range_sha256": "88abe309ac3d577f1c3184599d8ef4bc4768979370f012d1215c3eb43c548d12",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.LF.0"
]
},
{
"id": "OA-FLOW.BS.2",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-2",
"start_line": 134,
"end_line": 163,
"range_sha256": "36791c9f4abee3dfe75382256a64437e34373640e6fe5db03adf5d426b5d01a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.LF.5"
]
},
{
"id": "OA-FLOW.BS.3",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-3",
"start_line": 164,
"end_line": 243,
"range_sha256": "2c6856da94b3d3633f97268003ba5ef737cf81344c05ffc2246b034784833626",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.BS.4",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-4",
"start_line": 244,
"end_line": 322,
"range_sha256": "790e3f6075d9ba01c3d47365e8eaf345547f39dc6d533090df61d36f3411a52e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.3",
"OA-FLOW.BS.2",
"OA-FLOW.LF.3",
"OA-FLOW.LF.4",
"OA-FLOW.LF.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.BS.5",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-5",
"start_line": 323,
"end_line": 349,
"range_sha256": "d7acdbe07ac7a041023c34345f2aed3990a1b20a5a62067d2f6216e209e2396d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.3",
"OA-FLOW.SS.1",
"OA-FLOW.SS.2",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6"
]
},
{
"id": "OA-FLOW.BS.6",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-6",
"start_line": 350,
"end_line": 402,
"range_sha256": "70c58734b468e79a86b4733c454b2d69eae8f163deb2fb9333b4176eccf8a935",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.5",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7"
]
},
{
"id": "OA-FLOW.BS.7",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-7",
"start_line": 403,
"end_line": 510,
"range_sha256": "b4aa71ab7baae38e3e8aecd25420923f14394c0b4730cba70979cf49e1d93022",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.3",
"OA-FLOW.BS.5",
"OA-FLOW.BS.6",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.BS.8",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#bs-8",
"start_line": 511,
"end_line": 550,
"range_sha256": "5ad2a5ba0579ae5e43a4f4459e5025b3cba7a68e9e4c9c901ed9939d695dbc99",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.1",
"OA-FLOW.BS.5",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.BS.FIGURE",
"unit": "OA-FLOW-BS",
"source_file": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b",
"reader": "OA-FLOW-BS.html#oa-flow.bs.figure",
"start_line": 551,
"end_line": 617,
"range_sha256": "60070191140094c48aba404de724357a9b4d02de337cda5ada5cd80ffc382b80",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.2",
"OA-FLOW.BS.7"
]
},
{
"id": "OA-FLOW.GEN.BOUNDS",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.bounds",
"start_line": 1,
"end_line": 28,
"range_sha256": "c79329ade2f028f964ed5910f5c44a075c913375f51a0bafa9f82dbc2830fadd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.3",
"OA-FLOW.CF.1",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1"
]
},
{
"id": "OA-FLOW.GEN.SMOOTHING",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.smoothing",
"start_line": 29,
"end_line": 82,
"range_sha256": "96df4bf62020e0fa9ea1974f2eb2331943ebc847bc00003cbb18bb9dda0309f9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.8",
"OA-FLOW.RF.1",
"OA-FLOW.BS.1",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GEN.DOMAIN",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.domain",
"start_line": 83,
"end_line": 112,
"range_sha256": "35b2e9b0cac930b5bc8d0928e9b6eb826b7987e8cdfd83f6f86cc99b2d4ef49d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.BS.0"
]
},
{
"id": "OA-FLOW.GEN.DENSITY",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.density",
"start_line": 113,
"end_line": 137,
"range_sha256": "7dfb76734291f478967dc02bc7a9e3c4c6645d26d2ba028ca13b99ca89a0bc67",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.BS.1"
]
},
{
"id": "OA-FLOW.GEN.CLOSED",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.closed",
"start_line": 138,
"end_line": 169,
"range_sha256": "f1a63c959b98c87ded530e0c6ffdfb0c118bad2211884e72c3a245465eadc3e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.BS.0"
]
},
{
"id": "OA-FLOW.GEN.INVARIANT",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.invariant",
"start_line": 170,
"end_line": 189,
"range_sha256": "1cce99f03e275ce5c6b1afea2540b36c62388ac1f0d17c589d885bae462f7e5b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.BS.0"
]
},
{
"id": "OA-FLOW.GEN.NORM",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.norm",
"start_line": 191,
"end_line": 242,
"range_sha256": "97d81df375c7f77beba666892ce680339fb37518831d2cddc78d8fc1f499d8a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.CF.1",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"proof_available_after_line": 242,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GEN.NORM-MODEL",
"unit": "OA-FLOW-L94",
"source_file": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd",
"reader": "OA-FLOW-L94.html#oa-flow.gen.norm-model",
"start_line": 243,
"end_line": 283,
"range_sha256": "301a35fb1e55f099b61e76e65b313af0c447eb6ec7bb5c4acef4a8914e546d0e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.NORM",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.SC.7",
"OA-FLOW.CF.1"
],
"proof_available_after_line": 283,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GDW.0",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-0",
"start_line": 1,
"end_line": 26,
"range_sha256": "8a2a15e06274e56f702d211f96d6c125ed85e63f3dc09a12ff35483d0639e1c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.GDW.1",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-1",
"start_line": 27,
"end_line": 100,
"range_sha256": "8411b7938df950d6a29330a1d549dd9723d7f2412597fcdfc164a5d9ae90177c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.4",
"OA-FLOW.AT.4",
"OA-FLOW.NR.3",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.WR.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9"
]
},
{
"id": "OA-FLOW.GDW.2",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-2",
"start_line": 101,
"end_line": 135,
"range_sha256": "6b01d166c171eccd2591a6cb2fe30941e4fd6d2ab2b26e31fed506a7163c174b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GW.4",
"OA-FLOW.NR.3",
"OA-FLOW.ST.2",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3"
]
},
{
"id": "OA-FLOW.GDW.3",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-3",
"start_line": 136,
"end_line": 179,
"range_sha256": "0ddb4c9e61bece576f41ad1acdbc92370a7c7cb162a56315f3a52fb136a46ed3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.NF.5",
"OA-FLOW.WR.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.AT.4",
"OA-FLOW.HR.8",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.GDW.4",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-4",
"start_line": 180,
"end_line": 191,
"range_sha256": "edbae64cc17f620045a3c728d156850b5778afcab60c3484d8d4e533c767a97b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.CI.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.8",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GDW.5",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-5",
"start_line": 192,
"end_line": 325,
"range_sha256": "f345caba71ada723e5856bc41c2751fac9b7e71f5f078995d7dde1741667e557",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.3",
"OA-FLOW.GDW.4",
"OA-FLOW.CI.1",
"OA-FLOW.CI.2",
"OA-FLOW.CI.3",
"OA-FLOW.BC.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.HR.8",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.MC.1",
"OA-FLOW.MC.5",
"OA-FLOW.NR.1",
"OA-FLOW.WR.3",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3"
],
"proof_available_after_line": 321
},
{
"id": "OA-FLOW.GDW.6",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-6",
"start_line": 326,
"end_line": 365,
"range_sha256": "77672fad39f4ad537e2bea7b64f44178b8bceb89aef6b916c89405019f723eb4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.GDW.3",
"OA-FLOW.GDW.5",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.5",
"OA-FLOW.HA-R.6",
"OA-FLOW.HA-R.7",
"OA-FLOW.NR.4"
]
},
{
"id": "OA-FLOW.GDW.7",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-7",
"start_line": 366,
"end_line": 412,
"range_sha256": "0958faf052c4d7a8cd24dcb10914da35ce8178de1bfea63344a67d47c6f3e29c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.MW.1",
"OA-FLOW.MW.2",
"OA-FLOW.MW.3",
"OA-FLOW.MW.4",
"OA-FLOW.AT.1",
"OA-FLOW.AT.4",
"OA-FLOW.NR.3",
"OA-FLOW.ST.2"
]
},
{
"id": "OA-FLOW.GDW.COEFFICIENT-FLOW",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-7a",
"start_line": 413,
"end_line": 534,
"range_sha256": "86c8b106274a1da5b66a45f84eee1439ee5c82813f0f98dca25abf2a35459046",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.GDW.7",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.MW.4",
"OA-FLOW.AT.4",
"OA-FLOW.CCM.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 534,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 5,
"through_line": 7,
"description": "Arbitrary LCH group, arbitrary von Neumann algebra, point-ultraweak normal action and faithful n.s.f. weight; no countability, invariant-weight, finite-state or unimodularity hypothesis."
}
]
},
{
"id": "OA-FLOW.GDW.8",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-8",
"start_line": 538,
"end_line": 557,
"range_sha256": "d7ca8f8620b194576a5895eb0b770927f7e92bc19a9372f907fc446fd3ddd464",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.GDW.7",
"OA-FLOW.GW.3",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.GDW.FIGURE",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-figure",
"start_line": 558,
"end_line": 614,
"range_sha256": "8ee4f6b8166f00deb9f5638bace6661c6446d1247870b8d45f0fa3f6b82c1e90",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.3",
"OA-FLOW.GDW.4",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.GDW.7",
"OA-FLOW.GDW.8"
]
},
{
"id": "OA-FLOW.GDW.REFERENCE-TRANSFER",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-reference-transfer",
"start_line": 616,
"end_line": 650,
"range_sha256": "412dfc429f31ace21e6f98c167e92b05e32fa83a8762f6ea33e2e4c83d1f6861",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.AT.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 650,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 616,
"through_line": 650,
"description": "Arbitrary locally compact Hausdorff group with left Haar measure and the locally determined Haar multiplication algebra; arbitrary nonzero von Neumann algebra and point-ultraweak continuous action; faithful normal semifinite input weights. No separability, countability, unimodularity, invariant input weight or finite-weight-at-one assumption. The final model is explicitly the three-element cyclic group acting on three-by-three matrices."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 616,
"through_line": 650,
"description": "Ordered invariant-reference derivative identity; strong multiplier approximation on finite vector tests; both Haar-coordinate normalizer inclusions and nonabelian action cancellation; the full faithful normal semifinite Haar weight; tensor-weight transport through full Hilbert-algebra completion; invariance on every positive element; and the exact finite cyclic covariance example. HRW18 is a derivative on the enlarged tensor algebra, not an unproved general tensor-factorization assertion."
}
]
},
{
"id": "OA-FLOW.GDW.HAAR-COORDINATE",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-haar-coordinate",
"start_line": 651,
"end_line": 727,
"range_sha256": "61c2b2448d8beaea66c05acb7d7d5e210de3072cc8e33081a4274e3e81c89170",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.GRP.HAARCONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 727,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 651,
"through_line": 727,
"description": "Arbitrary locally compact Hausdorff group with left Haar measure and the locally determined Haar multiplication algebra; arbitrary nonzero von Neumann algebra and point-ultraweak continuous action; faithful normal semifinite input weights. No separability, countability, unimodularity, invariant input weight or finite-weight-at-one assumption. The final model is explicitly the three-element cyclic group acting on three-by-three matrices."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 651,
"through_line": 727,
"description": "Ordered invariant-reference derivative identity; strong multiplier approximation on finite vector tests; both Haar-coordinate normalizer inclusions and nonabelian action cancellation; the full faithful normal semifinite Haar weight; tensor-weight transport through full Hilbert-algebra completion; invariance on every positive element; and the exact finite cyclic covariance example. HRW18 is a derivative on the enlarged tensor algebra, not an unproved general tensor-factorization assertion."
}
]
},
{
"id": "OA-FLOW.GDW.INVARIANT-HAAR-WEIGHT",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-invariant-haar-weight",
"start_line": 728,
"end_line": 810,
"range_sha256": "dbf1905a491f7b94cf0f9a3116f4fb041991655ecfe8ff57a1f96c52c8d426cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.REFERENCE-TRANSFER",
"OA-FLOW.GDW.HAAR-COORDINATE",
"OA-FLOW.GW.4",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-03",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-04",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-05",
"../../OA-MOD/OA-MOD-WH.html#oa-mod-wh-11"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 810,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 728,
"through_line": 810,
"description": "Arbitrary locally compact Hausdorff group with left Haar measure and the locally determined Haar multiplication algebra; arbitrary nonzero von Neumann algebra and point-ultraweak continuous action; faithful normal semifinite input weights. No separability, countability, unimodularity, invariant input weight or finite-weight-at-one assumption. The final model is explicitly the three-element cyclic group acting on three-by-three matrices."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 728,
"through_line": 810,
"description": "Ordered invariant-reference derivative identity; strong multiplier approximation on finite vector tests; both Haar-coordinate normalizer inclusions and nonabelian action cancellation; the full faithful normal semifinite Haar weight; tensor-weight transport through full Hilbert-algebra completion; invariance on every positive element; and the exact finite cyclic covariance example. HRW18 is a derivative on the enlarged tensor algebra, not an unproved general tensor-factorization assertion."
}
]
},
{
"id": "OA-FLOW.GDW.HAAR-MODEL",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-haar-model",
"start_line": 811,
"end_line": 883,
"range_sha256": "252c36d16b31dced26d1322f57f039dca20791eaf4ddb476e4bca4586ba75cfd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.REFERENCE-TRANSFER",
"OA-FLOW.GDW.HAAR-COORDINATE",
"OA-FLOW.GDW.INVARIANT-HAAR-WEIGHT",
"OA-FLOW.BC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 883,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 811,
"through_line": 883,
"description": "Arbitrary locally compact Hausdorff group with left Haar measure and the locally determined Haar multiplication algebra; arbitrary nonzero von Neumann algebra and point-ultraweak continuous action; faithful normal semifinite input weights. No separability, countability, unimodularity, invariant input weight or finite-weight-at-one assumption. The final model is explicitly the three-element cyclic group acting on three-by-three matrices."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 811,
"through_line": 883,
"description": "Ordered invariant-reference derivative identity; strong multiplier approximation on finite vector tests; both Haar-coordinate normalizer inclusions and nonabelian action cancellation; the full faithful normal semifinite Haar weight; tensor-weight transport through full Hilbert-algebra completion; invariance on every positive element; and the exact finite cyclic covariance example. HRW18 is a derivative on the enlarged tensor algebra, not an unproved general tensor-factorization assertion."
}
]
},
{
"id": "OA-FLOW.GDW.ANALYTIC-CUTOFFS",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-analytic-cutoffs",
"start_line": 885,
"end_line": 956,
"range_sha256": "f3c260d983b9088b7ff5b1188563dc6d956c2612731bd9812c5e48a5efcd9782",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.MW.4",
"../../OA-MOD/OA-MOD-NW.html#oa-mod-nw-12",
"../../OA-MOD/OA-MOD-CX.html#oa-mod-cx-03"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 956,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 885,
"through_line": 956,
"description": "Arbitrary nonzero von Neumann algebras and faithful normal semifinite weights, including a nontracial second tensor factor. Analytic approximations use nets and full finite ideals. The final application has an arbitrary locally compact Hausdorff group, left Haar measure, point-ultraweak continuous action, and locally determined Haar algebra. No separability, countability, unimodularity or finite mass at one."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 885,
"through_line": 956,
"description": "Gaussian analytic finite cutoffs with uniformly bounded right GNS multiplication; identification of both entire finite ideals and GNS maps of the balanced tensor weight; all finite and infinite positive values; normalized one- and two-factor cocycles; exact relative graph, adjoint, positive product and complex-power domains; the full Haar tensor identity and solved scalar and nontracial normalization tests."
}
]
},
{
"id": "OA-FLOW.GDW.BALANCED-TENSOR",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-balanced-tensor",
"start_line": 957,
"end_line": 1070,
"range_sha256": "c49df056720cf850cb3c4839f3b9908681df403f9834d1798aaaa9e20bc0e4f3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.ANALYTIC-CUTOFFS",
"OA-FLOW.BC.1",
"OA-FLOW.BC.3",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-03",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-04",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-05"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1070,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 957,
"through_line": 1070,
"description": "Arbitrary nonzero von Neumann algebras and faithful normal semifinite weights, including a nontracial second tensor factor. Analytic approximations use nets and full finite ideals. The final application has an arbitrary locally compact Hausdorff group, left Haar measure, point-ultraweak continuous action, and locally determined Haar algebra. No separability, countability, unimodularity or finite mass at one."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 957,
"through_line": 1070,
"description": "Gaussian analytic finite cutoffs with uniformly bounded right GNS multiplication; identification of both entire finite ideals and GNS maps of the balanced tensor weight; all finite and infinite positive values; normalized one- and two-factor cocycles; exact relative graph, adjoint, positive product and complex-power domains; the full Haar tensor identity and solved scalar and nontracial normalization tests."
}
]
},
{
"id": "OA-FLOW.GDW.NORMALIZED-TENSOR-COCYCLE",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-normalized-tensor-cocycle",
"start_line": 1071,
"end_line": 1154,
"range_sha256": "9be8c32ca35fdc0f3af64f683c0cb465547e5a922be4615dd94347c4f9f90364",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.BALANCED-TENSOR",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-01",
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-02"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1154,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1071,
"through_line": 1154,
"description": "Arbitrary nonzero von Neumann algebras and faithful normal semifinite weights, including a nontracial second tensor factor. Analytic approximations use nets and full finite ideals. The final application has an arbitrary locally compact Hausdorff group, left Haar measure, point-ultraweak continuous action, and locally determined Haar algebra. No separability, countability, unimodularity or finite mass at one."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 1071,
"through_line": 1154,
"description": "Gaussian analytic finite cutoffs with uniformly bounded right GNS multiplication; identification of both entire finite ideals and GNS maps of the balanced tensor weight; all finite and infinite positive values; normalized one- and two-factor cocycles; exact relative graph, adjoint, positive product and complex-power domains; the full Haar tensor identity and solved scalar and nontracial normalization tests."
}
]
},
{
"id": "OA-FLOW.GDW.HAAR-TENSOR-IDENTITY",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-haar-tensor-identity",
"start_line": 1155,
"end_line": 1216,
"range_sha256": "5a0dd101c35c9dccf6d0527b5cd221fa494c475c6cc9b0bd1468952784c72b0f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.NORMALIZED-TENSOR-COCYCLE",
"OA-FLOW.GDW.REFERENCE-TRANSFER",
"OA-FLOW.GDW.HAAR-COORDINATE",
"OA-FLOW.GDW.INVARIANT-HAAR-WEIGHT",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.BC.5",
"OA-FLOW.BC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1216,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1155,
"through_line": 1216,
"description": "Arbitrary nonzero von Neumann algebras and faithful normal semifinite weights, including a nontracial second tensor factor. Analytic approximations use nets and full finite ideals. The final application has an arbitrary locally compact Hausdorff group, left Haar measure, point-ultraweak continuous action, and locally determined Haar algebra. No separability, countability, unimodularity or finite mass at one."
},
{
"source": "src/OA-FLOW-GDW.md",
"line": 1155,
"through_line": 1216,
"description": "Gaussian analytic finite cutoffs with uniformly bounded right GNS multiplication; identification of both entire finite ideals and GNS maps of the balanced tensor weight; all finite and infinite positive values; normalized one- and two-factor cocycles; exact relative graph, adjoint, positive product and complex-power domains; the full Haar tensor identity and solved scalar and nontracial normalization tests."
}
]
},
{
"id": "OA-FLOW.GDW.COEFFICIENT-ORDERS",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-coefficient-orders",
"start_line": 1218,
"end_line": 1296,
"range_sha256": "31152cb7cf3516af0b21454111ee03956ec858e99349e739ace3a1f72187a842",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.7",
"OA-FLOW.GDW.COEFFICIENT-FLOW",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.AT.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1296,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1218,
"through_line": 1296,
"description": "The coefficient-order theorem retains arbitrary locally compact groups, normal continuous actions and faithful normal semifinite weights. The scalar affine example has the complete multiplication-operator power domain. The Haar cancellation model works on every locally compact group and proves the whole-cone trace conclusion through KT5. The finite Hadamard model uses the two-element group, density diag(1,4), exact time pi/(2log4), and an exact sqrt(2) norm error for reversed factors."
}
]
},
{
"id": "OA-FLOW.GDW.AFFINE-SCALAR",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-affine-scalar",
"start_line": 1297,
"end_line": 1365,
"range_sha256": "6bbfbaecfe7d6a27b565c50ff6955d5d977aa363c431e471d691bbdf4ed6edc1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.7",
"OA-FLOW.BC.4",
"OA-FLOW.KT.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.SC.8",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.PROBLEMS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1365,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1297,
"through_line": 1365,
"description": "The coefficient-order theorem retains arbitrary locally compact groups, normal continuous actions and faithful normal semifinite weights. The scalar affine example has the complete multiplication-operator power domain. The Haar cancellation model works on every locally compact group and proves the whole-cone trace conclusion through KT5. The finite Hadamard model uses the two-element group, density diag(1,4), exact time pi/(2log4), and an exact sqrt(2) norm error for reversed factors."
}
]
},
{
"id": "OA-FLOW.GDW.HAAR-CANCELLATION",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-haar-cancellation",
"start_line": 1366,
"end_line": 1421,
"range_sha256": "8fa554444d7f5d51ec2e86f7ad50858f05ad73324f9b9975c760076e865bea4f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.7",
"OA-FLOW.GDW.INVARIANT-HAAR-WEIGHT",
"OA-FLOW.BC.5",
"OA-FLOW.KT.5",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GDW.AFFINE-SCALAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1421,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1366,
"through_line": 1421,
"description": "The coefficient-order theorem retains arbitrary locally compact groups, normal continuous actions and faithful normal semifinite weights. The scalar affine example has the complete multiplication-operator power domain. The Haar cancellation model works on every locally compact group and proves the whole-cone trace conclusion through KT5. The finite Hadamard model uses the two-element group, density diag(1,4), exact time pi/(2log4), and an exact sqrt(2) norm error for reversed factors."
}
]
},
{
"id": "OA-FLOW.GDW.HADAMARD-ORDER",
"unit": "OA-FLOW-GDW",
"source_file": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112",
"reader": "OA-FLOW-GDW.html#gdw-hadamard-order",
"start_line": 1422,
"end_line": 1488,
"range_sha256": "6fc991c8f7c2c47eee1b57577b74b87d28f7f6461cf489a374e92223ef3e3fc2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.COEFFICIENT-ORDERS",
"OA-FLOW.BC.6",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1488,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDW.md",
"line": 1422,
"through_line": 1488,
"description": "The coefficient-order theorem retains arbitrary locally compact groups, normal continuous actions and faithful normal semifinite weights. The scalar affine example has the complete multiplication-operator power domain. The Haar cancellation model works on every locally compact group and proves the whole-cone trace conclusion through KT5. The finite Hadamard model uses the two-element group, density diag(1,4), exact time pi/(2log4), and an exact sqrt(2) norm error for reversed factors."
}
]
},
{
"id": "OA-FLOW.CCM.0",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-0",
"start_line": 1,
"end_line": 36,
"range_sha256": "cd6def9c9d16a8a15a2f8ebbb3e7860ca83179520c2342e3b00e13f38c2d9d02",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.CCM.5",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-5",
"start_line": 39,
"end_line": 59,
"range_sha256": "e3f8cc9c0260efd3877139af83b1e868ddb2665bc7c014b8ce29afd6a0c0dd5e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.CP.4",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.CCM.TRANSPORT",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-transport",
"start_line": 62,
"end_line": 131,
"range_sha256": "ebfff9b36a33ab4c5c982b07a875404a26fc4b5df4009ac85f5b1e743361c4ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.CCM.STANDARD",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-standard",
"start_line": 134,
"end_line": 173,
"range_sha256": "e01bf8d5795ae9b859ceda05f310d475c1b203b8c8aa754357b21f0c817e1bf9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.9",
"OA-FLOW.NC.1",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.NF.5",
"OA-FLOW.NR.1",
"OA-FLOW.WR.1",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.MW.1",
"OA-FLOW.GRP.HAARCONVENTIONS"
]
},
{
"id": "OA-FLOW.CCM.7",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-7",
"start_line": 176,
"end_line": 199,
"range_sha256": "a9e569cd6c33a32feddb27f244f88c8476a725786916aa32e112b590079cd456",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.5",
"OA-FLOW.CCM.TRANSPORT",
"OA-FLOW.CCM.STANDARD",
"OA-FLOW.CP.4"
]
},
{
"id": "OA-FLOW.CCM.8",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-8",
"start_line": 202,
"end_line": 226,
"range_sha256": "26d8f5dc1bccee39dd005bbe5f9cb074a4104e1dea719bdf876309a42670c169",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.7",
"OA-FLOW.NR.1",
"OA-FLOW.NR.4",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4"
]
},
{
"id": "OA-FLOW.CCM.9",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#ccm-9",
"start_line": 229,
"end_line": 252,
"range_sha256": "6f34cd6cedeb3b079f6c9992d42fe8d49f44df065c1efd5ff37ffb1cffa67f2f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.7",
"OA-FLOW.CCM.8",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.CCM.FIGURE",
"unit": "OA-FLOW-CCM",
"source_file": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a",
"reader": "OA-FLOW-CCM.html#oa-flow.ccm.figure",
"start_line": 255,
"end_line": 269,
"range_sha256": "9a67462356db3e5aacbcd681ba68e14ed0f57be2dba77307bac84f1cfaff0d2b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCM.5",
"OA-FLOW.CCM.TRANSPORT",
"OA-FLOW.CCM.STANDARD",
"OA-FLOW.CCM.7",
"OA-FLOW.CCM.8",
"OA-FLOW.CCM.9"
]
},
{
"id": "OA-FLOW.QF.1",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-1",
"start_line": 10,
"end_line": 27,
"range_sha256": "f25ebac86eba914b49708b0f297d5381150a4989e280b0d940857ccaf6414281",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.QF.2",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-2",
"start_line": 30,
"end_line": 52,
"range_sha256": "ae28f3d2d3df7a26eb99b657a5cf4d83a07700619f7aa42ac7505b6884a50c21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.1",
"OA-FLOW.HR.7",
"OA-FLOW.HR.TOPOLOGY"
]
},
{
"id": "OA-FLOW.QF.3",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-3",
"start_line": 55,
"end_line": 70,
"range_sha256": "9154d3707e8b92276a83652bf183d3173d3b2df6942d8183a443a127ddf23c90",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.2",
"OA-FLOW.HR.8",
"OA-FLOW.HR.TOPOLOGY"
]
},
{
"id": "OA-FLOW.QF.4",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-4",
"start_line": 73,
"end_line": 86,
"range_sha256": "a5db2632b430d5803ce7ba04753e829470e3f158a33e0e66cdc6c47655ae6c14",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.TOPOLOGY"
]
},
{
"id": "OA-FLOW.QF.5",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-5",
"start_line": 89,
"end_line": 105,
"range_sha256": "18b568e8014a1c8677bdfd0eb40681388853e0fcd178b7dd2e021307711bf793",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.4",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.QF.6",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-6",
"start_line": 108,
"end_line": 123,
"range_sha256": "d7c05cec3f712c7d29471f1b01b19147c95d681f473b871ec82c13665687c1e6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.3",
"OA-FLOW.HR.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.7",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.QF.7",
"unit": "OA-FLOW-QF",
"source_file": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39",
"reader": "OA-FLOW-QF.html#qf-7",
"start_line": 126,
"end_line": 130,
"range_sha256": "4d1118ccbeca13ee12be43908059b01855ef1b0825acdaaa41619aaca22446be",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.7"
]
},
{
"id": "OA-FLOW.QM.SETTING",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.setting",
"start_line": 9,
"end_line": 81,
"range_sha256": "0d1da15c81b4050a5acd9bfeb9955eafb5dbde35ffe1f03dae157854ce89724b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.QF.3",
"OA-FLOW.QF.6",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.TOPOLOGY"
]
},
{
"id": "OA-FLOW.QM.CUTOFF",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.cutoff",
"start_line": 85,
"end_line": 102,
"range_sha256": "d5d738f4536376424654d75507de6e89369a557b75934e86958b28a9e54978c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.QF.2",
"OA-FLOW.QF.3"
]
},
{
"id": "OA-FLOW.QM.RHO",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.rho",
"start_line": 106,
"end_line": 143,
"range_sha256": "21ecb1f8acae7924bbe504f768605271452459546ee1510db6c7eff2586b0b64",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CUTOFF",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.HR.2"
]
},
{
"id": "OA-FLOW.QM.DESCENT",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.descent",
"start_line": 147,
"end_line": 167,
"range_sha256": "7404e77dd6d794030a7efb21c557e1c62dc5d8435ab2b6f15e68990dc4ad1a6e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.RHO",
"OA-FLOW.QF.6",
"OA-FLOW.HR.5",
"OA-FLOW.HR.3"
]
},
{
"id": "OA-FLOW.QM.CLASS",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.class",
"start_line": 171,
"end_line": 301,
"range_sha256": "ffc46d214dc854d8ca5465c5360218f10249372358041cbb33e0e155e69b1240",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.DESCENT",
"OA-FLOW.QM.RHO",
"OA-FLOW.QF.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.QM.DERIVATIVE"
]
},
{
"id": "OA-FLOW.QM.DERIVATIVE",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.derivative",
"start_line": 305,
"end_line": 349,
"range_sha256": "cd428ab5acea985c77ee17d039b913865fdf365ef88f9139d06c61545478c3f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.RHO",
"OA-FLOW.SC.8",
"OA-FLOW.HR.5"
]
},
{
"id": "OA-FLOW.IND.SETTING",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.setting",
"start_line": 11,
"end_line": 55,
"range_sha256": "e43a24fdd6f4ffd4d151c2de79f3184b4fb56c09e3ea0953e79a0ed6de3ae4f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.DESCENT",
"OA-FLOW.QF.6",
"OA-FLOW.CF.8",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
]
},
{
"id": "OA-FLOW.IND.COMPLETE",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.complete",
"start_line": 59,
"end_line": 92,
"range_sha256": "819d5438eed5b50d8a26f1a2c8819befbf9f348eccc69c2422b43bd2174f66da",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.SETTING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.IND.ELEMENTARY",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.elementary",
"start_line": 96,
"end_line": 131,
"range_sha256": "f374cb5323a01c74b555aa7bc03b44f87eccb04ea99012127a897be5991481cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.COMPLETE",
"OA-FLOW.QM.RHO",
"OA-FLOW.QM.CLASS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.QF.6",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.IND.TRANSLATION",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.translation",
"start_line": 135,
"end_line": 159,
"range_sha256": "7fd3c27e7892935a80b7c0c476b102987e525eedd20c16db0f823213be448f1f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.IND.IMPRIMITIVITY",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.imprimitivity",
"start_line": 163,
"end_line": 185,
"range_sha256": "f7058eb1831dc25471004effe85478754140f7ab26b24cf3b5e62a9ecf843a84",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.TRANSLATION",
"OA-FLOW.QM.DESCENT",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.SC.7"
]
},
{
"id": "OA-FLOW.IND.SMOOTHING",
"unit": "OA-FLOW-L44",
"source_file": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6",
"reader": "OA-FLOW-L44.html#oa-flow.ind.smoothing",
"start_line": 189,
"end_line": 215,
"range_sha256": "7b2be1ee245fd7a554bbd466adb8948c642328db22d8b3f3613a669f734c8cc6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.IMPRIMITIVITY",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.QF.6"
]
},
{
"id": "OA-FLOW.IMPR.SETTING",
"unit": "OA-FLOW-L45",
"source_file": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9",
"reader": "OA-FLOW-L45.html#oa-flow.impr.setting",
"start_line": 11,
"end_line": 38,
"range_sha256": "94076296e9d974160cfdf3de4bee6a647a111b500eda5f49bd51d814b79c94f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QF.5",
"OA-FLOW.QF.4",
"OA-FLOW.QF.2",
"OA-FLOW.IND.IMPRIMITIVITY",
"OA-FLOW.QM.CLASS",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.IMPR.SMOOTH",
"unit": "OA-FLOW-L45",
"source_file": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9",
"reader": "OA-FLOW-L45.html#oa-flow.impr.smooth",
"start_line": 42,
"end_line": 68,
"range_sha256": "b243a9fc26ad501dfd0bfa78cfd9d8045173244f6daa4b8114e80c13a69a8a66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.QF.4",
"OA-FLOW.QM.CLASS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.IMPR.FIBER",
"unit": "OA-FLOW-L45",
"source_file": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9",
"reader": "OA-FLOW-L45.html#oa-flow.impr.fiber",
"start_line": 72,
"end_line": 113,
"range_sha256": "5f2fabb4babdec749d4b648f658408dd1de5281db1c29ccf3af4a81d713d1a5c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.QM.DESCENT",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
]
},
{
"id": "OA-FLOW.IMPR.UNITARY",
"unit": "OA-FLOW-L45",
"source_file": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9",
"reader": "OA-FLOW-L45.html#oa-flow.impr.unitary",
"start_line": 117,
"end_line": 178,
"range_sha256": "a0e20ca985459d16c820e07f20612f435681f26935e28c564ae0f44b94550a39",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IMPR.FIBER",
"OA-FLOW.IND.COMPLETE",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IND.IMPRIMITIVITY",
"OA-FLOW.QF.5",
"OA-FLOW.CF.8",
"OA-FLOW.QF.6"
]
},
{
"id": "OA-FLOW.IMPR.UNIQUENESS",
"unit": "OA-FLOW-L45",
"source_file": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9",
"reader": "OA-FLOW-L45.html#oa-flow.impr.uniqueness",
"start_line": 182,
"end_line": 207,
"range_sha256": "ecfe57281a5c015e36b3a8e43f136b20ea1bf81ac4df55750b4ed4251e1ef3fe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.IND.SMOOTHING",
"OA-FLOW.QF.1"
]
},
{
"id": "OA-FLOW.IINT.SETTING",
"unit": "OA-FLOW-L46",
"source_file": "src/OA-FLOW-L46.md",
"source_sha256": "c52be1d9da5197056272f66299112e0bd1da9c4a570107fc98f0cfb98a4e426d",
"reader": "OA-FLOW-L46.html#oa-flow.iint.setting",
"start_line": 11,
"end_line": 34,
"range_sha256": "819e34e5a0666db31e10e91e226a9900447643cde74bc9e04e001a808f5f1336",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IND.IMPRIMITIVITY"
]
},
{
"id": "OA-FLOW.IINT.SMOOTH",
"unit": "OA-FLOW-L46",
"source_file": "src/OA-FLOW-L46.md",
"source_sha256": "c52be1d9da5197056272f66299112e0bd1da9c4a570107fc98f0cfb98a4e426d",
"reader": "OA-FLOW-L46.html#oa-flow.iint.smooth",
"start_line": 38,
"end_line": 56,
"range_sha256": "8a5cd53bbe72b3e82575a9869845b22724c0a60ca6a35cef20b7280d64707167",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IINT.SETTING",
"OA-FLOW.IND.SMOOTHING"
]
},
{
"id": "OA-FLOW.IINT.RECOVERY",
"unit": "OA-FLOW-L46",
"source_file": "src/OA-FLOW-L46.md",
"source_sha256": "c52be1d9da5197056272f66299112e0bd1da9c4a570107fc98f0cfb98a4e426d",
"reader": "OA-FLOW-L46.html#oa-flow.iint.recovery",
"start_line": 60,
"end_line": 106,
"range_sha256": "28d605148505d5c86b7686bc34e815232dc6c8113d823ff69526539dde6e693b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.IMPR.UNIQUENESS",
"OA-FLOW.QF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
]
},
{
"id": "OA-FLOW.IINT.NORMALITY",
"unit": "OA-FLOW-L46",
"source_file": "src/OA-FLOW-L46.md",
"source_sha256": "c52be1d9da5197056272f66299112e0bd1da9c4a570107fc98f0cfb98a4e426d",
"reader": "OA-FLOW-L46.html#oa-flow.iint.normality",
"start_line": 110,
"end_line": 153,
"range_sha256": "3196b2f61cb4104f5ed0ee75a8d3664c9dc913f705f94043c2d0881c8547ca98",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.QF.6",
"OA-FLOW.QF.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.RES.TWIST",
"unit": "OA-FLOW-L95",
"source_file": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf",
"reader": "OA-FLOW-L95.html#oa-flow.res.twist",
"start_line": 9,
"end_line": 21,
"range_sha256": "6a33ee0ab1aa2f1d98c90686b8ee216ea0cd6446b65ac5476ec39fdbb8512b32",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT"
]
},
{
"id": "OA-FLOW.RES.FORWARD",
"unit": "OA-FLOW-L95",
"source_file": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf",
"reader": "OA-FLOW-L95.html#oa-flow.res.forward",
"start_line": 22,
"end_line": 62,
"range_sha256": "331a97fcaf7fdb71fc25c9ba7cbf1e4420d4e3f659d1b4378bea4d3c343bbee7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT"
]
},
{
"id": "OA-FLOW.RES.BACKWARD",
"unit": "OA-FLOW-L95",
"source_file": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf",
"reader": "OA-FLOW-L95.html#oa-flow.res.backward",
"start_line": 63,
"end_line": 83,
"range_sha256": "182906440075215127987c05f9001b5f23d2abd84b0545cd760c959e7764515a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT"
]
},
{
"id": "OA-FLOW.RES.IMAGINARY",
"unit": "OA-FLOW-L95",
"source_file": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf",
"reader": "OA-FLOW-L95.html#oa-flow.res.imaginary",
"start_line": 84,
"end_line": 100,
"range_sha256": "314761252a921f482b6c1a395f5b2346d2ba86fcb829800f1433dbcf21410bac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT"
]
},
{
"id": "OA-FLOW.RES.NORM",
"unit": "OA-FLOW-L95",
"source_file": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf",
"reader": "OA-FLOW-L95.html#oa-flow.res.norm",
"start_line": 102,
"end_line": 177,
"range_sha256": "931882bc9cffce9b73bce7124a24e79f90801be8248fa6be4817bfdc47644c5b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.NORM",
"OA-FLOW.GEN.NORM-MODEL",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CF.1"
],
"proof_available_after_line": 177,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.HY.PRECONTINUITY",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.precontinuity",
"start_line": 9,
"end_line": 27,
"range_sha256": "098e630ebe1b0446372d3a8db79b813599442a1c47dbebe697bce0945beef281",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT"
]
},
{
"id": "OA-FLOW.HY.RESOLVENTS",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.resolvents",
"start_line": 30,
"end_line": 96,
"range_sha256": "a59ca5c40b7afcc6bc9851eaae3f5aac8b9ed5b7d7b95abbd50e85e197b224a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT",
"OA-FLOW.HY.PRECONTINUITY",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.BACKWARD"
]
},
{
"id": "OA-FLOW.HY.CRITERION",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.criterion",
"start_line": 30,
"end_line": 221,
"range_sha256": "feb8e2960fd42f96bdd908d354fb5502900970563eef9a95c44030684a71401d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT",
"OA-FLOW.HY.PRECONTINUITY",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.BACKWARD"
]
},
{
"id": "OA-FLOW.HY.PREADJOINT",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.preadjoint",
"start_line": 97,
"end_line": 221,
"range_sha256": "015081ea9a61bf6d04a50c017d6f2224495c3da48ad6d89e4a5d6df815d2e35d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT",
"OA-FLOW.HY.PRECONTINUITY",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.BACKWARD",
"OA-FLOW.HY.RESOLVENTS"
]
},
{
"id": "OA-FLOW.HY.EULER",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.euler",
"start_line": 222,
"end_line": 290,
"range_sha256": "d90c3424349d87c1ba7053cf749f3ae019aa859e59112d52fe2f0685e82b6ab0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT",
"OA-FLOW.HY.PRECONTINUITY",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.BACKWARD",
"OA-FLOW.HY.CRITERION",
"OA-FLOW.HY.PREADJOINT"
]
},
{
"id": "OA-FLOW.HY.YOSIDA",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.yosida",
"start_line": 291,
"end_line": 357,
"range_sha256": "692498bf50133fddafa179fd3ca6b75d018c3c5a1eb3a16c0237f432854ca5de",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.8",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8",
"OA-FLOW.ST.1",
"OA-FLOW.L34.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.GEN.SMOOTHING",
"OA-FLOW.GEN.DOMAIN",
"OA-FLOW.GEN.DENSITY",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.INVARIANT",
"OA-FLOW.HY.PRECONTINUITY",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.BACKWARD",
"OA-FLOW.HY.CRITERION",
"OA-FLOW.HY.PREADJOINT",
"OA-FLOW.HY.EULER"
]
},
{
"id": "OA-FLOW.HY.NORM",
"unit": "OA-FLOW-L96",
"source_file": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b",
"reader": "OA-FLOW-L96.html#oa-flow.hy.norm",
"start_line": 359,
"end_line": 437,
"range_sha256": "244b81672ddef1853deba02d6009fad0ac3cc4587350edbfb2672792a85a91e1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GEN.NORM",
"OA-FLOW.RES.NORM",
"OA-FLOW.HY.CRITERION",
"OA-FLOW.HY.PREADJOINT",
"OA-FLOW.HY.EULER",
"OA-FLOW.HY.YOSIDA",
"OA-FLOW.CF.1"
],
"proof_available_after_line": 437,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GDA.0",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-0",
"start_line": 1,
"end_line": 47,
"range_sha256": "0954a7816ade420bbbb50f4e5167b571fa1f1446078b7af2c1f455c0a44937ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.GDA.1",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-1",
"start_line": 49,
"end_line": 86,
"range_sha256": "1014c780af930beb6875dbd80603f78409e1d58fe53625f36f0a70cea9c1a489",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.WF.2",
"OA-FLOW.WF.5",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.EW.2",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.GDA.2",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-2",
"start_line": 88,
"end_line": 135,
"range_sha256": "7a6e2a4a1b1a5e6741a365967fd84bf0135d3516e440e314cf21ab4adbfc279c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.5",
"OA-FLOW.EW.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.GDA.3",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-3",
"start_line": 137,
"end_line": 234,
"range_sha256": "591de8cb9554fcf23beececca43653aff873aa0dc72d52a25419bab598ebfadc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.1",
"OA-FLOW.GDA.2",
"OA-FLOW.HA-R.4",
"OA-FLOW.WH04.3",
"OA-FLOW.EW.2",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.GDA.4",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-4",
"start_line": 236,
"end_line": 296,
"range_sha256": "5e24e6c398f14e7b3729cb2880295d8a6134403f005baefe876f443ac15a1961",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.2",
"OA-FLOW.GDA.3",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB6"
]
},
{
"id": "OA-FLOW.GDA.5",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-5",
"start_line": 298,
"end_line": 333,
"range_sha256": "ded3aa105ca565b7f589c4a1988200db3fd3a98d4f10c1db6a931a87bae6bc60",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.4",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.CCM.4",
"OA-FLOW.CCM.5",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GDA.6",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-6",
"start_line": 335,
"end_line": 372,
"range_sha256": "829d337ed8e866a7edbe536cae5b173fba061817d51b00fd3631f3bcd72e76be",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.1",
"OA-FLOW.GDA.3",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.CCM.4",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.6",
"OA-FLOW.NR.4",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.GDA.7",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-7",
"start_line": 374,
"end_line": 436,
"range_sha256": "b6c3289f4797ef57c08d4fafd664173766dbe0adaab3cbe24462cac0f92a0533",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MA.4",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.NC.4",
"OA-FLOW.RF.5",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.3",
"OA-FLOW.CZ.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.6",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.GDW.6",
"OA-FLOW.GDW.7",
"OA-FLOW.GW.3",
"OA-FLOW.NF.5",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CP.4",
"OA-FLOW.SF.SB4"
]
},
{
"id": "OA-FLOW.GDA.8",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-8",
"start_line": 438,
"end_line": 491,
"range_sha256": "7d3768eeeda7ad3c662f5f51ce0b84fe97d2c0accadbb0dffd679c388fa2e05a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.7",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.OT.1",
"OA-FLOW.OT.2",
"OA-FLOW.OT.3",
"OA-FLOW.OT.4",
"OA-FLOW.OT.5",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.6",
"OA-FLOW.GDW.7",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.NF.5",
"OA-FLOW.WS.5",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.GDA.9",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-9",
"start_line": 493,
"end_line": 550,
"range_sha256": "6cea4aee96c452311bd303bb0f3c0697c9e99d3c3dc0b1f42acddb96e1941d00",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.8",
"OA-FLOW.OT.5",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.7",
"OA-FLOW.RF.5",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.GDA.UNIQUE-MAP",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-10",
"start_line": 552,
"end_line": 608,
"range_sha256": "4243bd875f1540c6cd3d8e493408b4fc1e5a7c35827c27acb9050e218cb63cee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SF1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 608,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"line": 5,
"through_line": 5,
"description": "Arbitrary LCH group and normal continuous action; faithful n.s.f. input weights only where stated; zero algebra separated, no countability or unimodularity restriction."
}
]
},
{
"id": "OA-FLOW.GDA.BALANCED-ACTION",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-11",
"start_line": 610,
"end_line": 640,
"range_sha256": "dfc59f8e5a2a93bd9643b49275c80963334c0fa1626efef38d6977a2d5451771",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 640,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"line": 5,
"through_line": 5,
"description": "Arbitrary LCH group and normal continuous action; faithful n.s.f. input weights only where stated; zero algebra separated, no countability or unimodularity restriction."
}
]
},
{
"id": "OA-FLOW.GDA.BALANCED-DUAL",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-12",
"start_line": 642,
"end_line": 722,
"range_sha256": "85dc05ceb72062efa330e5946f5b5e70c8ce4e8f989add3115383a43339ef739",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.4",
"OA-FLOW.GDA.1",
"OA-FLOW.GDA.2",
"OA-FLOW.GDA.3",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.BC.1",
"OA-FLOW.GDW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 722,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"line": 5,
"through_line": 5,
"description": "Arbitrary LCH group and normal continuous action; faithful n.s.f. input weights only where stated; zero algebra separated, no countability or unimodularity restriction."
}
]
},
{
"id": "OA-FLOW.GDA.FIGURE",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-figure",
"start_line": 726,
"end_line": 830,
"range_sha256": "bf955f5114b916d6bf9ecf5c5baec6a8f9bdf3001ff898adb62a61583068cb45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.1",
"OA-FLOW.GDA.2",
"OA-FLOW.GDA.3",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.7",
"OA-FLOW.GDA.8",
"OA-FLOW.GDA.9",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.NCF.0",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-0",
"start_line": 1,
"end_line": 37,
"range_sha256": "984d3f8b9e8aeece195834def4a6853c3c8f51641b59f6c7ac4747649136f65b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"range_bytes": 2553
},
{
"id": "OA-FLOW.NCF.1",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-1",
"start_line": 38,
"end_line": 111,
"range_sha256": "efeaf7490ef47e6eb6a0b897177c9f57785f89f311eafbc8ea0fc5a19d50583e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2"
],
"range_bytes": 4929
},
{
"id": "OA-FLOW.NCF.2",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-2",
"start_line": 112,
"end_line": 186,
"range_sha256": "9e4e3b4dd43c4a0d270478b79c063948ead191631f3a1901fdc378620863e14d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.NR.1",
"OA-FLOW.NR.2",
"OA-FLOW.NR.3",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.CP.4",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CCM.4",
"OA-FLOW.CCM.5",
"OA-FLOW.CCM.7",
"OA-FLOW.CCM.8"
],
"range_bytes": 2994
},
{
"id": "OA-FLOW.NCF.3",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-3",
"start_line": 187,
"end_line": 241,
"range_sha256": "9331d934bbfe4e7489746630a0ebef3efe17d60d130dd7c2bbe3a5f371fbefc8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.2",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.ST.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"range_bytes": 2699
},
{
"id": "OA-FLOW.NCF.4",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-4",
"start_line": 242,
"end_line": 274,
"range_sha256": "e9aaab2fb78b9653374e1c3cd2f9ee49231559713e2dc9d58fbcaabee91e58f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.AT.1"
],
"range_bytes": 2100
},
{
"id": "OA-FLOW.NCF.5",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-5",
"start_line": 275,
"end_line": 304,
"range_sha256": "450184e86b41127790addffe1050408f513905e8ce4f0a587a1cc56ed6303ec2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.BD.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF1",
"OA-FLOW.ST.2"
],
"range_bytes": 2100
},
{
"id": "OA-FLOW.NCF.6",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-6",
"start_line": 305,
"end_line": 340,
"range_sha256": "44c527ba2af8cd355784e719c006d67682e249eadbc60a4f14562a56103672d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.GDA.5"
],
"range_bytes": 2257
},
{
"id": "OA-FLOW.NCF.FIGURE",
"unit": "OA-FLOW-NCF",
"source_file": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b",
"reader": "OA-FLOW-NCF.html#ncf-figure",
"start_line": 341,
"end_line": 395,
"range_sha256": "19f082cb5bc912dc8b17ab418eb5733fc504e23a201300a5113ccdb62ac5fc2d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.NCF.6",
"OA-FLOW.GDA.5"
],
"range_bytes": 3279
},
{
"id": "OA-FLOW.AF.0",
"unit": "OA-FLOW-AF",
"source_file": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7",
"reader": "OA-FLOW-AF.html#af-0",
"start_line": 1,
"end_line": 40,
"range_sha256": "3cc62da05bba2120629d37b73c882a737e5d978715c244a51fd27933977c684e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.7",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.AF.1",
"unit": "OA-FLOW-AF",
"source_file": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7",
"reader": "OA-FLOW-AF.html#af-1",
"start_line": 41,
"end_line": 129,
"range_sha256": "b8dd69b6b84c02c6c50df696cb5ebbbceae5cd717ce630c32a33fc3288cc18c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.CF.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.LF.5",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7",
"OA-FLOW.BS.2"
]
},
{
"id": "OA-FLOW.AF.2",
"unit": "OA-FLOW-AF",
"source_file": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7",
"reader": "OA-FLOW-AF.html#af-2",
"start_line": 130,
"end_line": 150,
"range_sha256": "3ec1163b8fa10991a74d1ccc571e3e1e0014d59beded6ed098af0921d344c62e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.AF.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.AF.3",
"unit": "OA-FLOW-AF",
"source_file": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7",
"reader": "OA-FLOW-AF.html#af-3",
"start_line": 151,
"end_line": 177,
"range_sha256": "1ceac008a68d39eb85409c52cf2b86f9ea6d04d89b6c4028141ab4a212b65e6d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.RF.1",
"OA-FLOW.CF.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.AF.4",
"unit": "OA-FLOW-AF",
"source_file": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7",
"reader": "OA-FLOW-AF.html#af-4",
"start_line": 178,
"end_line": 231,
"range_sha256": "8c62d216fe8ab3b2ef2abfd29a6a2ea184847149d66ffce4f4feb46b4d47727c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CF.4",
"OA-FLOW.SC.0",
"OA-FLOW.SC.8"
]
},
{
"id": "OA-FLOW.BAND.FLATTOP",
"unit": "OA-FLOW-L87",
"source_file": "src/OA-FLOW-L87.md",
"source_sha256": "cbf1d211411f51d9d4ae882c4193e513102c099d37869b69d47783b81ad353af",
"reader": "OA-FLOW-L87.html#oa-flow.band.flattop",
"start_line": 1,
"end_line": 43,
"range_sha256": "dcf4ba89c5295d6bbaacd36794df3416ace3981a0c6f9c1ec7bb803940ebc8c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.LF.0",
"OA-FLOW.LF.1"
]
},
{
"id": "OA-FLOW.BAND.SMALLPRODUCT",
"unit": "OA-FLOW-L87",
"source_file": "src/OA-FLOW-L87.md",
"source_sha256": "cbf1d211411f51d9d4ae882c4193e513102c099d37869b69d47783b81ad353af",
"reader": "OA-FLOW-L87.html#oa-flow.band.smallproduct",
"start_line": 44,
"end_line": 57,
"range_sha256": "95f2ea2cb9797fa6482fed7e699e84a25f181cd53303fa00c7c598b6ef1ba9fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BAND.FLATTOP",
"OA-FLOW.SS.1"
]
},
{
"id": "OA-FLOW.BAND.ORBIT",
"unit": "OA-FLOW-L87",
"source_file": "src/OA-FLOW-L87.md",
"source_sha256": "cbf1d211411f51d9d4ae882c4193e513102c099d37869b69d47783b81ad353af",
"reader": "OA-FLOW-L87.html#oa-flow.band.orbit",
"start_line": 58,
"end_line": 76,
"range_sha256": "600ae5432c2904ed58b7d8551ad61334e4e63d04a9d61dcd1b29b02e30446467",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.LF.4",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.BAND.UNIFORM",
"unit": "OA-FLOW-L87",
"source_file": "src/OA-FLOW-L87.md",
"source_sha256": "cbf1d211411f51d9d4ae882c4193e513102c099d37869b69d47783b81ad353af",
"reader": "OA-FLOW-L87.html#oa-flow.band.uniform",
"start_line": 77,
"end_line": 106,
"range_sha256": "32eae885c4f0e9534e6ad8c438deb957547ab5365d967eee38051c90bfa9b8d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BAND.SMALLPRODUCT",
"OA-FLOW.BAND.ORBIT",
"OA-FLOW.BS.4"
]
},
{
"id": "OA-FLOW.ACTSPEC.APPROX",
"unit": "OA-FLOW-L88",
"source_file": "src/OA-FLOW-L88.md",
"source_sha256": "de48f95a572282f90ce153d59d1e028acfe7b4fdd008615fb3fab8632392ee37",
"reader": "OA-FLOW-L88.html#oa-flow.actspec.approx",
"start_line": 1,
"end_line": 30,
"range_sha256": "52c0a62a0669825ba7bb12067d5f24068513f6d7efe6af0dcf878852a3f1edf6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BAND.UNIFORM",
"OA-FLOW.AF.0",
"OA-FLOW.LF.1",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4"
]
},
{
"id": "OA-FLOW.ACTSPEC.MEASURE",
"unit": "OA-FLOW-L88",
"source_file": "src/OA-FLOW-L88.md",
"source_sha256": "de48f95a572282f90ce153d59d1e028acfe7b4fdd008615fb3fab8632392ee37",
"reader": "OA-FLOW-L88.html#oa-flow.actspec.measure",
"start_line": 31,
"end_line": 50,
"range_sha256": "8644d23f1d11883a4084c53f6dfe14a849536d22263a6e550bca44d981388316",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.1"
]
},
{
"id": "OA-FLOW.ACTSPEC.TAIL",
"unit": "OA-FLOW-L88",
"source_file": "src/OA-FLOW-L88.md",
"source_sha256": "de48f95a572282f90ce153d59d1e028acfe7b4fdd008615fb3fab8632392ee37",
"reader": "OA-FLOW-L88.html#oa-flow.actspec.tail",
"start_line": 51,
"end_line": 74,
"range_sha256": "5470f6b324233fed4c427af7c8a734678e03c934d6812bbad4452d535e59a9c2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ACTSPEC.APPROX",
"OA-FLOW.ACTSPEC.MEASURE",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.ACTSPEC.NORM",
"unit": "OA-FLOW-L88",
"source_file": "src/OA-FLOW-L88.md",
"source_sha256": "de48f95a572282f90ce153d59d1e028acfe7b4fdd008615fb3fab8632392ee37",
"reader": "OA-FLOW-L88.html#oa-flow.actspec.norm",
"start_line": 75,
"end_line": 101,
"range_sha256": "bd363d96b41f17605933bb90263e905558e8ce1eee04173908920966225baa2f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ACTSPEC.APPROX",
"OA-FLOW.ACTSPEC.MEASURE",
"OA-FLOW.ACTSPEC.TAIL",
"OA-FLOW.AF.1",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.BS.3"
]
},
{
"id": "OA-FLOW.OPSP.CIRCLE",
"unit": "OA-FLOW-L89",
"source_file": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a",
"reader": "OA-FLOW-L89.html#oa-flow.opsp.circle",
"start_line": 1,
"end_line": 33,
"range_sha256": "76327e522dc66100985053b60a581a77dc2290e0650651be98f32457bda9489f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ACTSPEC.APPROX",
"OA-FLOW.AF.0",
"OA-FLOW.CF.2"
]
},
{
"id": "OA-FLOW.OPSP.RECIPROCAL",
"unit": "OA-FLOW-L89",
"source_file": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a",
"reader": "OA-FLOW-L89.html#oa-flow.opsp.reciprocal",
"start_line": 34,
"end_line": 47,
"range_sha256": "3be5e9681acaf46c04df21b537a0601f4ada4ce6a5aae749406bd10b5b697205",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.3",
"OA-FLOW.OPSP.CIRCLE"
]
},
{
"id": "OA-FLOW.OPSP.MEASURE-SIGN",
"unit": "OA-FLOW-L89",
"source_file": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a",
"reader": "OA-FLOW-L89.html#oa-flow.opsp.measure-sign",
"start_line": 48,
"end_line": 68,
"range_sha256": "572a63903160b4c09956a5c323cedc7ef66d76ab7c7da40ab1432d010ee28259",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.1",
"OA-FLOW.OPSP.CIRCLE",
"OA-FLOW.OPSP.RECIPROCAL"
]
},
{
"id": "OA-FLOW.OPSP.LOCAL-IDENTITY",
"unit": "OA-FLOW-L89",
"source_file": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a",
"reader": "OA-FLOW-L89.html#oa-flow.opsp.local-identity",
"start_line": 69,
"end_line": 89,
"range_sha256": "ffa7198d769f6e6eda18530189eb00dca7edb3fba3cc31d0b7c1fd8bceb3c000",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.1",
"OA-FLOW.LF.5",
"OA-FLOW.LF.6",
"OA-FLOW.BS.2"
]
},
{
"id": "OA-FLOW.OPSP.FORMULA",
"unit": "OA-FLOW-L89",
"source_file": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a",
"reader": "OA-FLOW-L89.html#oa-flow.opsp.formula",
"start_line": 90,
"end_line": 105,
"range_sha256": "b74df4430028f3b2729a5cc0eaf36e9b63281d5786f2ef781671967fb9d26bd8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OPSP.CIRCLE",
"OA-FLOW.OPSP.RECIPROCAL",
"OA-FLOW.OPSP.MEASURE-SIGN",
"OA-FLOW.OPSP.LOCAL-IDENTITY",
"OA-FLOW.ACTSPEC.APPROX"
]
},
{
"id": "OA-FLOW.ISOM.INVERTIBLE",
"unit": "OA-FLOW-L90",
"source_file": "src/OA-FLOW-L90.md",
"source_sha256": "9357fd6e25831a65f756198acdfbaa9ac81f29ddff5d998348a2bd006f04d003",
"reader": "OA-FLOW-L90.html#oa-flow.isom.invertible",
"start_line": 1,
"end_line": 29,
"range_sha256": "55016417bf7500765d9514a2d612ced0d9450adc85e9f7e594afe76b3ddd7e79",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.2"
]
},
{
"id": "OA-FLOW.ISOM.BIDUAL",
"unit": "OA-FLOW-L90",
"source_file": "src/OA-FLOW-L90.md",
"source_sha256": "9357fd6e25831a65f756198acdfbaa9ac81f29ddff5d998348a2bd006f04d003",
"reader": "OA-FLOW-L90.html#oa-flow.isom.bidual",
"start_line": 30,
"end_line": 44,
"range_sha256": "7bcbebe256f93960e463fe431994bc532f0d8d9274f6954b7cb68550d77ade86",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISOM.INVERTIBLE",
"OA-FLOW.CF.1",
"OA-FLOW.BS.0",
"OA-FLOW.AF.0"
]
},
{
"id": "OA-FLOW.ISOM.RIGIDITY",
"unit": "OA-FLOW-L90",
"source_file": "src/OA-FLOW-L90.md",
"source_sha256": "9357fd6e25831a65f756198acdfbaa9ac81f29ddff5d998348a2bd006f04d003",
"reader": "OA-FLOW-L90.html#oa-flow.isom.rigidity",
"start_line": 45,
"end_line": 66,
"range_sha256": "7d318d07477680802b29864ce3dc705a17a6c83a06573727fe2ecf7d14054736",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISOM.BIDUAL",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.BS.5",
"OA-FLOW.AF.0"
]
},
{
"id": "OA-FLOW.CHAR.ALGEBRA",
"unit": "OA-FLOW-L91",
"source_file": "src/OA-FLOW-L91.md",
"source_sha256": "90ede6465bc20c581118ae53318b3ca8cbdba492b0c98e6bb537e39748489ce8",
"reader": "OA-FLOW-L91.html#oa-flow.char.algebra",
"start_line": 1,
"end_line": 23,
"range_sha256": "81f880b31218243a624e27c5ebdb708821990f78688da865934dee1e8f8330cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.BS.1"
]
},
{
"id": "OA-FLOW.CHAR.FROM-FREQUENCY",
"unit": "OA-FLOW-L91",
"source_file": "src/OA-FLOW-L91.md",
"source_sha256": "90ede6465bc20c581118ae53318b3ca8cbdba492b0c98e6bb537e39748489ce8",
"reader": "OA-FLOW-L91.html#oa-flow.char.from-frequency",
"start_line": 24,
"end_line": 41,
"range_sha256": "332360d42c69023e86bed94019c1eeda06ec305f39997daf7666a79161ca51c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CHAR.ALGEBRA",
"OA-FLOW.ACTSPEC.NORM",
"OA-FLOW.LF.1"
]
},
{
"id": "OA-FLOW.CHAR.TO-FREQUENCY",
"unit": "OA-FLOW-L91",
"source_file": "src/OA-FLOW-L91.md",
"source_sha256": "90ede6465bc20c581118ae53318b3ca8cbdba492b0c98e6bb537e39748489ce8",
"reader": "OA-FLOW-L91.html#oa-flow.char.to-frequency",
"start_line": 42,
"end_line": 60,
"range_sha256": "8232a914805620d9ef56efa6cba4c959de9e53e9bfc2b3e66407777fafc4cd45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CHAR.FROM-FREQUENCY",
"OA-FLOW.AF.2",
"OA-FLOW.LF.1"
]
},
{
"id": "OA-FLOW.CHAR.TOPOLOGY",
"unit": "OA-FLOW-L91",
"source_file": "src/OA-FLOW-L91.md",
"source_sha256": "90ede6465bc20c581118ae53318b3ca8cbdba492b0c98e6bb537e39748489ce8",
"reader": "OA-FLOW-L91.html#oa-flow.char.topology",
"start_line": 61,
"end_line": 80,
"range_sha256": "fa2ac25228be85b845b557db7a2ccbce6f587da90a90e44e8be7b64cbddba7ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CHAR.FROM-FREQUENCY",
"OA-FLOW.CHAR.TO-FREQUENCY",
"OA-FLOW.AF.2",
"OA-FLOW.LF.1"
]
},
{
"id": "OA-FLOW.NORMCONT.FORWARD",
"unit": "OA-FLOW-L92",
"source_file": "src/OA-FLOW-L92.md",
"source_sha256": "2334a9b3a2faf564c3a972d16a576d97a067bf2b764379c4baef9e43974e4940",
"reader": "OA-FLOW-L92.html#oa-flow.normcont.forward",
"start_line": 1,
"end_line": 30,
"range_sha256": "9e0b54ae1887216856ac2fdf769b5fc2fa855a02f433e1c35820ef129591ffe0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CHAR.TOPOLOGY",
"OA-FLOW.CF.4",
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.NORMCONT.CUTOFF",
"unit": "OA-FLOW-L92",
"source_file": "src/OA-FLOW-L92.md",
"source_sha256": "2334a9b3a2faf564c3a972d16a576d97a067bf2b764379c4baef9e43974e4940",
"reader": "OA-FLOW-L92.html#oa-flow.normcont.cutoff",
"start_line": 31,
"end_line": 42,
"range_sha256": "32c7b9090bb09422795a26dd60cece6c70bf9993ad8b3553742fb12345d11d94",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.4",
"OA-FLOW.OPSP.LOCAL-IDENTITY",
"OA-FLOW.AF.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.3"
]
},
{
"id": "OA-FLOW.NORMCONT.REVERSE",
"unit": "OA-FLOW-L92",
"source_file": "src/OA-FLOW-L92.md",
"source_sha256": "2334a9b3a2faf564c3a972d16a576d97a067bf2b764379c4baef9e43974e4940",
"reader": "OA-FLOW-L92.html#oa-flow.normcont.reverse",
"start_line": 43,
"end_line": 75,
"range_sha256": "6c50a3d302bcc4ec5ee59476f17d2937cf0202dabf85f00d83177f1854508ed1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NORMCONT.FORWARD",
"OA-FLOW.NORMCONT.CUTOFF",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.BS.1"
]
},
{
"id": "OA-FLOW.BGROUP.EXPONENTIAL",
"unit": "OA-FLOW-L93",
"source_file": "src/OA-FLOW-L93.md",
"source_sha256": "6e4804e32e126460af0b6b9bd31090881b596fe3627bbf830ee77656dcd6691a",
"reader": "OA-FLOW-L93.html#oa-flow.bgroup.exponential",
"start_line": 1,
"end_line": 30,
"range_sha256": "fa28465d747d6a3e3d6fd49af5a1826eaaf5b0e0e3265a2bab4c667f67787dac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.BGROUP.AVERAGING",
"unit": "OA-FLOW-L93",
"source_file": "src/OA-FLOW-L93.md",
"source_sha256": "6e4804e32e126460af0b6b9bd31090881b596fe3627bbf830ee77656dcd6691a",
"reader": "OA-FLOW-L93.html#oa-flow.bgroup.averaging",
"start_line": 31,
"end_line": 60,
"range_sha256": "7dbd8c3f95be28cdbbef88433bcce1205872e98d9e320b460cf7a1e723b5c28a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BGROUP.EXPONENTIAL",
"OA-FLOW.CF.1",
"OA-FLOW.AF.4"
]
},
{
"id": "OA-FLOW.BGROUP.LOGARITHM",
"unit": "OA-FLOW-L93",
"source_file": "src/OA-FLOW-L93.md",
"source_sha256": "6e4804e32e126460af0b6b9bd31090881b596fe3627bbf830ee77656dcd6691a",
"reader": "OA-FLOW-L93.html#oa-flow.bgroup.logarithm",
"start_line": 61,
"end_line": 83,
"range_sha256": "9e0da18ae46b6378639d385c2452ba443307311fb41b5be23ad09b347e56143d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BGROUP.EXPONENTIAL",
"OA-FLOW.BGROUP.AVERAGING",
"OA-FLOW.AF.4",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.BGROUP.IMAGINARY",
"unit": "OA-FLOW-L93",
"source_file": "src/OA-FLOW-L93.md",
"source_sha256": "6e4804e32e126460af0b6b9bd31090881b596fe3627bbf830ee77656dcd6691a",
"reader": "OA-FLOW-L93.html#oa-flow.bgroup.imaginary",
"start_line": 84,
"end_line": 103,
"range_sha256": "c2321abc9982005c55560ddddaddd39cac66a57cbc8fe906cfcbb7a246c7b17a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BGROUP.AVERAGING",
"OA-FLOW.AF.4",
"OA-FLOW.CF.2"
]
},
{
"id": "OA-FLOW.RECIP.MODULATION",
"unit": "OA-FLOW-L97",
"source_file": "src/OA-FLOW-L97.md",
"source_sha256": "05e07b1120b5bf0b61ce848aa216bc6b6366cb3f29ecce35c4c20ced8e3a4e15",
"reader": "OA-FLOW-L97.html#oa-flow.recip.modulation",
"start_line": 1,
"end_line": 33,
"range_sha256": "0524324f53a3d2438c7cba279c0377abb078046a622a78a74c5dd9413e03d5dd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AF.0",
"OA-FLOW.RF.1",
"OA-FLOW.SC.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.BS.1"
],
"proof_available_after_line": 33,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.RECIP.TAIL",
"unit": "OA-FLOW-L97",
"source_file": "src/OA-FLOW-L97.md",
"source_sha256": "05e07b1120b5bf0b61ce848aa216bc6b6366cb3f29ecce35c4c20ced8e3a4e15",
"reader": "OA-FLOW-L97.html#oa-flow.recip.tail",
"start_line": 34,
"end_line": 81,
"range_sha256": "4e52640f840554fb72983807ecd4d2cfa80426beb67cb8d3587ffd3ea031c093",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RECIP.MODULATION",
"OA-FLOW.RF.1",
"OA-FLOW.CF.1"
],
"proof_available_after_line": 81,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.RECIP.DOMAIN",
"unit": "OA-FLOW-L97",
"source_file": "src/OA-FLOW-L97.md",
"source_sha256": "05e07b1120b5bf0b61ce848aa216bc6b6366cb3f29ecce35c4c20ced8e3a4e15",
"reader": "OA-FLOW-L97.html#oa-flow.recip.domain",
"start_line": 82,
"end_line": 112,
"range_sha256": "60c6ce6e433dd9c5529d62c70edae83dafb9d67bdae36c2037a766293915d908",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RECIP.MODULATION",
"OA-FLOW.RECIP.TAIL",
"OA-FLOW.GEN.NORM",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.RF.3"
],
"proof_available_after_line": 112,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.QSPEC.QUOTIENT",
"unit": "OA-FLOW-L98",
"source_file": "src/OA-FLOW-L98.md",
"source_sha256": "a9323d88f72e4eea49b84a19eaed84bb4dab5af9e7b9bf90a99ea9bf1e00d24c",
"reader": "OA-FLOW-L98.html#oa-flow.qspec.quotient",
"start_line": 1,
"end_line": 40,
"range_sha256": "7034cc3cfcf57add5d07246bc60fa02ded5089f20eeab72d10c7f7d6f8749c04",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RECIP.MODULATION",
"OA-FLOW.AF.2",
"OA-FLOW.RF.1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CF.1"
],
"proof_available_after_line": 40,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.QSPEC.GENERATOR",
"unit": "OA-FLOW-L98",
"source_file": "src/OA-FLOW-L98.md",
"source_sha256": "a9323d88f72e4eea49b84a19eaed84bb4dab5af9e7b9bf90a99ea9bf1e00d24c",
"reader": "OA-FLOW-L98.html#oa-flow.qspec.generator",
"start_line": 41,
"end_line": 70,
"range_sha256": "320fdade3a18b6ddab862833918352f3e82c042cf18ea5b0556dd8a7fb266f96",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QSPEC.QUOTIENT",
"OA-FLOW.RECIP.DOMAIN",
"OA-FLOW.RES.NORM"
],
"proof_available_after_line": 70,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.QSPEC.RESOLVENT",
"unit": "OA-FLOW-L98",
"source_file": "src/OA-FLOW-L98.md",
"source_sha256": "a9323d88f72e4eea49b84a19eaed84bb4dab5af9e7b9bf90a99ea9bf1e00d24c",
"reader": "OA-FLOW-L98.html#oa-flow.qspec.resolvent",
"start_line": 71,
"end_line": 103,
"range_sha256": "c0f5501b6d89859ec5ce64320ea9736900701de5a93844088036c79fa8f0e31f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QSPEC.GENERATOR",
"OA-FLOW.RECIP.TAIL",
"OA-FLOW.RECIP.DOMAIN",
"OA-FLOW.RES.NORM"
],
"proof_available_after_line": 103,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GENSPEC.INTERTWINING",
"unit": "OA-FLOW-L99",
"source_file": "src/OA-FLOW-L99.md",
"source_sha256": "5288902aad3740e08ff40a074e9001113773a65bc7af66148d94597eabf99250",
"reader": "OA-FLOW-L99.html#oa-flow.genspec.intertwining",
"start_line": 1,
"end_line": 26,
"range_sha256": "296cc703a16e4f1161c763f9befd8652fe611c31f4741225c0b61651c87d5597",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.AF.0",
"OA-FLOW.GEN.CLOSED",
"OA-FLOW.GEN.NORM",
"OA-FLOW.RECIP.DOMAIN"
],
"proof_available_after_line": 26,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GENSPEC.RECIPROCAL",
"unit": "OA-FLOW-L99",
"source_file": "src/OA-FLOW-L99.md",
"source_sha256": "5288902aad3740e08ff40a074e9001113773a65bc7af66148d94597eabf99250",
"reader": "OA-FLOW-L99.html#oa-flow.genspec.reciprocal",
"start_line": 27,
"end_line": 67,
"range_sha256": "f7168d72602eed61e1b33083f2c52b1855743acaff663f3bfafde153123a5394",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GENSPEC.INTERTWINING",
"OA-FLOW.LF.6",
"OA-FLOW.RF.3",
"OA-FLOW.BS.2",
"OA-FLOW.RECIP.TAIL"
],
"proof_available_after_line": 67,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GENSPEC.BUMP",
"unit": "OA-FLOW-L99",
"source_file": "src/OA-FLOW-L99.md",
"source_sha256": "5288902aad3740e08ff40a074e9001113773a65bc7af66148d94597eabf99250",
"reader": "OA-FLOW-L99.html#oa-flow.genspec.bump",
"start_line": 68,
"end_line": 96,
"range_sha256": "6ef854b31fc69560bd82304677a3b8208062a23220034f0f3d678c7c71e7ebdc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GENSPEC.INTERTWINING",
"OA-FLOW.BS.3",
"OA-FLOW.RF.1",
"OA-FLOW.RECIP.TAIL"
],
"proof_available_after_line": 96,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.GENSPEC.EQUALITY",
"unit": "OA-FLOW-L99",
"source_file": "src/OA-FLOW-L99.md",
"source_sha256": "5288902aad3740e08ff40a074e9001113773a65bc7af66148d94597eabf99250",
"reader": "OA-FLOW-L99.html#oa-flow.genspec.equality",
"start_line": 97,
"end_line": 112,
"range_sha256": "9e89b3ea64973fa850b79b7513e80aed123e94ea96bc59787f294f0a662ac316",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GENSPEC.RECIPROCAL",
"OA-FLOW.GENSPEC.BUMP",
"OA-FLOW.RES.FORWARD",
"OA-FLOW.RES.NORM",
"OA-FLOW.GEN.NORM-MODEL"
],
"proof_available_after_line": 112,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.STDWT.CONTRACT",
"unit": "OA-FLOW-L100",
"source_file": "src/OA-FLOW-L100.md",
"source_sha256": "d9968e792d8d0af011cec1fc1905e6ad6a7e5d41c385abbb9940cf6c05bc0e9c",
"reader": "OA-FLOW-L100.html#oa-flow.stdwt.contract",
"start_line": 1,
"end_line": 52,
"range_sha256": "585c0eeedb31b88a365874ef54f71c22a6980661e5b10e9d748d825c1d2b4e12",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.EW.3",
"OA-FLOW.CV.3",
"OA-FLOW.CF.1",
"OA-FLOW.WR.4",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4",
"OA-FLOW.NR.1",
"OA-FLOW.SF.SF1"
],
"proof_available_after_line": 52,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.STDWT.FILTER",
"unit": "OA-FLOW-L100",
"source_file": "src/OA-FLOW-L100.md",
"source_sha256": "d9968e792d8d0af011cec1fc1905e6ad6a7e5d41c385abbb9940cf6c05bc0e9c",
"reader": "OA-FLOW-L100.html#oa-flow.stdwt.filter",
"start_line": 53,
"end_line": 97,
"range_sha256": "aad57c60631d7a02743578e87dba5c07e9b2fc0135399c16f8567ede7cc87f15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.STDWT.CONTRACT",
"OA-FLOW.AF.0",
"OA-FLOW.AT.3",
"OA-FLOW.BS.1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.EW.3"
],
"proof_available_after_line": 97,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.STDWT.IDEALS",
"unit": "OA-FLOW-L100",
"source_file": "src/OA-FLOW-L100.md",
"source_sha256": "d9968e792d8d0af011cec1fc1905e6ad6a7e5d41c385abbb9940cf6c05bc0e9c",
"reader": "OA-FLOW-L100.html#oa-flow.stdwt.ideals",
"start_line": 98,
"end_line": 119,
"range_sha256": "ec568fac3aa1dd012009e70502b2cc60302e20fe00cd3f651643babf20f92c91",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.STDWT.FILTER",
"OA-FLOW.BS.3",
"OA-FLOW.RF.5",
"OA-FLOW.RF.1"
],
"proof_available_after_line": 119,
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.ORBIT.COUNTEREXAMPLE",
"unit": "OA-FLOW-L101",
"source_file": "src/OA-FLOW-L101.md",
"source_sha256": "8ddb869f398ddc5f9d61f99c8f070fda24ad6a4b26f778c7f2f2ca4028583cb7",
"reader": "OA-FLOW-L101.html#oa-flow.orbit.counterexample",
"start_line": 1,
"end_line": 51,
"range_sha256": "1fa752dd813fc47cecb844e809e3a04c4b7d94f905c9fd43bcff922496e3879b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 51
},
{
"id": "OA-FLOW.ORBIT.RELATIVE",
"unit": "OA-FLOW-L101",
"source_file": "src/OA-FLOW-L101.md",
"source_sha256": "8ddb869f398ddc5f9d61f99c8f070fda24ad6a4b26f778c7f2f2ca4028583cb7",
"reader": "OA-FLOW-L101.html#oa-flow.orbit.relative",
"start_line": 52,
"end_line": 90,
"range_sha256": "61bcb823788919d4e05a4c375499ae65cd84594db47004d49c146778110b4ecc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBIT.COUNTEREXAMPLE",
"OA-FLOW.ST.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 90
},
{
"id": "OA-FLOW.ORBIT.UNIFORM",
"unit": "OA-FLOW-L101",
"source_file": "src/OA-FLOW-L101.md",
"source_sha256": "8ddb869f398ddc5f9d61f99c8f070fda24ad6a4b26f778c7f2f2ca4028583cb7",
"reader": "OA-FLOW-L101.html#oa-flow.orbit.uniform",
"start_line": 91,
"end_line": 124,
"range_sha256": "147725d29d43866e65bd0453cb7e72e53fd3a54cb6c62692b54e28f8724a9ef7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBIT.RELATIVE",
"OA-FLOW.NORMCONT.REVERSE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 124
},
{
"id": "OA-FLOW.CIRCLE.TRANSLATION",
"unit": "OA-FLOW-L102",
"source_file": "src/OA-FLOW-L102.md",
"source_sha256": "e425c70b13b0b2cbb732b6a7e0e25ec129e409dcd8e2a47b27c7705bef8ada89",
"reader": "OA-FLOW-L102.html#oa-flow.circle.translation",
"start_line": 1,
"end_line": 40,
"range_sha256": "edc80718b8e184425b6b9ef0d3f618e7b513f02e9420f9572641472762f03e88",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.CC.0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 40
},
{
"id": "OA-FLOW.CIRCLE.PROJECTIONS",
"unit": "OA-FLOW-L102",
"source_file": "src/OA-FLOW-L102.md",
"source_sha256": "e425c70b13b0b2cbb732b6a7e0e25ec129e409dcd8e2a47b27c7705bef8ada89",
"reader": "OA-FLOW-L102.html#oa-flow.circle.projections",
"start_line": 41,
"end_line": 79,
"range_sha256": "73b3122c4a572f0b0115a9f6c355447b583e666d88a76284406d5ba040a2fe21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.AT.5",
"OA-FLOW.CC.0",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 79
},
{
"id": "OA-FLOW.CIRCLE.MULTIPLIERS",
"unit": "OA-FLOW-L102",
"source_file": "src/OA-FLOW-L102.md",
"source_sha256": "e425c70b13b0b2cbb732b6a7e0e25ec129e409dcd8e2a47b27c7705bef8ada89",
"reader": "OA-FLOW-L102.html#oa-flow.circle.multipliers",
"start_line": 80,
"end_line": 95,
"range_sha256": "27f62b4ddc6efb1cd22ead484ff2d070f7ee98a288153c814000a79ba67c0f5e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.CIRCLE.PROJECTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 95
},
{
"id": "OA-FLOW.CIRCLE.SIGN",
"unit": "OA-FLOW-L102",
"source_file": "src/OA-FLOW-L102.md",
"source_sha256": "e425c70b13b0b2cbb732b6a7e0e25ec129e409dcd8e2a47b27c7705bef8ada89",
"reader": "OA-FLOW-L102.html#oa-flow.circle.sign",
"start_line": 96,
"end_line": 107,
"range_sha256": "467a635a271c16c45b83f35f5aeff7a3a9a454a700fbc75368ec6317b3bf10b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.CIRCLE.PROJECTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 107
},
{
"id": "OA-FLOW.ERGCIRCLE.FEJER",
"unit": "OA-FLOW-L103",
"source_file": "src/OA-FLOW-L103.md",
"source_sha256": "d369a8bb7d3e0a271c2d6e1ca479e86ae42633612b759900fe861708a03f0eb7",
"reader": "OA-FLOW-L103.html#oa-flow.ergcircle.fejer",
"start_line": 1,
"end_line": 33,
"range_sha256": "ba3d4084e5fd6b46e40db7959f3a85d918d94f12508f2ae1a3cbe483344ea9ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.CC.0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 33
},
{
"id": "OA-FLOW.ERGCIRCLE.DEGREES",
"unit": "OA-FLOW-L103",
"source_file": "src/OA-FLOW-L103.md",
"source_sha256": "d369a8bb7d3e0a271c2d6e1ca479e86ae42633612b759900fe861708a03f0eb7",
"reader": "OA-FLOW-L103.html#oa-flow.ergcircle.degrees",
"start_line": 34,
"end_line": 64,
"range_sha256": "7749e836861ef7e7148630797299e63c0102a74798fef91e51280dc6550dd546",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ERGCIRCLE.FEJER",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 64
},
{
"id": "OA-FLOW.ERGCIRCLE.MODEL",
"unit": "OA-FLOW-L103",
"source_file": "src/OA-FLOW-L103.md",
"source_sha256": "d369a8bb7d3e0a271c2d6e1ca479e86ae42633612b759900fe861708a03f0eb7",
"reader": "OA-FLOW-L103.html#oa-flow.ergcircle.model",
"start_line": 65,
"end_line": 121,
"range_sha256": "8a78a78107d8da5e0d73d05974d2f2e3c21b3bd96db71ccaa14e83c6a373de24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ERGCIRCLE.FEJER",
"OA-FLOW.ERGCIRCLE.DEGREES",
"OA-FLOW.CF.8",
"OA-FLOW.AT.5",
"OA-FLOW.GW.3",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.CC.0",
"OA-FLOW.CIRCLE.MULTIPLIERS",
"OA-FLOW.CIRCLE.TRANSLATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 121
},
{
"id": "OA-FLOW.ERGCIRCLE.NONFAITHFUL",
"unit": "OA-FLOW-L103",
"source_file": "src/OA-FLOW-L103.md",
"source_sha256": "d369a8bb7d3e0a271c2d6e1ca479e86ae42633612b759900fe861708a03f0eb7",
"reader": "OA-FLOW-L103.html#oa-flow.ergcircle.nonfaithful",
"start_line": 122,
"end_line": 143,
"range_sha256": "3569acb030e9e41cc9bcaf95b093ae82b20157fba92f8bba4a0058fe5c67ab12",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ERGCIRCLE.DEGREES",
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.CC.0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 143
},
{
"id": "OA-FLOW.COMPACT.TRANSLATION",
"unit": "OA-FLOW-L104",
"source_file": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3",
"reader": "OA-FLOW-L104.html#oa-flow.compact.translation",
"start_line": 1,
"end_line": 41,
"range_sha256": "e837a66306973c892d31ba7c60e3581f41968582128e42a22716dcff79ecc1f9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.AT.5",
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.SC.6",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 41
},
{
"id": "OA-FLOW.COMPACT.COCYCLE",
"unit": "OA-FLOW-L104",
"source_file": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3",
"reader": "OA-FLOW-L104.html#oa-flow.compact.cocycle",
"start_line": 42,
"end_line": 63,
"range_sha256": "a07c155eea42ce3d89ea92eac78bf75e91ea0a77708f59e13945e45354f1adb3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ERGCIRCLE.DEGREES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 63
},
{
"id": "OA-FLOW.TORUS.REGULAR",
"unit": "OA-FLOW-L104",
"source_file": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3",
"reader": "OA-FLOW-L104.html#oa-flow.torus.regular",
"start_line": 64,
"end_line": 107,
"range_sha256": "67863b19d9813cb2470face72c0821ebf1ec72155ce229158e4687a7c5eeea24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACT.COCYCLE",
"OA-FLOW.SC.7",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.GW.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 107
},
{
"id": "OA-FLOW.TORUS.GAUGE",
"unit": "OA-FLOW-L104",
"source_file": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3",
"reader": "OA-FLOW-L104.html#oa-flow.torus.gauge",
"start_line": 108,
"end_line": 144,
"range_sha256": "6f5e0cee6b74b6bff14d9c3120f29885a8c043c4ae415804b3afed476c52a75d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TORUS.REGULAR",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 144
},
{
"id": "OA-FLOW.TORUS.UNTWISTED",
"unit": "OA-FLOW-L104",
"source_file": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3",
"reader": "OA-FLOW-L104.html#oa-flow.torus.untwisted",
"start_line": 145,
"end_line": 167,
"range_sha256": "8184f859f1490a97923a1a2a841a071e6c9d7868ea9a65809141226c3acf0f40",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TORUS.REGULAR",
"OA-FLOW.TORUS.GAUGE",
"OA-FLOW.CC.0",
"OA-FLOW.CIRCLE.TRANSLATION",
"OA-FLOW.CIRCLE.MULTIPLIERS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 167
},
{
"id": "OA-FLOW.IS.0",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-0",
"start_line": 1,
"end_line": 66,
"range_sha256": "da94a8fe14c65cc169a5504b67834d71f9f7cba8869ab6808aeb45c77687a816",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.1",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-1",
"start_line": 67,
"end_line": 153,
"range_sha256": "77500bc59a18b3b141dcc914e6ad0b60351d03a36029dd1507dce1d8a54a8946",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.QF.1"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.2",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-2",
"start_line": 154,
"end_line": 236,
"range_sha256": "126f00b42eb16f5be6d8059e46aae038a1707cfe6cc128eb85f3d3328db221b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.1",
"OA-FLOW.QM.SETTING",
"OA-FLOW.QM.CUTOFF",
"OA-FLOW.QM.RHO",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.QF.3",
"OA-FLOW.QF.6",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.3",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-3",
"start_line": 237,
"end_line": 265,
"range_sha256": "eb08521eb52b87237e5e2b60d147e65f9a726b3762a79f81b11b13d249d2c9f3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.2",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.GRP.TRANSLATIONS"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.4",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-4",
"start_line": 266,
"end_line": 447,
"range_sha256": "d381f32cff5664723fad7558b51afca8a9e9b3f4115696b48a29670c9b0ac4bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.QF.6",
"OA-FLOW.BD.1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CF.1",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.NCF.1"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.5",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-5",
"start_line": 448,
"end_line": 483,
"range_sha256": "cdbf2cd7683c76a17b79094f64d8ab7d52581e9267bc364d00118fff6b51519e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.3",
"OA-FLOW.IS.4",
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.3",
"OA-FLOW.NCF.4",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.HR.TOPOLOGY"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.6",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-6",
"start_line": 484,
"end_line": 531,
"range_sha256": "eebe7d4f7214c20234616997d87abd27bffa92284a621d7adb20cc735449f92f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.2",
"OA-FLOW.IS.5",
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.4"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.7",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-7",
"start_line": 532,
"end_line": 596,
"range_sha256": "eb25ed391b192fd2e9ae41ce07d8eb3db53e717d60438b6bb24f3aab23406429",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.2",
"OA-FLOW.IS.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.IS.6",
"OA-FLOW.QM.RHO",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.SC.8"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.8",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-8",
"start_line": 597,
"end_line": 614,
"range_sha256": "29175b110019b8807d87f74e29cf40010fe52662400a9c9fa6b629808da6ee77",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IS.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.IS.6",
"OA-FLOW.IS.7"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.IS.FIGURE",
"unit": "OA-FLOW-IS",
"source_file": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37",
"reader": "OA-FLOW-IS.html#is-figure",
"start_line": 615,
"end_line": 701,
"range_sha256": "c8a9cc91804aab18efdac36a1b871956ca684ddecddd0e1971a24580f306a4b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IS.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.IS.6",
"OA-FLOW.IS.7",
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.NCF.4"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IS.md",
"line": 7,
"through_line": 14,
"description": "Locally compact Hausdorff second-countable G; closed H; arbitrary von Neumann N with point-ultraweak continuous action; fixed left Haar convention and nonzero case."
}
]
},
{
"id": "OA-FLOW.MX.0",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-0",
"start_line": 1,
"end_line": 65,
"range_sha256": "0c6eb24bae1490f3035387b86aa99c0fbf17654c8e252c9ca08be5c9d1955670",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.BS.0"
],
"range_bytes": 6996
},
{
"id": "OA-FLOW.MX.1",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-1",
"start_line": 66,
"end_line": 97,
"range_sha256": "503d636c0a52c221097defc1dcb3992ce50a18b2eaf56047fce124aa99411212",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.0",
"OA-FLOW.BS.0"
],
"range_bytes": 2121
},
{
"id": "OA-FLOW.MX.2",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-2",
"start_line": 98,
"end_line": 243,
"range_sha256": "20b6e72f1924834facb08d7c132a9fccd5aae4f1408ebfcfd6365fd3b40746fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.0",
"OA-FLOW.MX.1",
"OA-FLOW.BS.1",
"OA-FLOW.BS.6",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.5"
],
"range_bytes": 7854
},
{
"id": "OA-FLOW.MX.3",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-3",
"start_line": 244,
"end_line": 297,
"range_sha256": "091d70868f2de3fede6ff39b8a9331d0d6bef6a0756d0f49835459ac7d9f81a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.2",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4",
"OA-FLOW.LF.5",
"OA-FLOW.LF.6"
],
"range_bytes": 3205
},
{
"id": "OA-FLOW.MX.4",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-4",
"start_line": 298,
"end_line": 332,
"range_sha256": "c95ee48ab8c01d0f62c3f469a5f616dd50aa5a8e22a5a5559e3048095483c2cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.LF.0"
],
"range_bytes": 2681
},
{
"id": "OA-FLOW.MX.COMPACT",
"unit": "OA-FLOW-MX",
"source_file": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43",
"reader": "OA-FLOW-MX.html#mx-compact-hulls",
"start_line": 334,
"end_line": 520,
"range_sha256": "d2c1704ee1f88b69cd22cd57a764d798ebe50a562bb544823a75f9833d12da27",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.0",
"OA-FLOW.GBAC.CONVEXHULL",
"OA-FND-HB-04",
"OA-FND-WT-03"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 520,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MX.md",
"line": 334,
"through_line": 520,
"description": "Arbitrary specified complex Banach duals X and Y. The normal operator space carries the relative weak-star topology from the full projective-tensor dual. K is compact in that topology. Probability barycentres require K nonempty and a Radon probability measure. No separability, group action, norm-continuous operator orbit or ambient weak-star closedness of all normal maps is assumed."
}
],
"range_bytes": 8374
},
{
"id": "OA-FLOW.SUM.CLAIM",
"unit": "OA-FLOW-L80",
"source_file": "src/OA-FLOW-L80.md",
"source_sha256": "67432774872bb8be0eee44deea134d3659c6733ef341c49100924c3103926fc4",
"reader": "OA-FLOW-L80.html#sum-claim",
"start_line": 1,
"end_line": 23,
"range_sha256": "bc2a5a80064a13058ad77b5cf033040689ea7011bf7a9b34981f872a2c852b2b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.0",
"OA-FLOW.MX.1",
"OA-FLOW.MX.2",
"OA-FLOW.MX.3",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4"
],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.SUM.KERNEL",
"unit": "OA-FLOW-L80",
"source_file": "src/OA-FLOW-L80.md",
"source_sha256": "67432774872bb8be0eee44deea134d3659c6733ef341c49100924c3103926fc4",
"reader": "OA-FLOW-L80.html#sum-kernel",
"start_line": 24,
"end_line": 77,
"range_sha256": "72f447ac9251c7c69fd10b455827413fb27a2ef6256459c8999c27a37f62c708",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.SUM.LOCAL",
"unit": "OA-FLOW-L80",
"source_file": "src/OA-FLOW-L80.md",
"source_sha256": "67432774872bb8be0eee44deea134d3659c6733ef341c49100924c3103926fc4",
"reader": "OA-FLOW-L80.html#sum-local",
"start_line": 78,
"end_line": 120,
"range_sha256": "0d3e9a593349fea6c0a1c3c961cb269001371aae278f0adcb59c37cecb08ecc1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUM.KERNEL",
"OA-FLOW.MX.3",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4",
"OA-FLOW.XGAPS.CSTAR.H0"
]
},
{
"id": "OA-FLOW.TRANSFER.CONE",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-claim",
"start_line": 1,
"end_line": 39,
"range_sha256": "ba6e5188d981be63894adfeba96476ea5e3c7a7bfab063e4258780a1a71eca5a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.0",
"OA-FLOW.MX.3"
],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.TRANSFER.FORWARD",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-forward",
"start_line": 40,
"end_line": 51,
"range_sha256": "491de94bbd0704874e74ccbcc73b1c03e082aa4fcc5e776147a77b56c43d006b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUM.LOCAL"
]
},
{
"id": "OA-FLOW.TRANSFER.KERNEL",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-kernel",
"start_line": 52,
"end_line": 118,
"range_sha256": "9d1272b59a9c9a718edf92bd14009509179c71a1f28ce5001065a3f4ea54b10e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MX.2",
"OA-FLOW.MX.4",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.XGAPS.CSTAR.H1"
]
},
{
"id": "OA-FLOW.TRANSFER.REVERSE",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-reverse",
"start_line": 119,
"end_line": 154,
"range_sha256": "999b30731d1b32fe617db99eda25f66255c7e9f737e8148ddd160e2142e8ff14",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRANSFER.KERNEL",
"OA-FLOW.MX.4",
"OA-FLOW.LF.0",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4"
]
},
{
"id": "OA-FLOW.TRANSFER.APPROX",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-approximation",
"start_line": 155,
"end_line": 173,
"range_sha256": "592322bce077678d6e043cdc15af4efd4669b162ccd1799bca92ef132bcf3634",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LF.5",
"OA-FLOW.GRP.ALGEBRA"
]
},
{
"id": "OA-FLOW.TRANSFER.BANACH-OPERATORS",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-banach-operators",
"start_line": 175,
"end_line": 289,
"range_sha256": "4c1edda759b1c724379d929fa199ed011b5165c96283adc25c413387ab58abdd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.6",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 289,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 196,
"description": "Arbitrary complex Banach X,Y and arbitrary LCA group, uniformly bounded strongly continuous actions. Positive Fourier convention; all vector integrals are norm integrals. No predual or separability is assumed."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 289,
"description": "Complete local statement and proof. Operator closure is uniform convergence on norm-compact input sets. Filtration requires E closed, regular closed and containing zero. Compact family and probability claims state their nonempty-set assumptions. The example uses counting Haar measure and bilateral shifts."
}
]
},
{
"id": "OA-FLOW.TRANSFER.BANACH-SPECTRA",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-banach-spectra",
"start_line": 290,
"end_line": 386,
"range_sha256": "44e0892d06eeaf1dc0af9f1700f3f8081a7abf107419449194011062b8ee2009",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRANSFER.BANACH-OPERATORS",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4",
"OA-FLOW.LF.5",
"OA-FLOW.LF.6",
"OA-FLOW.SS.1",
"OA-FLOW.SS.2",
"OA-FLOW.XGAPS.CSTAR.H0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 386,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 196,
"description": "Arbitrary complex Banach X,Y and arbitrary LCA group, uniformly bounded strongly continuous actions. Positive Fourier convention; all vector integrals are norm integrals. No predual or separability is assumed."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 290,
"through_line": 386,
"description": "Complete local statement and proof. Operator closure is uniform convergence on norm-compact input sets. Filtration requires E closed, regular closed and containing zero. Compact family and probability claims state their nonempty-set assumptions. The example uses counting Haar measure and bilateral shifts."
}
]
},
{
"id": "OA-FLOW.TRANSFER.BANACH-COMPACT",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-banach-compact",
"start_line": 387,
"end_line": 451,
"range_sha256": "4ee3f1bcdbb6991fb263f1e2cfb5ff89a17417bd7f0e0b6ea38fae9fc188306f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRANSFER.BANACH-OPERATORS",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FND-HB-04"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 451,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 196,
"description": "Arbitrary complex Banach X,Y and arbitrary LCA group, uniformly bounded strongly continuous actions. Positive Fourier convention; all vector integrals are norm integrals. No predual or separability is assumed."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 387,
"through_line": 451,
"description": "Complete local statement and proof. Operator closure is uniform convergence on norm-compact input sets. Filtration requires E closed, regular closed and containing zero. Compact family and probability claims state their nonempty-set assumptions. The example uses counting Haar measure and bilateral shifts."
}
]
},
{
"id": "OA-FLOW.TRANSFER.BANACH-TRANSFER",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-banach-theorems",
"start_line": 452,
"end_line": 567,
"range_sha256": "95dc59ceb658420913cd8dedef70de8c82f54dd0342892007d4edd01f0417fad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRANSFER.BANACH-OPERATORS",
"OA-FLOW.TRANSFER.BANACH-SPECTRA",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4",
"OA-FLOW.MX.4",
"OA-FLOW.SUM.KERNEL",
"OA-FLOW.SUM.LOCAL",
"OA-FLOW.TRANSFER.CONE",
"OA-FLOW.TRANSFER.KERNEL",
"OA-FLOW.TRANSFER.REVERSE",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 567,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 196,
"description": "Arbitrary complex Banach X,Y and arbitrary LCA group, uniformly bounded strongly continuous actions. Positive Fourier convention; all vector integrals are norm integrals. No predual or separability is assumed."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 452,
"through_line": 567,
"description": "Complete local statement and proof. Operator closure is uniform convergence on norm-compact input sets. Filtration requires E closed, regular closed and containing zero. Compact family and probability claims state their nonempty-set assumptions. The example uses counting Haar measure and bilateral shifts."
}
]
},
{
"id": "OA-FLOW.TRANSFER.BANACH-TOPOLOGY",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-banach-topology-example",
"start_line": 568,
"end_line": 642,
"range_sha256": "41aea80c4701b5d5f982f40930915f9f2bb3a3e103e0eb28896e3fccf5284c25",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRANSFER.BANACH-OPERATORS",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 642,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 175,
"through_line": 196,
"description": "Arbitrary complex Banach X,Y and arbitrary LCA group, uniformly bounded strongly continuous actions. Positive Fourier convention; all vector integrals are norm integrals. No predual or separability is assumed."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 568,
"through_line": 642,
"description": "Complete local statement and proof. Operator closure is uniform convergence on norm-compact input sets. Filtration requires E closed, regular closed and containing zero. Compact family and probability claims state their nonempty-set assumptions. The example uses counting Haar measure and bilateral shifts."
}
]
},
{
"id": "OA-FLOW.TRANSFER.MULTIPLIERS",
"unit": "OA-FLOW-L81",
"source_file": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b",
"reader": "OA-FLOW-L81.html#transfer-multipliers",
"start_line": 644,
"end_line": 810,
"range_sha256": "30b88ebc0377ce2662971b83223925825626be0014981ac2f7a58a885bdc5894",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.GL.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.MX.0",
"OA-FLOW.MX.1",
"OA-FLOW.MX.2",
"OA-FLOW.MX.3",
"OA-FLOW.SUM.CLAIM",
"OA-FLOW.SUM.LOCAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 810,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L81.md",
"line": 644,
"through_line": 661,
"description": "Arbitrary von Neumann algebra with its actual predual and arbitrary LCA group; point-ultraweakly continuous normal star-automorphism action. Positive Fourier convention and full projective-tensor relative topology."
},
{
"source": "src/OA-FLOW-L81.md",
"line": 644,
"through_line": 810,
"description": "Complete normal left-multiplier proof: continuous retraction by evaluation at one, full-domain filter intertwining, identical annihilator ideals, spectral equality and all-subset product inclusion. No separability or arbitrary-net boundedness assumption."
}
]
},
{
"id": "OA-FLOW.POSFLOW.SETTING",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.setting",
"start_line": 1,
"end_line": 38,
"range_sha256": "f2f7d137352a3157b91b1522068bdd3b15385f16d73aaa32769266ab5602b864",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.MX.0",
"OA-FLOW.FS.2",
"OA-FLOW.GL.0",
"OA-FLOW.BS.1",
"OA-FLOW.BS.3",
"OA-FLOW.GL.6",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 38
},
{
"id": "OA-FLOW.POSFLOW.MODELS",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.models",
"start_line": 39,
"end_line": 59,
"range_sha256": "606b41c57d633e1c76b699d9d2fdce5deccc4a9e91b67b18f00bdca11620e412",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.SETTING",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF2",
"OA-FLOW.LF.1",
"OA-FLOW.BS.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.SUM.CLAIM",
"OA-FLOW.TRANSFER.CONE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 59
},
{
"id": "OA-FLOW.POSFLOW.NECESSITY",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.necessity",
"start_line": 60,
"end_line": 73,
"range_sha256": "b7844253291a35c6da05212454580ba728532fe6650200f347c182fbbeaef7df",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.MODELS",
"OA-FLOW.SUM.CLAIM",
"OA-FLOW.FS.2",
"OA-FLOW.SF.SF1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 73
},
{
"id": "OA-FLOW.POSFLOW.MEASURE",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.measure",
"start_line": 74,
"end_line": 148,
"range_sha256": "2e210931219539fb618d3e87524179e46d9e500a33fd6082beb5a3cd729ac293",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.SETTING",
"OA-FLOW.POSFLOW.MODELS",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.SF.SF4",
"OA-FLOW.CP.4",
"OA-FLOW.ST.2",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 148
},
{
"id": "OA-FLOW.POSFLOW.SHIFT",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.shift",
"start_line": 149,
"end_line": 168,
"range_sha256": "69f89e44ec1e1dd8307358ceb8348408db86d9dd1fbcb383279880451f697177",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.MEASURE",
"OA-FLOW.GL.6",
"OA-FLOW.FS.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 168
},
{
"id": "OA-FLOW.POSFLOW.INCLUSION",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.inclusion",
"start_line": 169,
"end_line": 191,
"range_sha256": "bc2870b90ad118824aa66287f34719f396c38a79e68dd39b81288e3fa190711c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.SHIFT",
"OA-FLOW.POSFLOW.MODELS",
"OA-FLOW.TRANSFER.REVERSE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 191
},
{
"id": "OA-FLOW.POSFLOW.RECOVERY",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.recovery",
"start_line": 192,
"end_line": 214,
"range_sha256": "511adaa3fe44fd91fb1f9417f95e768ee324283f4efef9cd23307e7d9a72f5c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.INCLUSION",
"OA-FLOW.BS.3",
"OA-FLOW.BS.7",
"OA-FLOW.LF.4",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7",
"OA-FLOW.GL.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 214
},
{
"id": "OA-FLOW.POSFLOW.CHOICE",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.choice",
"start_line": 215,
"end_line": 221,
"range_sha256": "ae9bbac6a617e446f612b6ce486e3ebcf2bab3fe03393d323d38164a014b909e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.RECOVERY",
"OA-FLOW.POSFLOW.MEASURE",
"OA-FLOW.LF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 221
},
{
"id": "OA-FLOW.POSFLOW.EXAMPLES",
"unit": "OA-FLOW-L86",
"source_file": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f",
"reader": "OA-FLOW-L86.html#positiveflow.examples",
"start_line": 222,
"end_line": 240,
"range_sha256": "f816203065b438b3ef3816bfcc78c598bd614efe6ab12fd6ae6970eb9c77af26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.RECOVERY",
"OA-FLOW.POSFLOW.MEASURE",
"OA-FLOW.SF.SF1",
"OA-FLOW.CP.4",
"OA-FLOW.LF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 240
},
{
"id": "OA-FLOW.ND.MULTIPLICATION",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-multiplication",
"start_line": 121,
"end_line": 133,
"range_sha256": "7baeaa23ced88c8e6a32d2b8c1dddd583ff1cddb5b7caca42a33eef8cce309ef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2"
]
},
{
"id": "OA-FLOW.ND.WEYL",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-weyl-proof",
"start_line": 137,
"end_line": 182,
"range_sha256": "49fb98b91f60ef441be12c62e070bd55163607bad559875e219ea10108141f9b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.ND.MULTIPLICATION"
]
},
{
"id": "OA-FLOW.ND.TENSOR",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-tensor",
"start_line": 186,
"end_line": 290,
"range_sha256": "3365f984c6b0ff447bf88ceaf0dfcdf43b374947b23e681cd4a0f406f1045e3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2"
]
},
{
"id": "OA-FLOW.ND.ORBITS",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-orbits",
"start_line": 296,
"end_line": 324,
"range_sha256": "89820da91dc01c193a3345dd12a5c904ffed25eb27130957f24dd6413cda19c3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR"
]
},
{
"id": "OA-FLOW.ND.DUALITY",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-construction",
"start_line": 328,
"end_line": 425,
"range_sha256": "46bc49da3d467857bef986c494ba0cf30e6e1c56a72e118c2f61ad8fd410bc0e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.ND.TENSOR"
]
},
{
"id": "OA-FLOW.ND.MODELS",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-models",
"start_line": 429,
"end_line": 456,
"range_sha256": "59608785fcc058e86009e3261130d635553db96ef1b1ccd77a1f1256b4e05149",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY"
]
},
{
"id": "OA-FLOW.ND.EXERCISES",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-exercises",
"start_line": 460,
"end_line": 486,
"range_sha256": "62193ef4e02ffffbdf4c3409dad2ddf1127efab787e44e0672a35ce0329c3b3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY"
]
},
{
"id": "OA-FLOW.ND.INVERSE",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-inverse",
"start_line": 489,
"end_line": 503,
"range_sha256": "1dd1ae1cec1fa3ef56971965ee3dd097599a56ba26c58ca4838b710154a003f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.TENSOR"
]
},
{
"id": "OA-FLOW.ND.FIGURE",
"unit": "OA-FLOW-ND",
"source_file": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5",
"reader": "OA-FLOW-ND.html#nd-figure",
"start_line": 511,
"end_line": 542,
"range_sha256": "16ff2e72d33494a6e2808040b507aa964e7a9072d3ba2b106e2e40e5af786cba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.MODELS",
"OA-FLOW.ND.EXERCISES",
"OA-FLOW.ND.INVERSE"
]
},
{
"id": "OA-FLOW.WRC.STATEMENT",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-statement",
"start_line": 9,
"end_line": 55,
"range_sha256": "0e1815a4042e9360ab287f84935052688e93ddf93b414f47b7178ac896d51b53",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5"
],
"may_be_used_as_earlier_proved_premise": false
},
{
"id": "OA-FLOW.WRC.SPECTRAL",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-spectral",
"start_line": 56,
"end_line": 165,
"range_sha256": "7ed14dd11e550906f41ff27cd2ffae3c97a7753223ee82732c6d1f9ee3bfde0d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.INTEGRATION",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-integration",
"start_line": 166,
"end_line": 184,
"range_sha256": "8abe39c1f93910117edbca32b4761a0bd5b11074334db26181953adb92e1f2a3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.SPECTRAL"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.COVARIANCE",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-covariance",
"start_line": 185,
"end_line": 206,
"range_sha256": "2982cae38e2d81822f9c5751665386bc342acd805ef8e46c1a000bad1a6bfc04",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.WRC.INTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.KERNELS",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-kernels",
"start_line": 207,
"end_line": 273,
"range_sha256": "a67f5cc2170b7aacebfef589fc1835f6a48447ca1192032521598794108e3c44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.INTEGRATION",
"OA-FLOW.WRC.COVARIANCE"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.NORM",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-norm",
"start_line": 274,
"end_line": 306,
"range_sha256": "dd5f494010b9d4ac0f3796b60bcca4954783d6dbd8fcc0530a6b6273b61dca7e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.KERNELS"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.NONDEGENERATE",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-nondegenerate",
"start_line": 307,
"end_line": 347,
"range_sha256": "833df5b17417dbbda42fefea645ab46e0c1fccd7e0b0a26bc7f77c1f6cdd51ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.WRC.INTEGRATION",
"OA-FLOW.WRC.COVARIANCE",
"OA-FLOW.WRC.KERNELS",
"OA-FLOW.WRC.NORM"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.UNITARY",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-unitary",
"start_line": 348,
"end_line": 395,
"range_sha256": "5ebd58ab51a1e9e8338550abe0d0f017ad966d98236ff0aac6c98065e7b1c923",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.KERNELS",
"OA-FLOW.WRC.NORM",
"OA-FLOW.WRC.NONDEGENERATE"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.MODELS",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-models",
"start_line": 396,
"end_line": 427,
"range_sha256": "8daf4068fb9bfa39fe2e0547b2f8b6f61e90150ef98bf3705eb271d06f65ebaf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.UNITARY"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.EXERCISES",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-exercises",
"start_line": 428,
"end_line": 461,
"range_sha256": "f78655d8ccaee0139c884c32348c1038c6c0697b1985b18c75dcf018f6f31720",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.KERNELS",
"OA-FLOW.WRC.UNITARY",
"OA-FLOW.WRC.MODELS"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.ALTERNATIVES",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-alternatives",
"start_line": 472,
"end_line": 502,
"range_sha256": "eaf83f319422885b98ede9b9df7a4c2e1d507074553753acc20e5bfff8b00eb6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.GRP.INTERTWINERS",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.SP.4",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.WY.1",
"OA-FLOW.WY.2",
"OA-FLOW.WY.3",
"OA-FLOW.WY.4",
"OA-FLOW.WY.5",
"OA-FLOW.WRC.UNITARY"
],
"may_be_used_as_earlier_proved_premise": true
},
{
"id": "OA-FLOW.WRC.FIGURE",
"unit": "OA-FLOW-WRC",
"source_file": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7",
"reader": "OA-FLOW-WRC.html#wr-figure",
"start_line": 506,
"end_line": 539,
"range_sha256": "729235ca6875817300a611d85b4c294fb471c37e449fc15ad1c9f3c82883bacc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WRC.UNITARY",
"OA-FLOW.WRC.MODELS",
"OA-FLOW.WRC.EXERCISES",
"OA-FLOW.WRC.ALTERNATIVES"
]
},
{
"id": "OA-FLOW.AM.1",
"unit": "OA-FLOW-AM",
"source_file": "src/OA-FLOW-AM.md",
"source_sha256": "90e1330c4d8bc3145a29af68d0f5d0b267adb866340aa432510553beda1219e1",
"reader": "OA-FLOW-AM.html#am-1",
"start_line": 1,
"end_line": 88,
"range_sha256": "b2dded452d564692891a46094b6b8a49d80ce5774bee7e2d4b8e87a13e4b1dee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.SP.3",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.GDA.1",
"OA-FLOW.GDA.3",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.CP.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 88
},
{
"id": "OA-FLOW.AM.2",
"unit": "OA-FLOW-AM",
"source_file": "src/OA-FLOW-AM.md",
"source_sha256": "90e1330c4d8bc3145a29af68d0f5d0b267adb866340aa432510553beda1219e1",
"reader": "OA-FLOW-AM.html#am-2",
"start_line": 92,
"end_line": 113,
"range_sha256": "311def4ea10b482e7df3a5f7383c97dc780a01b1da043084daa9c3e58493d8e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AM.1",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.AT.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.BD.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 113
},
{
"id": "OA-FLOW.AM.3",
"unit": "OA-FLOW-AM",
"source_file": "src/OA-FLOW-AM.md",
"source_sha256": "90e1330c4d8bc3145a29af68d0f5d0b267adb866340aa432510553beda1219e1",
"reader": "OA-FLOW-AM.html#am-3",
"start_line": 117,
"end_line": 147,
"range_sha256": "62ded03e252d50392bb57af534027a22c2c3d8cdf925c86342b69e2450930f67",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AM.1",
"OA-FLOW.AM.2",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 147
},
{
"id": "OA-FLOW.DA.ACTION",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-action",
"start_line": 87,
"end_line": 108,
"range_sha256": "f8f81e950b65c3e21c84c43cdcb5cdc26070fe3d3339187cf72d1d0de2a7449b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.AT.4",
"OA-FLOW.CCM.4",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 108
},
{
"id": "OA-FLOW.DA.FIXED",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-fixed",
"start_line": 112,
"end_line": 152,
"range_sha256": "54b5138b365af454b45482bf1982b7e23bd7c3e0b893500f7698772a8f53ccf5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.ACTION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.HR.3",
"OA-FLOW.NR.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 152
},
{
"id": "OA-FLOW.DA.AVERAGE",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-compact",
"start_line": 156,
"end_line": 236,
"range_sha256": "a98500205b179782f1a6010735c8a6acf979360b779d5e1421415c5ecd7b923d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.ACTION",
"OA-FLOW.DA.FIXED",
"OA-FLOW.AT.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.FF.6",
"OA-FLOW.HR.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 236
},
{
"id": "OA-FLOW.DA.VECTOR",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-vector",
"start_line": 240,
"end_line": 254,
"range_sha256": "9d5fb5a813cc9c4cb981ae66ec7121d861c3b0c77897c668f466b010f77e1ff3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.9",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.GNS.4.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 254
},
{
"id": "OA-FLOW.DA.SQUARES",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-squares",
"start_line": 258,
"end_line": 292,
"range_sha256": "7542cb56826193f5c4df71ec4eb69e18d879353a412f4c2d6b3166a412b1ec70",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.CCM.4",
"OA-FLOW.NR.3",
"OA-FLOW.AT.4",
"OA-FLOW.GEN.BOUNDS",
"OA-FLOW.EP.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 292
},
{
"id": "OA-FLOW.DA.EQUALITY",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-equality",
"start_line": 296,
"end_line": 300,
"range_sha256": "5cca80546e0399f3c63e39d56a7e023eb5c97587eecb4dca18526c36caa87a1f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AM.3",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.EP.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 300
},
{
"id": "OA-FLOW.DA.PAIRS",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-pairs",
"start_line": 313,
"end_line": 341,
"range_sha256": "823baa8db2033bfc6deae16aa42d9c1381321aaf77fb1cd49b2bb3d6e1d77219",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.SQUARES",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.6",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.SC.6",
"OA-FLOW.EP.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 341
},
{
"id": "OA-FLOW.DA.DOMAINS",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-domains",
"start_line": 345,
"end_line": 381,
"range_sha256": "4977b22b7a642b9990da18d08f0c2e042f8b7e94c748ebbc0fd2e9da41994305",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.DA.PAIRS",
"OA-FLOW.EP.1",
"OA-FLOW.EP.3",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.GDA.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 381
},
{
"id": "OA-FLOW.DA.MODULAR",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-inner",
"start_line": 385,
"end_line": 459,
"range_sha256": "d540eb730098eec762709e038048c5bc84fd3369e9eecf5f61764ebdd815174c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.SQUARES",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.EP.6",
"OA-FLOW.OT.5",
"OA-FLOW.GDW.7",
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.7",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 459
},
{
"id": "OA-FLOW.DA.INVARIANCE",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-invariant",
"start_line": 463,
"end_line": 473,
"range_sha256": "30e47f4b565f708b0110d4986d0557d26f7bb0797274e6aa319685f9cc2ee806",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.DA.DOMAINS",
"OA-FLOW.DA.MODULAR",
"OA-FLOW.GDA.7",
"OA-FLOW.GDA.9",
"OA-FLOW.BC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 473
},
{
"id": "OA-FLOW.DA.HAAR",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-haar",
"start_line": 477,
"end_line": 483,
"range_sha256": "965c44eba0d010f848f5389d3f6f8ef35a1cfe50e40b023e7bf1fb5fc46e08bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.SP.3",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.XGAPS.CSTAR.H4",
"OA-FLOW.CF.4",
"OA-FLOW.HR.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 483
},
{
"id": "OA-FLOW.DA.MODELS",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-models",
"start_line": 487,
"end_line": 534,
"range_sha256": "62a267ad599d92537d596429331558a0d11534a9214e5912f0fb5537ae41e802",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.HAAR",
"OA-FLOW.DA.SQUARES",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FF.3",
"OA-FLOW.SP.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 534
},
{
"id": "OA-FLOW.DA.EXERCISES",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-problems",
"start_line": 538,
"end_line": 573,
"range_sha256": "6c1d4bc49be2e76f9f8404afce61c36cf634c2eb53c481d802777d8f58e4f23e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.MODELS",
"OA-FLOW.DA.MODULAR",
"OA-FLOW.DA.PAIRS",
"OA-FLOW.DA.AVERAGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 573
},
{
"id": "OA-FLOW.DA.RECOGNITION.CONDITIONAL",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-conditional",
"start_line": 575,
"end_line": 594,
"range_sha256": "bd54ac2fa6098684f3c873dccb3e27cb8b929cf1e7945472b79b5c1348d16d27",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.DA.MODULAR",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.CR.9",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.GDA.7",
"OA-FLOW.GDA.9",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 594,
"scope_boundary": "The cocycle reduction and injectivity of existing dual weights are unconditional; invariant-target surjectivity is conditional on UR."
},
{
"id": "OA-FLOW.DA.FIGURE",
"unit": "OA-FLOW-DA",
"source_file": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c",
"reader": "OA-FLOW-DA.html#da-figure",
"start_line": 606,
"end_line": 660,
"range_sha256": "9850c5bcef6636579e1e6975ee2b8a06a6d4b6041dbbde2c511e6d26bc0d2cce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AM.3",
"OA-FLOW.DA.DOMAINS",
"OA-FLOW.DA.HAAR",
"OA-FLOW.DA.MODELS"
],
"proof_available_after_line": 660
},
{
"id": "OA-FLOW.UR.0",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.0",
"start_line": 32,
"end_line": 88,
"range_sha256": "f592875f42a03925555eeb8d1ae1c99757b80f70f3dd7d5677ac90b4b88b6749",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RF.5",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF4",
"OA-FLOW.CI.3",
"OA-FLOW.FF.2",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.MW.1",
"OA-FLOW.MW.4",
"OA-FLOW.NC.3",
"OA-FLOW.NC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 88,
"range_bytes": 3839
},
{
"id": "OA-FLOW.UR.1",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.1",
"start_line": 94,
"end_line": 198,
"range_sha256": "5184f4fbd07a3de61b52850f511bec0739740bd65b99e70e1059aea0ccc9c141",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.WR.4",
"OA-FLOW.WF.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.MA.2",
"OA-FLOW.GF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 198,
"range_bytes": 5048
},
{
"id": "OA-FLOW.UR.2",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.2",
"start_line": 204,
"end_line": 304,
"range_sha256": "b685cc9bb221f71feca849f620085204db866a7c7f8799bda7496328f1d20a9e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.UR.1",
"OA-FLOW.BD.3",
"OA-FLOW.WH04.3",
"OA-FLOW.MW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 304,
"range_bytes": 4085
},
{
"id": "OA-FLOW.UR.3",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.3",
"start_line": 310,
"end_line": 357,
"range_sha256": "2751dd9c1befe841da1e497b9dd700de64a83b0a07b76f52609599db4997944f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.2",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4",
"OA-FLOW.WF.1",
"OA-FLOW.WF.2",
"OA-FLOW.WF.3",
"OA-FLOW.WF.4",
"OA-FLOW.WF.5",
"OA-FLOW.WF.6",
"OA-FLOW.MW.4",
"OA-FLOW.CI.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 357,
"range_bytes": 1886
},
{
"id": "OA-FLOW.UR.4",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.4",
"start_line": 363,
"end_line": 414,
"range_sha256": "07fb481d664b8620f11732af25dcc977686d17b4ff1f39d1ba707824bd9f7332",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.UR.1",
"OA-FLOW.UR.2",
"OA-FLOW.UR.3",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.KT.3",
"OA-FLOW.KT.4",
"OA-FLOW.GW.4",
"OA-FLOW.GW.5",
"OA-FLOW.CR.6",
"OA-FLOW.CR.8",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 414,
"range_bytes": 2468
},
{
"id": "OA-FLOW.UR.5",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.5",
"start_line": 420,
"end_line": 504,
"range_sha256": "04d51673797b089c864197134b036c3c8cfd0a32f5a948193a488dfe77ef3f48",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.UR.1",
"OA-FLOW.UR.3",
"OA-FLOW.UR.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.MC.5",
"OA-FLOW.MA.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.WF.1",
"OA-FLOW.UR.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 504,
"range_bytes": 3901
},
{
"id": "OA-FLOW.UR.6",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.6",
"start_line": 510,
"end_line": 614,
"range_sha256": "9cde11cc74f106e3a3f376708cc85cb2d82bc01bd2c2936fe655f622ab5a487e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.UR.1",
"OA-FLOW.UR.2",
"OA-FLOW.UR.3",
"OA-FLOW.UR.4",
"OA-FLOW.UR.5",
"OA-FLOW.BC.5",
"OA-FLOW.NC.4",
"OA-FLOW.MC.1",
"OA-FLOW.MC.5",
"OA-FLOW.WR.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 614,
"range_bytes": 6904
},
{
"id": "OA-FLOW.UR.FIGURE",
"unit": "OA-FLOW-UR",
"source_file": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402",
"reader": "OA-FLOW-UR.html#oa-flow.ur.figure",
"start_line": 618,
"end_line": 671,
"range_sha256": "706b6a5517fb5a41a0bc075e55dd7cb3ad7b2531c02bb36618d15c205738c790",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.0",
"OA-FLOW.UR.1",
"OA-FLOW.UR.2",
"OA-FLOW.UR.3",
"OA-FLOW.UR.4",
"OA-FLOW.UR.5",
"OA-FLOW.UR.6"
],
"proof_available_after_line": 671,
"range_bytes": 3306
},
{
"id": "OA-FLOW.FR.1",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.1",
"start_line": 16,
"end_line": 45,
"range_sha256": "e5377e22f1d064e350e360f089113b233e9415abf7901e977ece9a7fdab5a335",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.WS.1",
"OA-FLOW.ST.2",
"OA-FLOW.GW.1",
"OA-FLOW.GW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 45
},
{
"id": "OA-FLOW.FR.2",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.2",
"start_line": 50,
"end_line": 122,
"range_sha256": "8a527f8f24513c9c115d38aba00409e898c65c0f71d0b318628067a6855d0f7d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.NR.4",
"OA-FLOW.GDA.7",
"OA-FLOW.GDA.8",
"OA-FLOW.GDA.9",
"OA-FLOW.DA.ACTION",
"OA-FLOW.DA.FIXED",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.ST.2",
"OA-FLOW.UR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 122
},
{
"id": "OA-FLOW.FR.3",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.3",
"start_line": 127,
"end_line": 141,
"range_sha256": "04409ec34429b466e6c7aafb4304683a09433c16ce3c3647b7684c42674193f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.2",
"OA-FLOW.BC.5",
"OA-FLOW.EP.5",
"OA-FLOW.DA.HAAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 141
},
{
"id": "OA-FLOW.FR.4",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.4",
"start_line": 146,
"end_line": 188,
"range_sha256": "7b2597dd0e4dde65ce6151e85113fe3646d2a14f1121509a8a75389e8810a5c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.2",
"OA-FLOW.BC.6",
"OA-FLOW.DA.HAAR",
"OA-FLOW.DA.ACTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 188
},
{
"id": "OA-FLOW.FR.5",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.5",
"start_line": 193,
"end_line": 265,
"range_sha256": "294ead9b6de7829cbbfc4f61e35bf4dc5545ec2fac44db40eff5cdb3249af95e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.FR.2",
"OA-FLOW.FR.3",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.UR.5",
"OA-FLOW.GDA.9",
"OA-FLOW.BC.6",
"OA-FLOW.FR.4",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 265
},
{
"id": "OA-FLOW.FR.6",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.6",
"start_line": 270,
"end_line": 274,
"range_sha256": "480cc9d065e212569dff3082d7f82030ddf4ecdb403feb335248c1064cd26b58",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 274
},
{
"id": "OA-FLOW.FR.FIGURE",
"unit": "OA-FLOW-FR",
"source_file": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7",
"reader": "OA-FLOW-FR.html#oa-flow.fr.figure",
"start_line": 278,
"end_line": 295,
"range_sha256": "75f05af7f54ff8476664242fe3bd185369232c7aa9f7c800f762ba2e6b1056f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.FR.2",
"OA-FLOW.FR.3",
"OA-FLOW.UR.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 295
},
{
"id": "OA-FLOW.NWR.0",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.0",
"start_line": 36,
"end_line": 38,
"range_sha256": "e11f1951191dbb82e2e4f2e666c9905096e3c77087dc9ff05d6a55a7eaf165bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.EW.5",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.4",
"OA-FLOW.NO.1",
"OA-FLOW.WS.1",
"OA-FLOW.WS.2",
"OA-FLOW.GF.1",
"OA-FLOW.NF.1",
"OA-FLOW.WF.2",
"OA-FLOW.NC.4",
"OA-FLOW.OT.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.8",
"OA-FLOW.DA.FIXED",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.ST.2",
"OA-FLOW.CP.6",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.FR.1",
"OA-FLOW.FR.2",
"OA-FLOW.FR.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 38
},
{
"id": "OA-FLOW.NWR.1",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.1",
"start_line": 44,
"end_line": 68,
"range_sha256": "c1dbb6f670c381eeaf85a188f32307b86d9dd27238492bfa6683259674256a45",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.5",
"OA-FLOW.EP.5",
"OA-FLOW.NF.1",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.4",
"OA-FLOW.NO.1",
"OA-FLOW.WF.2",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 68
},
{
"id": "OA-FLOW.NWR.2",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.2",
"start_line": 74,
"end_line": 105,
"range_sha256": "7ba654663b8068c85a4d715b0a023463251420548196fd0440ce6e1fd30aed3c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.EP.6",
"OA-FLOW.NC.4",
"OA-FLOW.OT.5",
"OA-FLOW.GF.1",
"OA-FLOW.GW.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 105
},
{
"id": "OA-FLOW.NWR.3",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.3",
"start_line": 111,
"end_line": 153,
"range_sha256": "679d83081856ab153b6b6abd9e4258d37674c40e169eb88b404c26ab391e8d3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.2",
"OA-FLOW.FR.2",
"OA-FLOW.DA.FIXED",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 153
},
{
"id": "OA-FLOW.NWR.4",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.4",
"start_line": 159,
"end_line": 171,
"range_sha256": "454d3884a07e9ce9ef1947c8f239940d560663ccdd17a3bccfd9dc600e7f2b11",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.2",
"OA-FLOW.FR.1",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.DA.INVARIANCE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 171
},
{
"id": "OA-FLOW.NWR.5",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.5",
"start_line": 177,
"end_line": 240,
"range_sha256": "ac40b735e4133ae5346d966bfe1f224869747508086105b00a5aa85d75ed3359",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.3",
"OA-FLOW.NO.1",
"OA-FLOW.WS.2",
"OA-FLOW.GW.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 240
},
{
"id": "OA-FLOW.NWR.6",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.6",
"start_line": 246,
"end_line": 265,
"range_sha256": "77f313a83b00774eb10335cf5dc93d0f76e776ec99a8db4728c2fe808b5dcee1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.3",
"OA-FLOW.NWR.4",
"OA-FLOW.NWR.5",
"OA-FLOW.GDA.6",
"OA-FLOW.EP.6",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.CF.7",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 265
},
{
"id": "OA-FLOW.NWR.7",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.7",
"start_line": 271,
"end_line": 392,
"range_sha256": "0af5c04c8bbacd22ade3bbb28499c393c60e6d4a6cd5aa23c3b21197a09af741",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.2",
"OA-FLOW.NWR.3",
"OA-FLOW.NWR.5",
"OA-FLOW.NWR.6",
"OA-FLOW.FR.4",
"OA-FLOW.WS.1",
"OA-FLOW.GW.4",
"OA-FLOW.NO.1",
"OA-FLOW.FR.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 392
},
{
"id": "OA-FLOW.NWR.8",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.8",
"start_line": 398,
"end_line": 402,
"range_sha256": "a1b8de41150646287e2b9f9f1ec95a882c8e1a10c3ff529ecc818e02d1d529df",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 402
},
{
"id": "OA-FLOW.NWR.FIGURE",
"unit": "OA-FLOW-NWR",
"source_file": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49",
"reader": "OA-FLOW-NWR.html#oa-flow.nwr.figure",
"start_line": 406,
"end_line": 458,
"range_sha256": "bd8f0f46475c61a03cabb254940c134d8dccc31316ec7385c85610de76834c49",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.3",
"OA-FLOW.NWR.5",
"OA-FLOW.NWR.6",
"OA-FLOW.NWR.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 458
},
{
"id": "OA-FLOW.ET.0",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.0",
"start_line": 4,
"end_line": 28,
"range_sha256": "d41cd058b280316b77d3f9c59302b4496831f57970c984d48a43b323352863ad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 28
},
{
"id": "OA-FLOW.ET.1",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.1",
"start_line": 32,
"end_line": 96,
"range_sha256": "ae42f7e0c0c18de303861f17863721123d6c1675bcf7e54cf4238d67b69652eb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 96
},
{
"id": "OA-FLOW.ET.2",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.2",
"start_line": 99,
"end_line": 211,
"range_sha256": "692ca52948de0cb13da5229a0efa52ff031923e7b855fd17e7fe7555c3f7fb63",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ET.1",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 211
},
{
"id": "OA-FLOW.ET.3",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.3",
"start_line": 214,
"end_line": 257,
"range_sha256": "aad3ec1267bc999ce15d255992f9b998168274dd500182f5fd92b304c759c4b6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ET.1",
"OA-FLOW.ET.2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 257
},
{
"id": "OA-FLOW.ET.4",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.4",
"start_line": 260,
"end_line": 265,
"range_sha256": "c2ac7d1dfe7a784acc17795e280ba48edb9188f3ad0d279103e0c7bdcc3b0d75",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 265
},
{
"id": "OA-FLOW.ET.FIGURE",
"unit": "OA-FLOW-ET",
"source_file": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd",
"reader": "OA-FLOW-ET.html#oa-flow.et.figure",
"start_line": 269,
"end_line": 279,
"range_sha256": "40683a938891f30856c9bd81285ef75219abb43efd6fd836bdd2c067d6ca09a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ET.1",
"OA-FLOW.ET.2",
"OA-FLOW.ET.3"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 279
},
{
"id": "OA-FLOW.HCR.0",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.0",
"start_line": 4,
"end_line": 22,
"range_sha256": "8f64e9b739d0e023cd2c1a9995b8c3651f756b762888648fa75203602f3df936",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF0",
"OA-FLOW.NF.1",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.AT.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 22
},
{
"id": "OA-FLOW.HCR.1",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.1",
"start_line": 25,
"end_line": 85,
"range_sha256": "58d7cdb92dc37474ced295391065ab580c3e58f18997dd40ab613f91bd0a82bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 85
},
{
"id": "OA-FLOW.HCR.2",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.2",
"start_line": 88,
"end_line": 141,
"range_sha256": "3014649e193757737732468832922f0ebb2851c191ae95eef8cdaa79e3686676",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HCR.1",
"OA-FLOW.NF.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 141
},
{
"id": "OA-FLOW.HCR.3",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.3",
"start_line": 144,
"end_line": 184,
"range_sha256": "6c8fd819ce57fb588205505f18ed663c88f1a26cc7e2929c0b8bad7202d8e20b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HCR.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.5",
"OA-FLOW.CF.8",
"OA-FLOW.AT.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184
},
{
"id": "OA-FLOW.HCR.4",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.4",
"start_line": 187,
"end_line": 219,
"range_sha256": "d44a01fa0bedbfddb927e3cbc44cabc976fb5cb329d11a7b492f67bfa7ce105f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HCR.1",
"OA-FLOW.HCR.2",
"OA-FLOW.HCR.3",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 219
},
{
"id": "OA-FLOW.HCR.5",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.5",
"start_line": 223,
"end_line": 227,
"range_sha256": "b4ea293359db40d2d1d04c0a0c953f04ad157e2af9cda501516c27ae868b351a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 227
},
{
"id": "OA-FLOW.HCR.FIGURE",
"unit": "OA-FLOW-HCR",
"source_file": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae",
"reader": "OA-FLOW-HCR.html#oa-flow.hcr.figure",
"start_line": 231,
"end_line": 243,
"range_sha256": "a6f0230e61afef6959c546553fa676c815e225e60a503c967489a6141c68b5e7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HCR.1",
"OA-FLOW.HCR.2",
"OA-FLOW.HCR.3",
"OA-FLOW.HCR.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 243
},
{
"id": "OA-FLOW.MATRIX.SUPPORTS",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.matrix.supports",
"start_line": 1,
"end_line": 42,
"range_sha256": "d875a20bdaea1361d17a1767bb249d1e05b4d3abd28c6145be128a9f3c32aaf8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HCR.2",
"OA-FLOW.HCR.3",
"OA-FLOW.PC.1",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.CF.5",
"OA-FLOW.AT.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 42
},
{
"id": "OA-FLOW.MATRIX.FIXED",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.matrix.fixed",
"start_line": 43,
"end_line": 75,
"range_sha256": "a3108613d89914d8cd19f4262ed7494a306c4a068a2096e4bdb37eb8c5417d8c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MATRIX.SUPPORTS",
"OA-FLOW.AT.5",
"OA-FLOW.HR.6",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 75
},
{
"id": "OA-FLOW.MATRIX.CARRIERS",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.matrix.carriers",
"start_line": 76,
"end_line": 98,
"range_sha256": "d0ea77ae63f817e15f739b884a117735b367e02f78e468a1648eeb7d89a02cf6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MATRIX.SUPPORTS",
"OA-FLOW.MATRIX.FIXED",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 98
},
{
"id": "OA-FLOW.MATRIX.UNITARIES",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.matrix.unitaries",
"start_line": 99,
"end_line": 118,
"range_sha256": "ea05fe6dbc4666b90085177824ee694ee1a3675f3ce45f7e19f011f0775da78f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MATRIX.CARRIERS",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 118
},
{
"id": "OA-FLOW.MATRIX.SCALARS",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.matrix.scalars",
"start_line": 119,
"end_line": 146,
"range_sha256": "f613d8a673be19a976464f43006073bdad730429ec0d6d1227bc379e438eb951",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MATRIX.SUPPORTS",
"OA-FLOW.MATRIX.UNITARIES",
"OA-FLOW.HCR.3",
"OA-FLOW.XGAPS.CSTAR.H3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 146
},
{
"id": "OA-FLOW.AMPLIFY.MATRICES",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.amplify.matrices",
"start_line": 147,
"end_line": 165,
"range_sha256": "2b4114be043519e99980515b94a3e7b4aa384bda72a50acdfaddb5159cfe1061",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 165
},
{
"id": "OA-FLOW.AMPLIFY.FIXED",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.amplify.fixed",
"start_line": 166,
"end_line": 192,
"range_sha256": "071f7fc83943a2c52c5aa95cf4a5bfbfe0485cf73522e1503736a3e5930c732d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AMPLIFY.MATRICES",
"OA-FLOW.MATRIX.UNITARIES",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.PC.7",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 192
},
{
"id": "OA-FLOW.AMPLIFY.SUBGROUP",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.amplify.subgroup",
"start_line": 193,
"end_line": 212,
"range_sha256": "87df770d264bd2a6cd44fc7cd255bcee5047243086fcfcbbb2e51f5c71060a0d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AMPLIFY.MATRICES",
"OA-FLOW.AMPLIFY.FIXED",
"OA-FLOW.HCR.0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 212
},
{
"id": "OA-FLOW.AMPLIFY.RIGIDITY",
"unit": "OA-FLOW-L106",
"source_file": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a",
"reader": "OA-FLOW-L106.html#oa-flow.amplify.rigidity",
"start_line": 213,
"end_line": 237,
"range_sha256": "8fc1c68470038cea8885f3130842b7da6a66f9d5fb45df2daf8b77a5e2f9e50f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MATRIX.SUPPORTS",
"OA-FLOW.MATRIX.UNITARIES",
"OA-FLOW.MATRIX.SCALARS",
"OA-FLOW.AMPLIFY.MATRICES",
"OA-FLOW.AMPLIFY.FIXED",
"OA-FLOW.AMPLIFY.SUBGROUP",
"OA-FLOW.HCR.3",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 237
},
{
"id": "OA-FLOW.FM.ALGEBRA",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.algebra",
"start_line": 1,
"end_line": 60,
"range_sha256": "3c503c704679a45b9101a88b1ef523c2e02974b1494de0a43b753cd941114ea1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.NCF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 60
},
{
"id": "OA-FLOW.FM.CUTOFFS",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.cutoffs",
"start_line": 61,
"end_line": 94,
"range_sha256": "51ae8130b62bdcd15155a108e6903b97358e6436e74c0d05311829e5dc79edbb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.ALGEBRA",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 94
},
{
"id": "OA-FLOW.FM.SCHUR",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.schur",
"start_line": 95,
"end_line": 149,
"range_sha256": "975757ea0726cdbf7454af436807a2cb55e7a9f09d7e7d11d48c9bc7bbe4ff95",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.ALGEBRA",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.5",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 149
},
{
"id": "OA-FLOW.FM.LOCAL",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.local",
"start_line": 150,
"end_line": 192,
"range_sha256": "f537039a087d22ac18b0f13275e2c6ff987cbd968544426ee8a992da3749742c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.SCHUR",
"OA-FLOW.CF.2",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.HAARCONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 192
},
{
"id": "OA-FLOW.FM.IDEALS",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.ideals",
"start_line": 193,
"end_line": 246,
"range_sha256": "9869039664ee991d6f2c9113da3de5ea4baa4141f6e7dcd80862cb6db2d8a8a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.ALGEBRA",
"OA-FLOW.FM.CUTOFFS",
"OA-FLOW.FM.SCHUR",
"OA-FLOW.FM.LOCAL",
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 246
},
{
"id": "OA-FLOW.FM.MODULE",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.module",
"start_line": 247,
"end_line": 293,
"range_sha256": "68ae25e6fb7e5b8e1069f3a6df8a7e713b93e4abb6985653e18913884c307d39",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.ALGEBRA",
"OA-FLOW.FM.SCHUR",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NCF.1",
"OA-FLOW.BD.1",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 293
},
{
"id": "OA-FLOW.FM.SPECTRUM",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.spectrum",
"start_line": 294,
"end_line": 328,
"range_sha256": "707e17fbf45dcca16c73a5e26137c70f87189472a879289123c23f1061800422",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.CUTOFFS",
"OA-FLOW.FM.LOCAL",
"OA-FLOW.FM.IDEALS",
"OA-FLOW.FM.MODULE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 328
},
{
"id": "OA-FLOW.FM.DIAGONAL",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.diagonal",
"start_line": 329,
"end_line": 362,
"range_sha256": "b27bc2014c2a3356ca40057d6fea4792ef2417661c9fde7f9aa8fd55694a9b43",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.LOCAL",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 362
},
{
"id": "OA-FLOW.FM.FIBERS",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.fibers",
"start_line": 363,
"end_line": 417,
"range_sha256": "cfe243d6964b73cdf8f521d051301d7cda3cdb5025615b65be84c418d12c3d1b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.LOCAL",
"OA-FLOW.FM.MODULE",
"OA-FLOW.FM.SPECTRUM",
"OA-FLOW.FM.DIAGONAL",
"OA-FLOW.NR.1",
"OA-FLOW.HR.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 417
},
{
"id": "OA-FLOW.FM.POINTS",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.points",
"start_line": 418,
"end_line": 432,
"range_sha256": "2d5e6ec8e2a371b535919311f2ae8183846a49c639b0f2324d3f74905ca9d8be",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.CUTOFFS",
"OA-FLOW.FM.IDEALS",
"OA-FLOW.FM.FIBERS",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 432
},
{
"id": "OA-FLOW.FM.FILTER",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.filter",
"start_line": 433,
"end_line": 472,
"range_sha256": "971a03c6ee2011ec298ac864e937a4ff3752be7e1253266adaf67cc6e072b8e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.CUTOFFS",
"OA-FLOW.FM.MODULE",
"OA-FLOW.FM.SPECTRUM",
"OA-FLOW.FF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 472
},
{
"id": "OA-FLOW.FM.PICTURE",
"unit": "OA-FLOW-L109",
"source_file": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5",
"reader": "OA-FLOW-L109.html#oa-flow.fm.picture",
"start_line": 473,
"end_line": 487,
"range_sha256": "0750c50c7537d558c4dc42d77b4e2d2ab99a7066c12434b20eb05d3001170de0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FM.LOCAL",
"OA-FLOW.FM.MODULE",
"OA-FLOW.FM.FIBERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 487
},
{
"id": "OA-FLOW.NBS.VECTORS",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.vectors",
"start_line": 1,
"end_line": 78,
"range_sha256": "36b5fd6a9d35d19e726377a8228ece001af0d1f9a1b05bddadb1b4ced0b7542f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BS.1",
"OA-FLOW.BS.2",
"OA-FLOW.BS.3",
"OA-FLOW.BS.4",
"OA-FLOW.LF.0",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.HR.5",
"OA-FLOW.CP.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 78
},
{
"id": "OA-FLOW.NBS.OPERATORS",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.operators",
"start_line": 79,
"end_line": 137,
"range_sha256": "4953826caf55cba3156c02349261a94b4622d2e9a6000ad41dc901b6cb8db221",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 137
},
{
"id": "OA-FLOW.NBS.TOPOLOGY",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.topology",
"start_line": 138,
"end_line": 185,
"range_sha256": "4e33cc0e7b9b32dad9b3148633bc4f920b904069bc3c0abe8c1f9ff946a21163",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.OPERATORS",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.HAARCONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 185
},
{
"id": "OA-FLOW.NBS.LOCALIZATION",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.localization",
"start_line": 186,
"end_line": 254,
"range_sha256": "b9094a484d2e490e8e6cd60164b2be83cd321ab99c3a557a2fa23cc7dd5ebbcd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.NBS.OPERATORS",
"OA-FLOW.NBS.TOPOLOGY",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4",
"OA-FLOW.LF.5",
"OA-FLOW.LF.6",
"OA-FLOW.LF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 254
},
{
"id": "OA-FLOW.NBS.KERNEL",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.kernel",
"start_line": 255,
"end_line": 300,
"range_sha256": "80a0b14a7502d63095869d37a4f4e708c2c0373832c15c3d4e1a5765acb0a102",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.NBS.OPERATORS",
"OA-FLOW.SUM.KERNEL",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.XGAPS.CSTAR.H1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 300
},
{
"id": "OA-FLOW.NBS.TRANSFER",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.transfer",
"start_line": 301,
"end_line": 329,
"range_sha256": "ddf7855ab37a3829b877db89a7dbed92cd99ccd4df2801d4cfff4cee2eadd861",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.NBS.LOCALIZATION",
"OA-FLOW.NBS.KERNEL",
"OA-FLOW.XGAPS.CSTAR.H0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 329
},
{
"id": "OA-FLOW.NBS.EXAMPLES",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.examples",
"start_line": 330,
"end_line": 363,
"range_sha256": "039b6a20517d66b74e92ed39b5d45068e3cde60254f3e362d69508e194e460d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.NBS.OPERATORS",
"OA-FLOW.NBS.TRANSFER",
"OA-FLOW.LF.1",
"OA-FLOW.FF.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 363
},
{
"id": "OA-FLOW.NBS.ADJOINTS",
"unit": "OA-FLOW-L110",
"source_file": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15",
"reader": "OA-FLOW-L110.html#oa-flow.nbs.adjoints",
"start_line": 364,
"end_line": 391,
"range_sha256": "c42adae430be07ce03e9adf9a81b5f7807c4f1af4d6a39c940ff00b56c9bb955",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NBS.VECTORS",
"OA-FLOW.AT.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 391
},
{
"id": "OA-FLOW.GCC.0",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.0",
"start_line": 4,
"end_line": 11,
"range_sha256": "5c6450a9ce009e819a67706f8624fe83d620398fdce29d5a6fed6a4a92807b44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 11
},
{
"id": "OA-FLOW.GCC.INPUTS",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.inputs",
"start_line": 14,
"end_line": 24,
"range_sha256": "bd64ed855c6b85fd02f8de48503163bca96b9a2e4225ee3fa0fc572299a8ec24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 24
},
{
"id": "OA-FLOW.GCC.SETTING",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.setting",
"start_line": 27,
"end_line": 75,
"range_sha256": "40b8d237a7778d2520e48e2169b917811099695655e1e12b3d97a5526a8bd276",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.LF.0",
"OA-FLOW.LF.1",
"OA-FLOW.LF.4",
"OA-FLOW.GL.0",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.4",
"OA-FLOW.GL.6",
"OA-FLOW.GL.7",
"OA-FLOW.SS.3",
"OA-FLOW.PC.1",
"OA-FLOW.SF.SF0",
"OA-FLOW.SF.SB0",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 75
},
{
"id": "OA-FLOW.GCC.DEFINITION",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.definition",
"start_line": 78,
"end_line": 118,
"range_sha256": "958a4428f5b800800f653a6310ea27ed21d111684a8947b439ff5b57b981982e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GL.2",
"OA-FLOW.GL.7",
"OA-FLOW.SS.3",
"OA-FLOW.LF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 118
},
{
"id": "OA-FLOW.GCC.TOOLS",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.tools",
"start_line": 121,
"end_line": 150,
"range_sha256": "7c0e29ef5a2deee4846628fc77133297d4b5ad957df0e54a54090beef3854893",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 150
},
{
"id": "OA-FLOW.GCC.EQUIVALENCE",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.equivalence",
"start_line": 153,
"end_line": 256,
"range_sha256": "0d3cc90bf5a69532a020b7b928ce9e32041a245aea97d9b7943e320d392a84ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.6",
"OA-FLOW.LF.1",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 256
},
{
"id": "OA-FLOW.GCC.CENTRAL",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.central",
"start_line": 259,
"end_line": 301,
"range_sha256": "e8df4c9088a49befdd6e9aaad5bcbb4b57001547f7a3784a186692bbdb8a8676",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.SS.3",
"OA-FLOW.GL.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 301
},
{
"id": "OA-FLOW.GCC.INTERSECTION",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.intersection",
"start_line": 304,
"end_line": 326,
"range_sha256": "62dba6cc4dae4134c9fe8bd7ea930551c596be9c0dd435c782c0885b2c44bc41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GCC.CENTRAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 326
},
{
"id": "OA-FLOW.GCC.MODEL",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.model",
"start_line": 329,
"end_line": 366,
"range_sha256": "678539d5a7150968853f87eb080dd2383c1ceef2f8e69fe36ceb6e910fd51830",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.GCC.EQUIVALENCE",
"OA-FLOW.LF.4",
"OA-FLOW.GL.3",
"OA-FLOW.SS.3",
"OA-FLOW.SF.SB0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 366
},
{
"id": "OA-FLOW.GCC.SOURCES",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.sources",
"start_line": 369,
"end_line": 376,
"range_sha256": "203fd84b3694e78c484939bf6677d7142b6b6b5be5eabf8babb356bbcd844ede",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 376
},
{
"id": "OA-FLOW.GCC.FIGURE",
"unit": "OA-FLOW-GCC",
"source_file": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42",
"reader": "OA-FLOW-GCC.html#oa-flow.gcc.figure",
"start_line": 380,
"end_line": 423,
"range_sha256": "3614aa86376736c085a287afaa28edecf8868b2d4ca62756f98f5330b05a45c6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.EQUIVALENCE",
"OA-FLOW.GCC.MODEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 423
},
{
"id": "OA-FLOW.L112.0",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.0",
"start_line": 4,
"end_line": 11,
"range_sha256": "76610263194eb4b8e4d2f732d332ec8f3aacdfecdd4ddb72ae090eb9ddd85dcc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 11
},
{
"id": "OA-FLOW.L112.INPUTS",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.inputs",
"start_line": 14,
"end_line": 22,
"range_sha256": "df0e13c21031f317b22a31a820b1ac728e358724e9eddb0dcfc9168d6a0c0e60",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 22
},
{
"id": "OA-FLOW.L112.PREDUAL",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.predual",
"start_line": 25,
"end_line": 50,
"range_sha256": "ac852c6027fe90aa48a4a2d45828c23a1e296c3001d9fc762a6a87f290c906c3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 50
},
{
"id": "OA-FLOW.L112.SYSTEMS",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.systems",
"start_line": 53,
"end_line": 80,
"range_sha256": "47b54c86948df60b6acf53ac03983adf7dd7af712d785db109d4626063019dd9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 80
},
{
"id": "OA-FLOW.L112.REPRESENTATIONS",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.representations",
"start_line": 83,
"end_line": 119,
"range_sha256": "34046a5e330718727d7670fa272ee6ca25eeff43ccf7e96a8afc40ec5c35df8a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.PREDUAL",
"OA-FLOW.L112.SYSTEMS",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 119
},
{
"id": "OA-FLOW.L112.QI1",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.qi1",
"start_line": 120,
"end_line": 163,
"range_sha256": "c606fa91a6fcd03df7dfc7a8b8c2946d3ed93b6bb20aa2c3414df5e4681baae0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.REPRESENTATIONS",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.L112.QI2",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.qi2",
"start_line": 164,
"end_line": 232,
"range_sha256": "f1af285ed79837a2bb3af3e1218766852790ec56dbf0e7acc96edcf647a17063",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.QI1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.NF.1",
"OA-FLOW.NF.2",
"OA-FLOW.NF.3",
"OA-FLOW.NF.4",
"OA-FLOW.SC.4",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.L112.QI3",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.qi3",
"start_line": 233,
"end_line": 279,
"range_sha256": "d312172d5c8021db2ed8100bfff2815e428dda5e629277de6aa018d1ccbf232c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.QI1",
"OA-FLOW.L112.QI2",
"OA-FLOW.CF.1",
"OA-FLOW.CP.6",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6"
]
},
{
"id": "OA-FLOW.L112.QI4",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.qi4",
"start_line": 280,
"end_line": 299,
"range_sha256": "e727a5bca9f77332ed10acc4c8fbdd757bc4e7be9ebe1792bd2a50b06df2fdb7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.QI3",
"OA-FLOW.GNS.5.1"
]
},
{
"id": "OA-FLOW.L112.OBSTRUCTION",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.obstruction",
"start_line": 302,
"end_line": 423,
"range_sha256": "b6dafac26eea1dad40e8f08c55a8ec6a64efa7f68b0b039059d3b0f329756d53",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.PREDUAL",
"OA-FLOW.L112.SYSTEMS",
"OA-FLOW.L112.REPRESENTATIONS",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.5.1",
"OA-FLOW.GNS.7.1",
"OA-FLOW.GNS.7.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 423
},
{
"id": "OA-FLOW.COCMAT.ACTION",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.cocmat.action",
"start_line": 426,
"end_line": 490,
"range_sha256": "8fa409608a2826e08593354dd51f62824998253036f53614b19a590f9117c794",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.PREDUAL",
"OA-FLOW.L112.SYSTEMS",
"OA-FLOW.NCF.1",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 490
},
{
"id": "OA-FLOW.COCMAT.CORNERS",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.cocmat.corners",
"start_line": 493,
"end_line": 549,
"range_sha256": "15bcfb511ec3fd7a6d61d00ef7b0923ff8b87e6a044766906b4305a40de17681",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COCMAT.ACTION",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.GCC.EQUIVALENCE",
"OA-FLOW.SS.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 549
},
{
"id": "OA-FLOW.L112.SOURCES",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.sources",
"start_line": 552,
"end_line": 559,
"range_sha256": "479c4660a03429e9e3ee22dd4d83f262b6bacd30908950eb70770d118eeee11d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 559
},
{
"id": "OA-FLOW.L112.FIGURE",
"unit": "OA-FLOW-L112",
"source_file": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2",
"reader": "OA-FLOW-L112.html#oa-flow.l112.figure",
"start_line": 563,
"end_line": 605,
"range_sha256": "abfb1e79f51818aadd1e40998d36a4c171166e288ea7cfa75b6732f14f437fcf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L112.PREDUAL",
"OA-FLOW.L112.OBSTRUCTION",
"OA-FLOW.COCMAT.ACTION",
"OA-FLOW.COCMAT.CORNERS"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 605
},
{
"id": "OA-FLOW.L113.0",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.0",
"start_line": 4,
"end_line": 9,
"range_sha256": "4d878aba1b663c60a3ab28d86034d2df5309cfcd76dbf432037ed05f21337251",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 9
},
{
"id": "OA-FLOW.L113.INPUTS",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.inputs",
"start_line": 12,
"end_line": 18,
"range_sha256": "a0ef457eb7d0a077e31d11308f244feb2a20526d0cb3df82dba72e80ec20a2ae",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 18
},
{
"id": "OA-FLOW.CONT.ALGEBRA",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.cont.algebra",
"start_line": 21,
"end_line": 61,
"range_sha256": "671aa291c9062d03206274501f261e656eef9b1175ee19d7ccabf4468ce9d58f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.3",
"OA-FLOW.GRP.CSTARQUOTIENTS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 61
},
{
"id": "OA-FLOW.CONT.SMOOTHING",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.cont.smoothing",
"start_line": 64,
"end_line": 108,
"range_sha256": "736f18b14b801e0f324ab05a556e84d84126a9da4384e300532c97caf4ccd747",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONT.ALGEBRA",
"OA-FLOW.AT.5",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 108
},
{
"id": "OA-FLOW.CONT.DENSITY",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.cont.density",
"start_line": 111,
"end_line": 175,
"range_sha256": "7c07f62451e8b437fc81423070a778f5f7ebea09a885e997bad8bbdfc78841c3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONT.ALGEBRA",
"OA-FLOW.CONT.SMOOTHING",
"OA-FLOW.AT.1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.PC.1",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 175
},
{
"id": "OA-FLOW.L113.TRANSLATION-MODEL",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.translation-model",
"start_line": 178,
"end_line": 209,
"range_sha256": "17980805f0c570450a77cbc580ffb0162886ba499e7b750dad59f7365349d152",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONT.ALGEBRA",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 209
},
{
"id": "OA-FLOW.L113.NONUNITAL",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.nonunital",
"start_line": 212,
"end_line": 243,
"range_sha256": "39b004b5464fd2bbff5b00d6bbef38b6236bf82181256dea5cff330d29b5e9d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.9",
"OA-FLOW.GRP.CSTARQUOTIENTS",
"OA-FLOW.QF.1",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 243
},
{
"id": "OA-FLOW.CONT.GELFAND",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.cont.gelfand",
"start_line": 246,
"end_line": 299,
"range_sha256": "f007aee900d1a57db80228f854c5a3fb8e41da0db3038d2c26435979792b7c6f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L113.NONUNITAL",
"OA-FLOW.CF.4",
"OA-FLOW.CF.9",
"OA-FLOW.GRP.CSTARQUOTIENTS",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 299
},
{
"id": "OA-FLOW.CONT.DISJOINT",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.cont.disjoint",
"start_line": 302,
"end_line": 354,
"range_sha256": "4a3367e50a1ca9462cbae7bc6b1b69941d57e289e93ada5ea17181c25325a9d1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONT.GELFAND",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.PC.1",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 354
},
{
"id": "OA-FLOW.L113.MODEL",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.model",
"start_line": 357,
"end_line": 382,
"range_sha256": "7beaf1496826463f888e9837b9ac189446d9af29a8822f2afd0969efcfb199b7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONT.ALGEBRA",
"OA-FLOW.L113.TRANSLATION-MODEL",
"OA-FLOW.CONT.GELFAND",
"OA-FLOW.CONT.DISJOINT",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 382
},
{
"id": "OA-FLOW.L113.SOURCES",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.sources",
"start_line": 385,
"end_line": 390,
"range_sha256": "5f1cc392699ccaba45dfab252f39bec7877367a34aa159dbcbdfda52433c12f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 390
},
{
"id": "OA-FLOW.L113.FIGURE",
"unit": "OA-FLOW-L113",
"source_file": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f",
"reader": "OA-FLOW-L113.html#oa-flow.l113.figure",
"start_line": 394,
"end_line": 415,
"range_sha256": "260709e216183c875ba3f6c0806632618fdd2d3930c30cbc95659cee8975e351",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L113.TRANSLATION-MODEL",
"OA-FLOW.L113.MODEL",
"OA-FLOW.CONT.DISJOINT"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 415
},
{
"id": "OA-FLOW.L114.0",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.0",
"start_line": 4,
"end_line": 11,
"range_sha256": "0f12cd7f2bd05c99b551b107f9291bd003a3cb1ef285e697680afc21d980bd9f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 11
},
{
"id": "OA-FLOW.L114.INPUTS",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.inputs",
"start_line": 14,
"end_line": 20,
"range_sha256": "16bbf92ea1c384116ca77594881f89f1fc30145a754ced6e2429d5be52722b4a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 20
},
{
"id": "OA-FLOW.L114.CONVENTIONS",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.conventions",
"start_line": 23,
"end_line": 45,
"range_sha256": "99747de64a68c79211d57238ef77c24262744167b64e7436466351fa67d0a64b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.7",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.SS.3",
"OA-FLOW.AT.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 45
},
{
"id": "OA-FLOW.L114.AVERAGE",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.average",
"start_line": 48,
"end_line": 116,
"range_sha256": "2fea1ab02e18f336b8413ccc2f25877e4fd3f7ac7345882d3300f2e4ef39755d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.FF.6",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.WF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 116
},
{
"id": "OA-FLOW.L114.DENSITY",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.density",
"start_line": 119,
"end_line": 141,
"range_sha256": "3efb6fca792ca4ad3e8165d29850436b151dc4a2ff017091c60caa79884efd24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.EP.6",
"OA-FLOW.GW.4",
"OA-FLOW.PC.1",
"OA-FLOW.CF.7",
"OA-FLOW.WF.2",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 141
},
{
"id": "OA-FLOW.INTSPEC.DOMAIN",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.domain",
"start_line": 48,
"end_line": 204,
"range_sha256": "8c56c0d0a07906259e90609980d873cdd8b7fee5f77f9e2b59c03ca6caf0b1af",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.FF.6",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.WF.2",
"OA-FLOW.EP.6",
"OA-FLOW.GW.4",
"OA-FLOW.PC.1",
"OA-FLOW.CF.7",
"OA-FLOW.CP.6",
"OA-FLOW.CP.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 204
},
{
"id": "OA-FLOW.L114.FIELD",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.field",
"start_line": 207,
"end_line": 253,
"range_sha256": "7cf2a21b42a5fcef7f73fddb26d60d7e5dd8225c5489f48b55b367ca0174ec28",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.CP.6",
"OA-FLOW.SS.3",
"OA-FLOW.XGAPS.CSTAR.H1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 253
},
{
"id": "OA-FLOW.INTSPEC.FOURIER",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.fourier",
"start_line": 207,
"end_line": 272,
"range_sha256": "44e5fa32067de676e5208a81ab0c8cb8ecb281b63c15e69103afe14b999f2fc7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.CP.6",
"OA-FLOW.SS.3",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.HR.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 272
},
{
"id": "OA-FLOW.L114.FILTERS",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.filters",
"start_line": 275,
"end_line": 323,
"range_sha256": "f70dc28abc717b75bf6430c5e28a4c4f741303487bbc8ecb451a1d2a0e239a25",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.L114.FIELD",
"OA-FLOW.AT.5",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 323
},
{
"id": "OA-FLOW.INTSPEC.SUPPORT",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.support",
"start_line": 326,
"end_line": 347,
"range_sha256": "0103049068e9b6cbba043922099ab334e265e041a37f91887e84829f7ed2d32b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.INTSPEC.FOURIER",
"OA-FLOW.L114.FILTERS",
"OA-FLOW.LF.1",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 347
},
{
"id": "OA-FLOW.INTSPEC.MULTIPLIER",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.multiplier",
"start_line": 275,
"end_line": 416,
"range_sha256": "ad60bd7427edb2eebae28037016d5dc078b429dbbbbba6642a7229c7bf006d7c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.L114.FIELD",
"OA-FLOW.AT.5",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.INTSPEC.FOURIER",
"OA-FLOW.LF.1",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.AF.4",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 416
},
{
"id": "OA-FLOW.INTSPEC.DETECTION",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.detection",
"start_line": 419,
"end_line": 477,
"range_sha256": "77c8f3c189296ac4854af4246072dd16e5f437e2d23a3f3e8c38338ba36b1f97",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.L114.DENSITY",
"OA-FLOW.L114.FILTERS",
"OA-FLOW.INTSPEC.FOURIER",
"OA-FLOW.INTSPEC.SUPPORT",
"OA-FLOW.SS.3",
"OA-FLOW.LF.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 477
},
{
"id": "OA-FLOW.INTSPEC.POINTSPECTRUM",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.intspec.pointspectrum",
"start_line": 480,
"end_line": 521,
"range_sha256": "758a51109faee96d5efb4531aa7886ccd6360fe0ea05b4d0953dc656bb81b8e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.INTSPEC.DETECTION",
"OA-FLOW.INTSPEC.SUPPORT",
"OA-FLOW.L114.DENSITY",
"OA-FLOW.GL.2",
"OA-FLOW.GL.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 521
},
{
"id": "OA-FLOW.L114.SOURCES",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.sources",
"start_line": 524,
"end_line": 531,
"range_sha256": "225c57005913be04b7d492dcfe7cd1589fea5c542e7682e3359ed79638305b15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 531
},
{
"id": "OA-FLOW.L114.FIGURE",
"unit": "OA-FLOW-L114",
"source_file": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43",
"reader": "OA-FLOW-L114.html#oa-flow.l114.figure",
"start_line": 535,
"end_line": 592,
"range_sha256": "51dbe9b3098ddde67a4e0e0eb1cfc8e201461f0b530aae8701e0006dfc9dfddb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.INTSPEC.FOURIER",
"OA-FLOW.INTSPEC.MULTIPLIER",
"OA-FLOW.INTSPEC.DETECTION",
"OA-FLOW.INTSPEC.POINTSPECTRUM"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 592
},
{
"id": "OA-FLOW.L117.0",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.l117.0",
"start_line": 4,
"end_line": 9,
"range_sha256": "08a0f2b29345e28e8830ccab09e5001ee0abb23ae060e160b46c2e66f297cd3b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 9
},
{
"id": "OA-FLOW.L117.INPUTS",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.l117.inputs",
"start_line": 12,
"end_line": 18,
"range_sha256": "c622fd1d1ca81582dee8bade6a58112c797cf4a1e3bc74921556bd4ffc6e62f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 18
},
{
"id": "OA-FLOW.FULLCORNER.COMMUTANT",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.fullcorner.commutant",
"start_line": 21,
"end_line": 149,
"range_sha256": "a45fbac2a9169aa94c15351001a54310a38e3c3471db61ec96250c997f59a916",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.PC.1",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 149,
"scope_boundary": "Normal injective commutant compression; explicit inverse and strong continuity on unitary groups. No full bounded-operator inverse assertion."
},
{
"id": "OA-FLOW.FULLCORNER.INNER",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.fullcorner.inner",
"start_line": 152,
"end_line": 234,
"range_sha256": "9a7f04033a689ac651093c21c257f5ff754cdaf6230cfe324facd184d6d08d71",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FULLCORNER.COMMUTANT",
"OA-FLOW.CR.9",
"OA-FLOW.MC.1",
"OA-FLOW.ST.2",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 234
},
{
"id": "OA-FLOW.FULLCORNER.CONTINUITY",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.fullcorner.continuity",
"start_line": 237,
"end_line": 291,
"range_sha256": "8f385e350b561ac5f188c5052ec55aa3c33bff7ff911d75dbddc83a419e384b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FULLCORNER.COMMUTANT",
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 291
},
{
"id": "OA-FLOW.FULLCORNER.COCYCLE",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.fullcorner.cocycle",
"start_line": 294,
"end_line": 355,
"range_sha256": "0c8b6ac8128e7f13f115f79a315244c9ee3796dc960384ee0776b1bb8de1c6ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.FULLCORNER.CONTINUITY",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 355
},
{
"id": "OA-FLOW.L117.MODEL",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.l117.model",
"start_line": 358,
"end_line": 388,
"range_sha256": "8e05236b22563e282ef354cf10c4e8f646a800ccd891ada87c67cf70621f9328",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FULLCORNER.COMMUTANT",
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.FULLCORNER.CONTINUITY",
"OA-FLOW.FULLCORNER.COCYCLE",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 388
},
{
"id": "OA-FLOW.L117.SOURCES",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.l117.sources",
"start_line": 391,
"end_line": 396,
"range_sha256": "ef35bb0abb2269aab5179613d19a3c268526ce0b605b7b4cb76e70940a194d73",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 396
},
{
"id": "OA-FLOW.L117.FIGURE",
"unit": "OA-FLOW-L117",
"source_file": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be",
"reader": "OA-FLOW-L117.html#oa-flow.l117.figure",
"start_line": 400,
"end_line": 413,
"range_sha256": "2155b45abb2541c3fe71d21ee0ba8552230e553fba33fc343245cf2909c15033",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FULLCORNER.COMMUTANT",
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.FULLCORNER.CONTINUITY",
"OA-FLOW.FULLCORNER.COCYCLE",
"OA-FLOW.L117.MODEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 413
},
{
"id": "OA-FLOW.L118.0",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.0",
"start_line": 4,
"end_line": 9,
"range_sha256": "e0aeef170b200ba60783f827e1d14eccd6f35f69e1ad89bb50aaa2cf034a05d0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 9
},
{
"id": "OA-FLOW.L118.INPUTS",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.inputs",
"start_line": 12,
"end_line": 18,
"range_sha256": "a691964a7f3f5834eaf7a92a0b93fb5b8af5e8dbace8a6bfdfcccfbc20cb9c44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 18
},
{
"id": "OA-FLOW.L118.STATEMENT",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.statement",
"start_line": 21,
"end_line": 58,
"range_sha256": "92067a024330ade3043504c817dedd8a28edfbfe14fb81a781539dda694815cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 58
},
{
"id": "OA-FLOW.L118.MULTIPLIERS",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.multipliers",
"start_line": 61,
"end_line": 77,
"range_sha256": "c650df83608ce39b0de078aabecaf47cc8b928be424ec075fe152eae792d46e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 77
},
{
"id": "OA-FLOW.L118.EXPECTATION",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.expectation",
"start_line": 80,
"end_line": 147,
"range_sha256": "ac42cf3f934e5ce0c9a2f8f22eabf863f1def46bd180abb96c971c33d4479c3c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.TD.8",
"OA-FLOW.EP.3",
"OA-FLOW.NF.1",
"OA-FLOW.NF.6",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 147
},
{
"id": "OA-FLOW.REPEXT.COMPACT",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.repext.compact",
"start_line": 150,
"end_line": 191,
"range_sha256": "4adbd8bb3799499503809cb8e0413e0b481fa26470ea2f4e5e652d28d0094311",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.3.3",
"OA-FLOW.GNS.4.1",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.QF.1",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 191,
"scope_boundary": "Finite Radon and compact-carrier reduction only; the full extension E4 is not yet established."
},
{
"id": "OA-FLOW.REPEXT.EXTREME",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.repext.extreme",
"start_line": 194,
"end_line": 250,
"range_sha256": "defba42b472b0735b65e900538161e36d1e5ded5213c59d12bca196b48a870d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.EXPECTATION",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.REPEXT.COMPACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 250
},
{
"id": "OA-FLOW.L118.KERNEL",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.kernel",
"start_line": 253,
"end_line": 268,
"range_sha256": "0aff6ef4eed7cef814c0346db591a34106418567cc7ef19d22dcb62b38212520",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.EXPECTATION",
"OA-FLOW.REPEXT.EXTREME",
"OA-FLOW.CP.6",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 268
},
{
"id": "OA-FLOW.REPEXT.COLLAPSE",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.repext.collapse",
"start_line": 271,
"end_line": 365,
"range_sha256": "a955b38d4635941a51c3282c67277581e59dd1b0ed1c22518ccdf73d7b19d9ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.EXPECTATION",
"OA-FLOW.REPEXT.EXTREME",
"OA-FLOW.REPEXT.COMPACT",
"OA-FLOW.L118.MULTIPLIERS",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF1",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 365
},
{
"id": "OA-FLOW.REPEXT.ASSEMBLE",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.repext.assemble",
"start_line": 368,
"end_line": 434,
"range_sha256": "50895f2876af8267264d33e6593656b8a03777641d38e466ec215aaf21ca6c4e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.MULTIPLIERS",
"OA-FLOW.REPEXT.COLLAPSE",
"OA-FLOW.NF.1",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.SF.SF2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.CF.1",
"OA-FLOW.GNS.5.1",
"OA-FLOW.GNS.3.1",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 434,
"scope_boundary": "Complete E4 at the full arbitrary-representation scope, available only after this endpoint."
},
{
"id": "OA-FLOW.L118.DUAL-MAP",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.dual-map",
"start_line": 437,
"end_line": 455,
"range_sha256": "61bd1eebb945ccb0019d8d0ec5d2f6a9d19161c7af86bf61a96dddd85d28a95a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 455
},
{
"id": "OA-FLOW.REPEXT.GROUP",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.repext.group",
"start_line": 458,
"end_line": 509,
"range_sha256": "2ed0ef61055ba3cd9d321c42070b8530c55f974aeaa91b7e66fc0abd66ab2215",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REPEXT.ASSEMBLE",
"OA-FLOW.L118.DUAL-MAP",
"OA-FLOW.L118.MULTIPLIERS",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.GRP.RECOVERY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 509
},
{
"id": "OA-FLOW.L118.MODEL",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.model",
"start_line": 512,
"end_line": 548,
"range_sha256": "8ddccda502595efa1fa708fc4ad3a48e479286c1bdd785b9e2151232a128f742",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.EXPECTATION",
"OA-FLOW.L118.KERNEL",
"OA-FLOW.REPEXT.ASSEMBLE",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 548
},
{
"id": "OA-FLOW.L118.SOURCES",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.sources",
"start_line": 551,
"end_line": 556,
"range_sha256": "37e397252963fa7b0b9860c8db9756a8afbbe771e7e556bab4b852dcbca4e9bf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 556
},
{
"id": "OA-FLOW.L118.FIGURE",
"unit": "OA-FLOW-L118",
"source_file": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026",
"reader": "OA-FLOW-L118.html#oa-flow.l118.figure",
"start_line": 560,
"end_line": 571,
"range_sha256": "28fb883b9a06a924f58a6f1e76aec8be8d8b5c1373eaa1a953a0f1371aa66a35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.EXPECTATION",
"OA-FLOW.L118.KERNEL",
"OA-FLOW.REPEXT.COLLAPSE",
"OA-FLOW.REPEXT.ASSEMBLE",
"OA-FLOW.L118.MODEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 571
},
{
"id": "OA-FLOW.L115.0",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.0",
"start_line": 4,
"end_line": 11,
"range_sha256": "04bbeccf846a2c136c47eb9512567d1df805108f33486710ff4fc50f4aa98928",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 11
},
{
"id": "OA-FLOW.L115.INPUTS",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.inputs",
"start_line": 14,
"end_line": 22,
"range_sha256": "18f637289fff6ac1261883c7df575bd57f095acc63914ddfe0e4cac8c716c03d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 22
},
{
"id": "OA-FLOW.L115.CONVENTIONS",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.conventions",
"start_line": 25,
"end_line": 44,
"range_sha256": "41cc36d74be8984ee429eb1f4ad36f335413ce9e3d099c2ecf87c69bdb11db03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.CONVENTIONS",
"OA-FLOW.AT.5",
"OA-FLOW.LF.1",
"OA-FLOW.GL.0",
"OA-FLOW.GL.2",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 44,
"scope_boundary": "Written complete reduction at actual earlier reflected Fourier/hull/plateau proofs; not an unproved heading or forward statement."
},
{
"id": "OA-FLOW.L115.LOCAL-DETECTION",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.local-detection",
"start_line": 48,
"end_line": 49,
"range_sha256": "812ac132c536f86ff4863ae22034b3b0d25aa33d65c472fee1eed619b9fb8846",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 49,
"scope_boundary": "Written complete reduction at actual earlier reflected Fourier/hull/plateau proofs; not an unproved heading or forward statement."
},
{
"id": "OA-FLOW.L115.CORNER-INTEGRABILITY",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.corner-integrability",
"start_line": 52,
"end_line": 67,
"range_sha256": "ec1704270a7bc379ad44f585f60fd6d023430be10597b171a2d6b9c43eb42ed1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.L114.DENSITY",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 67
},
{
"id": "OA-FLOW.L115.OVERLAP",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.overlap",
"start_line": 70,
"end_line": 121,
"range_sha256": "687f3e13ffb6737144f97f5f094a84ef6acbdf3ae6d5994fce0fe60fdcce3c9c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.L115.LOCAL-DETECTION",
"OA-FLOW.L115.CORNER-INTEGRABILITY",
"OA-FLOW.CP.6",
"OA-FLOW.AT.1",
"OA-FLOW.AT.5",
"OA-FLOW.SS.3",
"OA-FLOW.LF.1",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.CONT.DENSITY",
"OA-FLOW.CONT.GELFAND",
"OA-FLOW.CONT.DISJOINT",
"OA-FLOW.INTSPEC.DETECTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 121
},
{
"id": "OA-FLOW.CONNES.STABILIZATION",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.stabilization",
"start_line": 124,
"end_line": 206,
"range_sha256": "571c09be9492881a57027ab698b8213c4fef0cdb4e67d877abc016909f2df21c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.NCF.1",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.AT.1",
"OA-FLOW.AT.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.COCMAT.ACTION",
"OA-FLOW.COCMAT.CORNERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 206
},
{
"id": "OA-FLOW.CONNES.REGULARINT",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.regularint",
"start_line": 209,
"end_line": 291,
"range_sha256": "7208d5ecc4c040e9232beb9d7b81712a536c6cb4d8985183a0a76178a508e3e5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CORNER-INTEGRABILITY",
"OA-FLOW.CONNES.STABILIZATION",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.NCF.1",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2",
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.L114.DENSITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 291
},
{
"id": "OA-FLOW.CONNES.FIXEDEIGEN",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.fixedeigen",
"start_line": 294,
"end_line": 369,
"range_sha256": "8a5f0f5ecfa557373fc119cf8758e0976347504be371e8468b2be34c8d5e1223",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CONNES.STABILIZATION",
"OA-FLOW.CONNES.REGULARINT",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.HR.3",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SS.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 369
},
{
"id": "OA-FLOW.CONNES.CENTRALOVERLAP",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.centraloverlap",
"start_line": 372,
"end_line": 410,
"range_sha256": "5863955d3a7057a9dd938101269cb84d24ed86d12eb899fb615a0ab82468f2a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.OVERLAP",
"OA-FLOW.CONNES.FIXEDEIGEN",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 410
},
{
"id": "OA-FLOW.CONNES.CENTERKERNEL",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.centerkernel",
"start_line": 413,
"end_line": 494,
"range_sha256": "c707c00b4beecec71b1e612be42df94eed2b54506f64ca6dd63547a55cee59f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.OVERLAP",
"OA-FLOW.CONNES.STABILIZATION",
"OA-FLOW.CONNES.REGULARINT",
"OA-FLOW.CONNES.FIXEDEIGEN",
"OA-FLOW.CONNES.CENTRALOVERLAP",
"OA-FLOW.NCF.2",
"OA-FLOW.NCF.3",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.CONT.DISJOINT",
"OA-FLOW.LF.1",
"OA-FLOW.GL.7",
"OA-FLOW.INTSPEC.DETECTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 494,
"scope_boundary": "Complete arbitrary-LCA kernel theorem after all regular-system and central-corner steps; original alpha need not be integrable."
},
{
"id": "OA-FLOW.CONNES.ABSORPTION",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.absorption",
"start_line": 497,
"end_line": 591,
"range_sha256": "417b34a9bf08da4bf5b8b94d0edc0db04396c81b91edafb899ed38dd6f9c3e13",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.L115.LOCAL-DETECTION",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.6",
"OA-FLOW.GL.7",
"OA-FLOW.LF.1",
"OA-FLOW.LF.6",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.XGAPS.CSTAR.H1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 591
},
{
"id": "OA-FLOW.CONNES.SUBGROUP",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.connes.subgroup",
"start_line": 594,
"end_line": 638,
"range_sha256": "a438238bce4a8529758b91bab17347cac739bf6dd20a439114c0ecc61a38d928",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CONNES.ABSORPTION",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.GL.2",
"OA-FLOW.SS.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 638
},
{
"id": "OA-FLOW.L115.SOURCES",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.sources",
"start_line": 641,
"end_line": 648,
"range_sha256": "1545baf4821fbd0539a7e661ebc78ce1ae5efb9d2ea60afa2950f5e6d44e3ebc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 648
},
{
"id": "OA-FLOW.L115.FIGURE",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#oa-flow.l115.figure",
"start_line": 652,
"end_line": 686,
"range_sha256": "53fbef6fbd3412a19b92c55b90843c716fcf8296f3a05e92656fd13e5eab1053",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CORNER-INTEGRABILITY",
"OA-FLOW.L115.OVERLAP",
"OA-FLOW.CONNES.STABILIZATION",
"OA-FLOW.CONNES.REGULARINT",
"OA-FLOW.CONNES.FIXEDEIGEN",
"OA-FLOW.CONNES.CENTRALOVERLAP",
"OA-FLOW.CONNES.CENTERKERNEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 686
},
{
"id": "OA-FLOW.L116.0",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.0",
"start_line": 4,
"end_line": 9,
"range_sha256": "7b6ea5dfa686ddfa7d6e7ab88d9334ddf2587fbb12d5f2a883583e6d3760b21f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 9,
"scope_boundary": "Setting and prerequisites; not a proof premise"
},
{
"id": "OA-FLOW.CONNES.CENTRALERGODIC",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.connes.centralergodic",
"start_line": 12,
"end_line": 61,
"range_sha256": "d5676597e60dda3f72907de8d4a468f0891f32cc209d65d1300a1b8094abd868",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 61,
"scope_boundary": "Setting, definition F2 and forward theorem statement F3; not an earlier proved premise"
},
{
"id": "OA-FLOW.L116.DIRECT",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.direct",
"start_line": 64,
"end_line": 104,
"range_sha256": "a6ab664d27eedab90846d9901ed2224f4c0b3196d6795666e9df63f4f5d90eb2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.NR.3",
"OA-FLOW.BD.1",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 104,
"scope_boundary": "Full fixed-center normal isomorphism, both inclusions"
},
{
"id": "OA-FLOW.CONNES.FACTORFORWARD",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.connes.factorforward",
"start_line": 107,
"end_line": 147,
"range_sha256": "accdf2f4b15d8bfeeb3b7ff43454c21e0424085d3ac701b676f216933e474bc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.DIRECT",
"OA-FLOW.CONNES.CENTERKERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 147,
"scope_boundary": "Both implications at complete conditional kernel; reverse scope separately retained"
},
{
"id": "OA-FLOW.L116.TENSORCENTER",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.tensorcenter",
"start_line": 150,
"end_line": 169,
"range_sha256": "8419f951933de1de1d51af78521c9fb951ca44cb80cdc3d296a7e256a0385ebe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.CP.6",
"OA-FLOW.HR.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 169,
"scope_boundary": "Arbitrary-basis tensor center, full normal amplification/inverse, L2G nonzero"
},
{
"id": "OA-FLOW.CONNES.BIDUALCENTER",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.connes.bidualcenter",
"start_line": 172,
"end_line": 257,
"range_sha256": "d1129870d7ca8d1e1a15116a4abfd5b9657e7d16beb0074cbd649968788429e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.DIRECT",
"OA-FLOW.L116.TENSORCENTER",
"OA-FLOW.NR.3",
"OA-FLOW.DA.ACTION",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 257,
"scope_boundary": "Full F7–15 surviving-action alternative; orbit constant only for invariant c"
},
{
"id": "OA-FLOW.L116.TRANSLATION",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.translation",
"start_line": 260,
"end_line": 327,
"range_sha256": "6a08b651610310a209d47d8800df4e2da38344e66201a664831686825676047e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.TENSORCENTER",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.AT.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 327,
"scope_boundary": "Arbitrary LCA S unitary on full L2G tensor, all multipliers, whole crossing and normal inverse"
},
{
"id": "OA-FLOW.L116.TWO_COMPONENTS",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.two_components",
"start_line": 330,
"end_line": 366,
"range_sha256": "9bb331fc35818c3bade2a90a6cab7ca2e2de9fbc91853e874c1b13fafbd4bf0f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.TRANSLATION",
"OA-FLOW.L116.TENSORCENTER",
"OA-FLOW.CONNES.CENTERKERNEL",
"OA-FLOW.ND.WEYL",
"OA-FLOW.NR.3",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 366,
"scope_boundary": "Whole doubled crossing, fullGamma/nonfactor and invariant center for arbitrary LCA"
},
{
"id": "OA-FLOW.L116.MODELS",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.models",
"start_line": 369,
"end_line": 381,
"range_sha256": "a690d126f1a81d162aadd61a078fa47f323521f138993b95af113c41150d72b6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.TRANSLATION",
"OA-FLOW.L116.TWO_COMPONENTS",
"OA-FLOW.L116.TENSORCENTER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 381,
"scope_boundary": "Exact Z/3 illustration and two solved exercises"
},
{
"id": "OA-FLOW.L116.SOURCES",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.sources",
"start_line": 384,
"end_line": 389,
"range_sha256": "03ae3371b1cb36f069eec11e06dce2c9528b5fe43e11611335b6c1a224ac33ab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 389,
"scope_boundary": "Source comparison and conclusion; not an earlier theorem premise"
},
{
"id": "OA-FLOW.CONNES.FACTORREVERSE",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.connes.factorreverse",
"start_line": 107,
"end_line": 147,
"range_sha256": "accdf2f4b15d8bfeeb3b7ff43454c21e0424085d3ac701b676f216933e474bc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.CENTERKERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 147,
"scope_boundary": "Factor=>fullGamma, without central ergodicity; full section context retained"
},
{
"id": "OA-FLOW.L116.FIGURE",
"unit": "OA-FLOW-L116",
"source_file": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a",
"reader": "OA-FLOW-L116.html#oa-flow.l116.figure",
"start_line": 393,
"end_line": 405,
"range_sha256": "542da8b6e260d42eeaba38055972e926f304d594adafa332f697f1d3599640d9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L116.DIRECT",
"OA-FLOW.CONNES.FACTORFORWARD",
"OA-FLOW.L116.TRANSLATION",
"OA-FLOW.L116.TWO_COMPONENTS",
"OA-FLOW.L116.MODELS"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 405
},
{
"id": "OA-FLOW.FCT.SETTING",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.setting",
"start_line": 4,
"end_line": 24,
"range_sha256": "a661d495148cf7373c5b5a78e58d16ec28cb04df6c84a2a3d8b9d6e2011d0920",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 24
},
{
"id": "OA-FLOW.FCT.0",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.0",
"start_line": 29,
"end_line": 73,
"range_sha256": "8ae65a9602d3559c88eb9bf9af9e80321d55d0c11b18474d8031823458732238",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 73
},
{
"id": "OA-FLOW.FCT.1",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.1",
"start_line": 79,
"end_line": 141,
"range_sha256": "4c637a91f0f5099f0fcf235fbad88e1af301727cd2d2f63feee73f904aaa7e99",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.3",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.PC.2",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 141
},
{
"id": "OA-FLOW.FCT.2",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.2",
"start_line": 147,
"end_line": 161,
"range_sha256": "ac92321b70fbc3902b434299ceb437c34f96b9ba3f138ccfcf3746a45cfdbd23",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 161
},
{
"id": "OA-FLOW.FCT.3",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.3",
"start_line": 167,
"end_line": 205,
"range_sha256": "04037b514bf090c1070d86f7991dfac805df48b0e8abda300fb1bbf713b6bfcd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.2",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 205
},
{
"id": "OA-FLOW.FCT.4",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.4",
"start_line": 211,
"end_line": 266,
"range_sha256": "dfe7417c7f00bbba6c0cd31398f7c2040f8e3fc732590c86d27542f36aa16c7b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1",
"OA-FLOW.PC.7",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 266
},
{
"id": "OA-FLOW.FCT.5",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.5",
"start_line": 272,
"end_line": 340,
"range_sha256": "8f50f6165753b3bc982f1fbbff5eae872c4ee201a6fab0a9b9b6ceb7034d806e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 340
},
{
"id": "OA-FLOW.FCT.6",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.6",
"start_line": 346,
"end_line": 392,
"range_sha256": "d3bd74db2bbb8da426d3ed5b28be4ca72003b3769371f88f704277d325e69d33",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 392
},
{
"id": "OA-FLOW.FCT.7",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.7",
"start_line": 398,
"end_line": 425,
"range_sha256": "12da6a4189a23c26e53e537c910dbe4080db12b58c0239dff02ec7bad9aff650",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.6",
"OA-FLOW.CF.6",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.SF.SB0",
"OA-FLOW.SF.SB1",
"OA-FLOW.SF.SB2",
"OA-FLOW.SF.SB3",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB5",
"OA-FLOW.SF.SB6",
"OA-FLOW.SF.SF0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 425,
"scope_boundary": "Faithful normalized normal trace; uniqueness among normal traces only. The converse existence assertion expressly concerns an arbitrary von Neumann algebra."
},
{
"id": "OA-FLOW.FCT.8",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.8",
"start_line": 431,
"end_line": 456,
"range_sha256": "214e66eaa355bcd078523bf2342873750b6838aca85e5740aa63eea61c052431",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.CF.6",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.NF.6",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 456
},
{
"id": "OA-FLOW.FCT.9",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.9",
"start_line": 462,
"end_line": 503,
"range_sha256": "da5c5d01866c27a88c2b6ff2af0474821523fd97b252e481dcdc0ebd8a70e4aa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.7",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 503
},
{
"id": "OA-FLOW.FCT.SOURCES",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.sources",
"start_line": 509,
"end_line": 515,
"range_sha256": "544b2ae08c0a741cf326846bec20b0e91d24d80f73ea09add28cfec63d5b27bb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 515
},
{
"id": "OA-FLOW.FCT.FIGURE",
"unit": "OA-FLOW-FCT",
"source_file": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda",
"reader": "OA-FLOW-FCT.html#oa-flow.fct.figure",
"start_line": 519,
"end_line": 572,
"range_sha256": "7ad5b8d94f7bc08b57ef1ad050a81b8e07aa6cc70f00fe77158f5946e6e2f2e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.FCT.9"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 572
},
{
"id": "OA-FLOW.L119.0",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.0",
"start_line": 4,
"end_line": 9,
"range_sha256": "00cc8d0a0a15348a51fc29dc237cef001e7cca5ed939412382f50d498f42b47a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 9,
"source_level_hypothesis_ranges": [],
"scope_boundary": "Setting and prerequisites; not an earlier proved premise"
},
{
"id": "OA-FLOW.L119.SETTING",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.setting",
"start_line": 12,
"end_line": 20,
"range_sha256": "44154ea902e00ee92abc6227b3e131d1c38ac43355533a732474f8afeff3be8e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 20,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Global setting, exact hypotheses and input announcements; not an earlier theorem"
},
{
"id": "OA-FLOW.L119.CONVENTIONS",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.conventions",
"start_line": 23,
"end_line": 60,
"range_sha256": "0b0f613331576d9ab5c6ab2722f7e83eb27289cd50bbbd2e399b4c8022b7214f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.ISOM.RIGIDITY",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.AF.0",
"OA-FLOW.CF.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QF.1",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.L118.DUAL-MAP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 60,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Positive convention, fixed-algebra closure, exact kernel detection and full double annihilator"
},
{
"id": "OA-FLOW.L119.CANCELLATION",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.cancellation",
"start_line": 63,
"end_line": 152,
"range_sha256": "06760b3e86086338ec4e0beba82ecabd6fee269eebdc5a3f0f610fcc679d9801",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.REPEXT.GROUP",
"OA-FLOW.CP.6",
"OA-FLOW.AT.1",
"OA-FLOW.QF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 152,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "General fixed-valued subgroup cancellation, whole continuity, quotient descent, spectral inclusion and a worked problem"
},
{
"id": "OA-FLOW.L119.REGULARCORNER",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.regularcorner",
"start_line": 155,
"end_line": 203,
"range_sha256": "fbd8f1f716b8ea088550b1180285b6cc40872baa2c5ceafbf4bd1aa7bd849bae",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.CP.6",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.1",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.DA.ACTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 203,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "All fixed projection regular corners, full normal inverse and exact dual equivariance"
},
{
"id": "OA-FLOW.MINFIX.FULLCONNES",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.minfix.fullconnes",
"start_line": 206,
"end_line": 251,
"range_sha256": "f53398acd43e5a394278fe51a94351d898592ca950720f7c5d6989ef8f023b59",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.REGULARCORNER",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CONNES.CENTERKERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 251,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Full ambient central support passes to crossing; full normal center equivalence; full-corner Gamma equality"
},
{
"id": "OA-FLOW.MINFIX.ANNIHILATOR",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.minfix.annihilator",
"start_line": 254,
"end_line": 306,
"range_sha256": "c15a3eef862fb3edd3403fd9abf0cf9bcdcdeed9afb64180bd8cf77e74f2ec72",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.MINFIX.FULLCONNES",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 306,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Minimal fixed corner, central fullness, exact reduced spectrum and identity action on its annihilator"
},
{
"id": "OA-FLOW.MINFIX.FIXEDREP",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.minfix.fixedrep",
"start_line": 309,
"end_line": 362,
"range_sha256": "58f95d444b046972b67cb48688374a51a6b58faeafa328cad814a95836856998",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MINFIX.ANNIHILATOR",
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.FULLCORNER.CONTINUITY",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 362,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Prescribed full-corner strong lift, actual subgroup representation and whole-group fixed values"
},
{
"id": "OA-FLOW.MINFIX.CANCEL",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.minfix.cancel",
"start_line": 365,
"end_line": 408,
"range_sha256": "2d544aac0f9924abbdd404c9d4fce5ce3825104f2237099fa3c731d76886998b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.L119.CANCELLATION",
"OA-FLOW.MINFIX.FIXEDREP",
"OA-FLOW.CONNES.CENTERKERNEL",
"OA-FLOW.COCMAT.ACTION",
"OA-FLOW.COCMAT.CORNERS",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 408,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Full theorem: strongly continuous M-valued cocycle and exact action spectrum/Gamma/kernel annihilator"
},
{
"id": "OA-FLOW.L119.MODEL",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.model",
"start_line": 411,
"end_line": 488,
"range_sha256": "d46541ad85a3f98af2c7a3ea7305efb38dd90d5cf1d403763f1730b6659030b7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.MINFIX.CANCEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 488,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Exact integer action on two M2 summands, every fixed projection and both complete spectra/kernels; second solved exercise"
},
{
"id": "OA-FLOW.L119.SOURCES",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.sources",
"start_line": 491,
"end_line": 496,
"range_sha256": "f3824904debd04a7df49fac118e7f7dce5330fb03f70a2ea5eeb6ef3c57b9e9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 496,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L119.md",
"start_line": 15,
"end_line": 15,
"range_sha256": "03f1960c6e3171d4a1bb9e3f51631c31552a100a2fc19b369833b575c8e724d3",
"full_statement_context": "Unless a local lemma says otherwise, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), \\(M\\ne0\\) is a von Neumann algebra faithfully normally represented on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is an action by normal unital automorphisms which is point-ultraweakly continuous. No separability, sigma-compactness, countable basis, sigma-finite algebra or faithful state is assumed. Put \\(F=M^\\alpha\\). In the main application \\(Z(M)^\\alpha=\\mathbb C1\\) and \\(e\\) is a nonzero minimal projection of \\(F\\). Minimal means that its only subprojections in \\(F\\) are zero and itself. This does not mean that \\(e\\) is minimal in \\(M\\).\n",
"usage": "Complete global setting applies subject to the local lemma hypotheses. Setting/context is not an earlier proved premise."
}
],
"scope_boundary": "Source and qualified completion context, never an earlier theorem premise"
},
{
"id": "OA-FLOW.L119.FIGURE",
"unit": "OA-FLOW-L119",
"source_file": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12",
"reader": "OA-FLOW-L119.html#oa-flow.l119.figure",
"start_line": 500,
"end_line": 510,
"range_sha256": "de217feb388d42135375f5ef3c1eec2144da7d93e66763c804d62e6e9b5dc92e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L119.MODEL",
"OA-FLOW.MINFIX.CANCEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 510
},
{
"id": "OA-FLOW.L120.SETTING",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.setting",
"start_line": 9,
"end_line": 19,
"range_sha256": "65cd5741d736249599bc680680aa0ca8ffebc179c35ea1719031a66cec91a388",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 19,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Declared arbitrary algebra/group setting and exact input convention; never a proved premise"
},
{
"id": "OA-FLOW.L120.TRANSPORT",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.transport",
"start_line": 22,
"end_line": 98,
"range_sha256": "581e8db6c1197cf6675ab11b66338428026365d00cd0c9bc0b2258978d64e433",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.AT.5",
"OA-FLOW.GL.7",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.FULLCORNER.CONTINUITY",
"OA-FLOW.FULLCORNER.COCYCLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 98,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Entire normal amplification, point-ultraweak continuity, filter ideal equality, normal conjugacy, fixed-first-corner lift and corner monotonicity"
},
{
"id": "OA-FLOW.COCINT.EQUIVATOMIC",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.equivatomic",
"start_line": 101,
"end_line": 184,
"range_sha256": "f849a652d654bc984f0b33765962530b09ccb316b7810dbbacd1b3f4d848d1f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.AT.1",
"OA-FLOW.CP.6",
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.L119.CANCELLATION",
"OA-FLOW.L119.REGULARCORNER",
"OA-FLOW.MINFIX.FULLCONNES",
"OA-FLOW.MINFIX.ANNIHILATOR",
"OA-FLOW.MINFIX.FIXEDREP",
"OA-FLOW.MINFIX.CANCEL",
"OA-FLOW.L119.MODEL",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Explicit equivalent-corner coboundary with continuity, exact spectra and all C6–11 identities; complete atomic alternative at the same full private L119 input"
},
{
"id": "OA-FLOW.L120.ATOMIC",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.atomic",
"start_line": 170,
"end_line": 184,
"range_sha256": "419708688c2d3207e455502aabf37442c64c6cf85b820f75e02d833bd0d01fa0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.L119.CONVENTIONS",
"OA-FLOW.L119.CANCELLATION",
"OA-FLOW.L119.REGULARCORNER",
"OA-FLOW.MINFIX.FULLCONNES",
"OA-FLOW.MINFIX.ANNIHILATOR",
"OA-FLOW.MINFIX.FIXEDREP",
"OA-FLOW.MINFIX.CANCEL",
"OA-FLOW.L119.MODEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Minimal fixed projection argument in L119, valid in finite and infinite M"
},
{
"id": "OA-FLOW.L120.DIFFUSE-SPLITTING",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.diffuse-splitting",
"start_line": 187,
"end_line": 197,
"range_sha256": "493d084317c4674e46889c3477dc39cd80364dadf7201025162e45c5b0c56136",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 197,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Every desired trace value in a diffuse algebra; full small-projection and maximal-family proof"
},
{
"id": "OA-FLOW.COCINT.FINITE",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.finite",
"start_line": 200,
"end_line": 226,
"range_sha256": "52b4e655fd74b0b65e0005008479adfc4e22fa5024605780f82514e60dd46f75",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.DIFFUSE-SPLITTING",
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.FCT.9",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 226,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Scalar faithful normal trace on the fixed algebra, full small fixed projection and exact equal finite corner tiling"
},
{
"id": "OA-FLOW.COCINT.MATRIX",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.matrix",
"start_line": 229,
"end_line": 366,
"range_sha256": "e9734425ab07ece69ea9d390c85924e500244be0b5f7efcc6d210d34cd6afd2c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COCINT.FINITE",
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.CP.6",
"OA-FLOW.NCF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 366,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Correctly oriented finite matrix units, unitary identification, normal inverse, exact first-corner action and continuous cocycle"
},
{
"id": "OA-FLOW.L120.CARDINAL-ARITHMETIC",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.cardinal-arithmetic",
"start_line": 369,
"end_line": 400,
"range_sha256": "dc29ac35a235b6545008bd2d2b729812ab415ae52da60d2b300e4e5d875377f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.PC.0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 400,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Complete least-failing-cardinal square/product, countable multiplication and block partition"
},
{
"id": "OA-FLOW.L120.RESIDUAL-ABSORPTION",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.residual-absorption",
"start_line": 403,
"end_line": 426,
"range_sha256": "199ecd79e1b1704a069c37eb4da37fa448ebdf99e802b1a5f7cfaf66322440b5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 426,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Residual absorption after comparison, without claiming pointwise preservation of transported members"
},
{
"id": "OA-FLOW.L120.HOMOGENEOUS-FILLING",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.homogeneous-filling",
"start_line": 429,
"end_line": 448,
"range_sha256": "8a9d1afe64aa8e3096f047bfaf81ed12288615fb85857e2fc217f661e9729ec2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.RESIDUAL-ABSORPTION",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 448,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Maximal enlarged homogeneous family, nonzero comparison piece and exhaustive central assembly; initial cardinal may increase"
},
{
"id": "OA-FLOW.L120.FULL-SIGMA-FINITE-CORNER",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.full-sigma-finite-corner",
"start_line": 451,
"end_line": 487,
"range_sha256": "ed261b4eb5e97d8db387a4b4d206efc3b64c1da1197653cb3c4eaabb3ba13441",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.NF.1",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 487,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Full state-support corner on sigma-finite center and explicit full properly infinite sigma-finite corner"
},
{
"id": "OA-FLOW.L120.COVERING-BOUND",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.covering-bound",
"start_line": 490,
"end_line": 516,
"range_sha256": "f33b49fd7f79b9e4248c5a78a7e6ed17305b9d62eef5ace68ebf22a75e3c1889",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.CARDINAL-ARITHMETIC",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 516,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Intrinsic cardinal bound from detecting corner states, with every nonzero central restriction"
},
{
"id": "OA-FLOW.L120.CENTRAL-CARDINAL-PARTITION",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.central-cardinal-partition",
"start_line": 519,
"end_line": 549,
"range_sha256": "8c48ec41c38b3321f1a6147d1998e343fd839a762c34c37d8b531f568d3e66ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.FULL-SIGMA-FINITE-CORNER",
"OA-FLOW.L120.HOMOGENEOUS-FILLING",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.CF.1",
"OA-FLOW.NF.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.5",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 549,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Existence on arbitrary center, grouping equal cardinals and every sigma-finite central subpiece, weighted states and central equivalences"
},
{
"id": "OA-FLOW.L120.CARDINAL-CLASSIFICATION",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.cardinal-classification",
"start_line": 552,
"end_line": 570,
"range_sha256": "054663f2f5187994ae9e962c527132cace45f5fb84b94c31fe2e540f82bc8a54",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.CENTRAL-CARDINAL-PARTITION",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.PC.1",
"OA-FLOW.PC.3",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 570,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Unique intrinsic partition, automorphism invariance and equivalence of full properly infinite projections with the same partition"
},
{
"id": "OA-FLOW.L120.FIXED-COMPLEMENTS",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.fixed-complements",
"start_line": 573,
"end_line": 657,
"range_sha256": "ac98b60cd0e2a2a575e56dabfbb1b04e266cacecef11f8a81e2c38f17251765b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.CARDINAL-CLASSIFICATION",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.L120.CARDINAL-ARITHMETIC",
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.FCT.9",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 657,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Ambient fullness, ergodic single cardinal, finite fixed complement and smaller infinite fixed complement of full unit cardinal"
},
{
"id": "OA-FLOW.COCINT.INFINITE",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.infinite",
"start_line": 660,
"end_line": 757,
"range_sha256": "84cab0bf377253a3d9f8029d71259df66dc1300d4b7e46c1a5aec2c1a4e94455",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.FIXED-COMPLEMENTS",
"OA-FLOW.L120.CARDINAL-CLASSIFICATION",
"OA-FLOW.L120.CARDINAL-ARITHMETIC",
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.FCT.9",
"OA-FLOW.NCF.1",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 757,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Both amplification complements are full of identical local cardinal; exact prescribed first corner v=r+v0, whole normal map and spectral cocycle construction"
},
{
"id": "OA-FLOW.COCINT.REDUCTION",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.reduction",
"start_line": 760,
"end_line": 833,
"range_sha256": "dd9911c277d505bbf70ebe25954150c850c9073cc8ebb4cebfa64ff0c524712a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COCINT.EQUIVATOMIC",
"OA-FLOW.L120.ATOMIC",
"OA-FLOW.COCINT.MATRIX",
"OA-FLOW.COCINT.INFINITE",
"OA-FLOW.PC.5",
"OA-FLOW.COCMAT.CORNERS",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 833,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Complete cornerwise construction and both cocycle-intersection inclusions, all arbitrary-cardinal cases"
},
{
"id": "OA-FLOW.L120.PROBLEM",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.problem",
"start_line": 836,
"end_line": 842,
"range_sha256": "d59e4260504fd9df956cb24a3f26d01647becc73e6af4f10730f8358c3e9935b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COCINT.MATRIX"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 842,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Complete retained matrix-unit orientation problem and exact original solution"
},
{
"id": "OA-FLOW.L120.MODEL",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.model",
"start_line": 845,
"end_line": 887,
"range_sha256": "c7e523e78471bea471d48af8bebadcebc5e4b74eacfd8f73aafccc8309ab5535",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.6",
"OA-FLOW.PC.0",
"OA-FLOW.NCF.1",
"OA-FLOW.GL.7",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 887,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Whole arbitrary-I diagonal action/cocycle and exact first-corner model, finite integer-matrix restriction and correctly signed spectra"
},
{
"id": "OA-FLOW.L120.SOURCES",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.sources",
"start_line": 890,
"end_line": 895,
"range_sha256": "6f5a83b6bfdcb6266e342f7dba61e848ec52ffed0c47ff23c41de00e2ae5e47f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 895,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Bounded actual source/context comparison, never an earlier proved premise"
},
{
"id": "OA-FLOW.COCINT.SIGMAFINITE",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.cocint.sigmafinite",
"start_line": 760,
"end_line": 833,
"range_sha256": "dd9911c277d505bbf70ebe25954150c850c9073cc8ebb4cebfa64ff0c524712a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COCINT.EQUIVATOMIC",
"OA-FLOW.L120.ATOMIC",
"OA-FLOW.COCINT.MATRIX",
"OA-FLOW.COCINT.INFINITE",
"OA-FLOW.PC.5",
"OA-FLOW.COCMAT.CORNERS",
"OA-FLOW.GCC.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 833,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L120.md",
"start_line": 12,
"end_line": 12,
"range_sha256": "3bb76bd3c1a44636d34b28ab5f18a07dc8ef3f3cce6a1fed3ea8d5b5c5b28fc5",
"full_statement_context": "Throughout the main theorem, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on a Hilbert space \\(\\mathcal H\\), \\(G\\) is an arbitrary locally compact Hausdorff abelian group, and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal unital star automorphisms. Assume \\(Z(M)^\\alpha=\\mathbb C1\\), and put \\(F=M^\\alpha\\). No separability, countable basis, sigma-compact group, sigma-finite algebra or faithful normal scalar state is assumed. Local projection lemmas expressly allow general algebras and dispense with this action. A sigma-finite projection means that its corner has a faithful normal state, as proved equivalent to countable decomposability in PC7. A sigma-finite central piece refers to its **center**, and need not make the whole piece sigma-finite.\n",
"usage": "Complete setting/context with locally stated qualifications; never an earlier proved premise."
}
],
"scope_boundary": "Legacy SIGMAFINITE identifier retained for the full unrestricted cocycle-intersection theorem; no countability hypothesis"
},
{
"id": "OA-FLOW.L120.FIGURE",
"unit": "OA-FLOW-L120",
"source_file": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b",
"reader": "OA-FLOW-L120.html#oa-flow.l120.figure",
"start_line": 899,
"end_line": 911,
"range_sha256": "150f84dcc712fcf2be6007f91f1df7cc3070d487e99fc2812a0e8110bf4f9970",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L120.MODEL",
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.COCINT.INFINITE",
"OA-FLOW.L120.CARDINAL-CLASSIFICATION"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 911
},
{
"id": "OA-FLOW.L121.SETTING",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.setting",
"start_line": 7,
"end_line": 22,
"range_sha256": "03a3b0414e5acea5d4644e542fb4e77e2a9d05232e537db4e921ce631b3fddb0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 22,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Global setting and announced complete earlier proofs; not a theorem premise"
},
{
"id": "OA-FLOW.L121.CONVENTIONS",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.conventions",
"start_line": 23,
"end_line": 29,
"range_sha256": "242bd642ef754d9c71e2e04b41522fcc7cee4184cdb0fc7e43da30e3d7f00d7d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 29,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Exact positive-label notation; context, not a separate proved result"
},
{
"id": "OA-FLOW.L121.SC0",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sc0",
"start_line": 30,
"end_line": 46,
"range_sha256": "148f7e3667a07aa8091fb2afee9c49da5f728f944c46a1ce9a5de556c2f3eedd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GL.6",
"OA-FLOW.GL.2",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.XGAPS.CSTAR.H0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 46,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Compact-plus-closed sums and exact two-step spectral sandwich bound"
},
{
"id": "OA-FLOW.L121.SC1",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sc1",
"start_line": 47,
"end_line": 64,
"range_sha256": "f2ca6f32629dcfd5e2db6d58fb812bb4b4feb9291640769c66aca5ca6292c2ef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 64,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Arbitrary family range detection in both corner orientations"
},
{
"id": "OA-FLOW.L121.SC2",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sc2",
"start_line": 65,
"end_line": 91,
"range_sha256": "e5ce576c3ee2d4b2e9eb68910395b6d0e3f723b5aab25129ae164a230cac1644",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L121.SC0",
"OA-FLOW.L121.SC1",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.7",
"OA-FLOW.L115.CONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 91,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "General compact-frequency family comparison; no central ergodicity"
},
{
"id": "OA-FLOW.L121.SC3",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sc3",
"start_line": 92,
"end_line": 347,
"range_sha256": "3ee79536ef78ec6e1ab12893221085787820c1eaedb2ca3a90de2d2014b86ffc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L121.SC0",
"OA-FLOW.L121.SC1",
"OA-FLOW.L121.SC2",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.TOOLS",
"OA-FLOW.PC.1",
"OA-FLOW.GL.1",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.GL.6",
"OA-FLOW.GL.7",
"OA-FLOW.PC.2",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.LF.1",
"OA-FLOW.CP.6",
"OA-FLOW.GL.0",
"OA-FLOW.XGAPS.CSTAR.H0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 347,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Complete centrally ergodic mutual-corner construction; all D2–D23"
},
{
"id": "OA-FLOW.MUTUAL.LOCALIZE",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.mutual.localize",
"start_line": 97,
"end_line": 158,
"range_sha256": "f6dfac573e4f891aeabd9cdaea7dca60e073259034f292a78f3e7b0b7d3c7c7e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.2",
"OA-FLOW.GL.0",
"OA-FLOW.GL.2",
"OA-FLOW.GL.3",
"OA-FLOW.LF.1",
"OA-FLOW.CP.6",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.L115.CONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 158,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Nonzero bridge with compact Fourier support K and K−K inside prescribed neighborhood"
},
{
"id": "OA-FLOW.MUTUAL.SUPPORTS",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.mutual.supports",
"start_line": 159,
"end_line": 218,
"range_sha256": "9641ad7596996be2f7dfdeccd4de6d097567ee2be0235580ab7b1e14341c9c74",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MUTUAL.LOCALIZE",
"OA-FLOW.L121.SC2",
"OA-FLOW.PC.1",
"OA-FLOW.GCC.TOOLS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 218,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Actual two nonzero central fixed support carriers subordinate to e1,e2"
},
{
"id": "OA-FLOW.MUTUAL.SANDWICH",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.mutual.sandwich",
"start_line": 219,
"end_line": 286,
"range_sha256": "4771c9cc9fae41a5bc92672150df5653db764d06de4b55c6040f3a78a2672c40",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MUTUAL.SUPPORTS",
"OA-FLOW.L121.SC0",
"OA-FLOW.L121.SC1",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GL.2",
"OA-FLOW.GL.6",
"OA-FLOW.GL.7",
"OA-FLOW.L115.CONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 286,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Nonzero sandwich and exact D18 product estimate from compact actual outer spectra"
},
{
"id": "OA-FLOW.MUTUAL.TWOWAY",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.mutual.twoway",
"start_line": 287,
"end_line": 347,
"range_sha256": "8d302bf4579959c72fc683d20e9b57e986bff5fe5f927e9f3f625367160121c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MUTUAL.SANDWICH",
"OA-FLOW.MUTUAL.SUPPORTS",
"OA-FLOW.L121.SC0",
"OA-FLOW.GL.2",
"OA-FLOW.GL.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 347,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Two spectral inclusions for the same pair of carriers and result D23"
},
{
"id": "OA-FLOW.DIRECTED.THICKENING",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.directed.thickening",
"start_line": 348,
"end_line": 430,
"range_sha256": "15f6a5c48aa5bb1a9a200e9d94b03e818e4eaaf27bf453de510fe89bbafa06b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L121.SC3",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GL.7",
"OA-FLOW.L115.CONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 430,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 95,
"end_line": 95,
"range_sha256": "d57f60f4bbc22026ed87865a21828bb2bc3a104bedd675a8eaa2bce2af76ed5b",
"full_statement_context": "Assume now \\(Z(M)^\\alpha=\\mathbb C1\\). We construct a family satisfying SC2 for any prescribed nonzero central fixed corners and any identity neighborhood. The retained equations (D2)–(D23) make every step explicit.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
},
{
"source": "src/OA-FLOW-L121.md",
"start_line": 100,
"end_line": 100,
"range_sha256": "0bf2e23d42f02ce3a67ecb105592f9f1cc6114db9b4bb61b0247bdffcf1a32a3",
"full_statement_context": "Fix nonzero \\(e_1,e_2\\in\\operatorname{Proj}(D)\\) and a neighborhood \\(U\\) of \\(0\\) in \\(H\\). The ambient central support \\(z_M(e_i)\\) is fixed by \\(\\alpha\\): fixedness of \\(e_i\\) and uniqueness of central support give\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "One actual member refines any two thickenings; arbitrary LCA directedness SC4"
},
{
"id": "OA-FLOW.L121.SC5",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sc5",
"start_line": 431,
"end_line": 455,
"range_sha256": "57ba17a9c076bb12a70e17e297c424819d08dcdb739e693684b7aab33ff82c77",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L121.SC2",
"OA-FLOW.DIRECTED.THICKENING",
"OA-FLOW.GL.7",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 455,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Worked problem: exact M2 action, supports, frequencies, sandwich and intervals"
},
{
"id": "OA-FLOW.L121.SOURCES",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.sources",
"start_line": 456,
"end_line": 463,
"range_sha256": "43a2f3f6a03e4ede8ca88d9823dc421d7d6a7c918fc3d44450396205ffbb391d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 463,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L121.md",
"start_line": 10,
"end_line": 10,
"range_sha256": "16080934b8ee6572dcec8f6800867622b668a7c7a24972a19b37ef61c90177b3",
"full_statement_context": "Throughout, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\) has its additive dual-group law, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary Hilbert space \\(\\mathcal H\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is a point-ultraweakly continuous action by normal automorphisms. Put \\(F=M^\\alpha\\). No separability, countability, sigma-finiteness or factor hypothesis is assumed. SC0–SC2 require no central ergodicity; SC3–SC4 additionally assume \\(Z(M)^\\alpha=\\mathbb C1\\), exactly where stated.\n",
"usage": "Exact source-level context applies subject to the local statement. Context is not an earlier proved premise."
}
],
"scope_boundary": "Mathematical/source provenance and bounded completion notice; not a proof premise"
},
{
"id": "OA-FLOW.L121.FIGURE",
"unit": "OA-FLOW-L121",
"source_file": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34",
"reader": "OA-FLOW-L121.html#oa-flow.l121.figure",
"start_line": 465,
"end_line": 475,
"range_sha256": "9c9eed8607d8777c73c93086a4b2f265f97f4f63d6e798fb6e13f9b47661a77e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L121.SC1",
"OA-FLOW.L121.SC2",
"OA-FLOW.L121.SC3",
"OA-FLOW.DIRECTED.THICKENING",
"OA-FLOW.L121.SC5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 475
},
{
"id": "OA-FLOW.L122.SETTING",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.setting",
"start_line": 4,
"end_line": 14,
"range_sha256": "eaf1add303297417eb752faf118573f00f57312f339390f61b8c90ae7b4cca06",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full arbitrary nonzero M, arbitrary H/LCH abelian G, normal point-ultraweak action and two distinct spectral domains"
},
{
"id": "OA-FLOW.L122.INPUTS",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.inputs",
"start_line": 17,
"end_line": 23,
"range_sha256": "1102f3a029613d4ee77630c21c4e44a854b2238e6f31ed0a818d80dd829f40df",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Reading locators and exact input convention; not a mathematical premise"
},
{
"id": "OA-FLOW.L122.SUPPORT",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.support",
"start_line": 26,
"end_line": 74,
"range_sha256": "af992dc70b7bd9c867b1a1802d9ad8ae486509a956148b9e59ff86bc5f357edd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.2",
"OA-FLOW.CF.6",
"OA-FLOW.PC.1",
"OA-FLOW.GCC.TOOLS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 74,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full continuous-calculus construction, norm estimate, corner spectrum, open support strong limit and invariance"
},
{
"id": "OA-FLOW.L122.DIRECTANNIHILATOR",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.directannihilator",
"start_line": 77,
"end_line": 131,
"range_sha256": "7786413924ee4fbeebbef7a817ceefd92f6514d1b4144b37815ed7cb45b62a5b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.SUPPORT",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.AT.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 131,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Direct operator-norm disc argument, actual specified predual continuity and annihilator conclusion without product theorem"
},
{
"id": "OA-FLOW.L122.PRODUCT",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.product",
"start_line": 134,
"end_line": 156,
"range_sha256": "09ea192b2d0afd7547ce73a953172ae9dfc9032a70bb22744e62cc7a7160cf85",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 156,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Complete maximal commutative closed inverse-closed Banach algebra proof of commuting product inclusion"
},
{
"id": "OA-FLOW.INNERSP.MULTIPLIERS",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.innersp.multipliers",
"start_line": 159,
"end_line": 232,
"range_sha256": "4f2c9fe318907fb00cf27b32855cc613e78d0245afb9a349e2e009eba3a4c6fb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.PRODUCT",
"OA-FLOW.CF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 232,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Exact left/right multiplication spectra, adjoint reflection and universal Ad quotient bound"
},
{
"id": "OA-FLOW.INNERSP.FACTOR",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.innersp.factor",
"start_line": 235,
"end_line": 308,
"range_sha256": "10d7ad0631733861acb8d81148119c3963facf3022f57eb344fac6693c6a6d76",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.SUPPORT",
"OA-FLOW.INNERSP.MULTIPLIERS",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 308,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Factor central supports, bounded polar nonzero bridge and approximate-eigenvector reverse inclusion"
},
{
"id": "OA-FLOW.INNERSP.CORNER",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.innersp.corner",
"start_line": 311,
"end_line": 356,
"range_sha256": "ab1585a0b29d47363bb021b3ac5149971f3a201d7797da7f1b0d472cf6aa34a7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.SUPPORT",
"OA-FLOW.INNERSP.MULTIPLIERS",
"OA-FLOW.GCC.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 356,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained I11-14 constructed nonzero fixed corner, compressed spectrum and quotient disc bound"
},
{
"id": "OA-FLOW.INNERSP.ANNIHILATOR",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.innersp.annihilator",
"start_line": 359,
"end_line": 412,
"range_sha256": "eba37b9f1fde616184b88c617182b8b02edbd51e2858d17f8d9f7c370f30bbd7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERSP.CORNER",
"OA-FLOW.L122.DIRECTANNIHILATOR",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.L115.CONVENTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 412,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained I15-19 complete OPSP/Connes route with actual earlier predual context"
},
{
"id": "OA-FLOW.L122.PROBLEM",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.problem",
"start_line": 415,
"end_line": 423,
"range_sha256": "0d22b2b3ed948349a33783a912882be93295690ec8e68ab4580d35f1119ea97e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERSP.MULTIPLIERS",
"OA-FLOW.INNERSP.FACTOR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 423,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Exact original solved nonfactor example and complete finite matrix factor contrast"
},
{
"id": "OA-FLOW.L122.MODEL",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.model",
"start_line": 426,
"end_line": 459,
"range_sha256": "009ba13e3abb28591c012039593c8b4f19d86d8118fc93917aa0a407aa37d1ae",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.SUPPORT",
"OA-FLOW.L122.DIRECTANNIHILATOR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 459,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Exact rational three-point support and four-dimensional corner conjugation spectrum"
},
{
"id": "OA-FLOW.L122.SOURCES",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.sources",
"start_line": 462,
"end_line": 466,
"range_sha256": "19a31415107c320f031354bb8d18de295bae2f5012f494e725da0238b0cc5173",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 4,
"end_line": 23,
"range_sha256": "f90d60be6f518b38e4d1e0d0339ebf1e01478790713ebec3253c62dbcf5633d5",
"full_statement_context": "\n# Small fixed corners, inner spectra and annihilating times\n\nA globally fixed unitary can implement an action at a time only if every Connes frequency evaluates to one there. The mechanism is a nonzero fixed corner on which conjugation is arbitrarily close to the identity in operator norm. We construct that corner first, so this application does not depend on the later product-spectrum theorem. We then prove the universal quotient bound and explain exactly how a factor supplies the missing lower bound.\n\nThroughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n\n## Exact earlier proof inputs\n\nThe eleven direct earlier scopes are [CF1](OA-FLOW-CF.md#oa-flow.cf.1) (choice and maximality), [CF2](OA-FLOW-CF.md#oa-flow.cf.2) (Banach spectrum and geometric series), [CF4](OA-FLOW-CF.md#oa-flow.cf.4) (characters), [CF6](OA-FLOW-CF.md#oa-flow.cf.6) (isometric continuous calculus and automorphism contractivity), [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1) (corners, supports and polar decomposition), [PC2](OA-FLOW-PC.md#oa-flow.projection.pc2) (central support and nonzero bridges), [GCC SETTING](OA-FLOW-GCC.md#gcc-setting), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [GCC DEFINITION](OA-FLOW-GCC.md#gcc-definition), [L115 CONVENTIONS](OA-FLOW-L115.md#oa-flow.l115.conventions), and [OPSP FORMULA](OA-FLOW-L89.md#oa-flow.opsp.formula).\n\nFor OPSP the scope includes its full source setting and proof O1–O9, the actual [AF0](OA-FLOW-AF.md#af-0) specified-dual hypotheses, and the full [AT1–3](OA-FLOW-AT.md#oa-flow.at.3) bridge. Those contexts are necessary: mere ultraweak continuity must be connected to norm continuity of every orbit in the actual predual. They are complete inherited contexts within the eleven-root closure, not extra direct proof roots. Opening descriptions, source comparisons and illustrations are never proof providers.\n\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L122.md",
"source_sha256": "a278604c4526b8790c06402575ac26cd221d46fd449345427661f66d467bc213",
"start_line": 9,
"end_line": 11,
"range_sha256": "d18f5c27ab6638a21351e3be403901c36349b9eab3e9e4debddb8942ebf72e55",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a von Neumann algebra on an arbitrary complex Hilbert space \\(\\mathcal H\\), and \\(u\\in\\mathcal U(M)\\). For the action statements, \\(G\\) is an arbitrary locally compact Hausdorff abelian group, \\(H=\\widehat G\\), and \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) is point-ultraweakly continuous with normal unital star automorphisms. Write \\(N=M^\\alpha\\). There is no separability, sigma-finiteness, countability, finite-Haar, central-ergodicity or factor hypothesis unless it is explicitly introduced.\n\nThe ordinary complex spectrum in the algebra \\(M\\), or in the Banach algebra of bounded operators on \\(M\\), will be denoted by \\(\\sigma\\) or \\(\\operatorname{Sp}\\) with the algebra shown. The action spectrum is a closed subset of \\(H\\). These are different objects. The positive frequency convention is the exact earlier [L115 convention](OA-FLOW-L115.md#oa-flow.l115.conventions): \\(p(s)=(s,p)\\). The earlier GCC negative-label spectrum reflects vector labels, but adjoint reflection makes each whole corner action spectrum symmetric. Its positive and negative versions, and their Connes intersections, consequently agree. No individual vector spectrum is assumed symmetric.\n",
"usage": "Full global setting and frequency convention, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Bounded source comparison and retained history; never a mathematical proof provider"
},
{
"id": "OA-FLOW.L122.FIGURE",
"unit": "OA-FLOW-L122",
"source_file": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399",
"reader": "OA-FLOW-L122.html#oa-flow.l122.figure",
"start_line": 469,
"end_line": 481,
"range_sha256": "d7769864e3e3fac0d9e4bcc28154a8c328f4099034199e60b71f5cf5eeaa01f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L122.SUPPORT",
"OA-FLOW.L122.DIRECTANNIHILATOR",
"OA-FLOW.INNERSP.FACTOR",
"OA-FLOW.L122.PROBLEM",
"OA-FLOW.L122.MODEL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null
},
{
"id": "OA-FLOW.L123.SETTING",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.setting",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full arbitrary concrete algebra/Hilbert setting and exact earlier inputs; not a theorem premise"
},
{
"id": "OA-FLOW.L123.IC0",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic0",
"start_line": 16,
"end_line": 49,
"range_sha256": "498bbb4cb87522d28f5fa9037a57b9cb23a66969ca6c66ba9a15af3d95c7f528",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.NCF.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.NF.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 49,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Constructed Hilbert tensor, automatic ultraweak normality and both-factor tensor automorphism"
},
{
"id": "OA-FLOW.L123.IC1",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic1",
"start_line": 50,
"end_line": 85,
"range_sha256": "a2f32a2208d4ba28afb6f9c658be1b81af88d98121bb2a85a4573d3a61b185f5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.CF.6",
"OA-FLOW.GCC.TOOLS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 85,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Every nonzero intertwiner has an invariant central carrier and implements that corner"
},
{
"id": "OA-FLOW.L123.IC2",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic2",
"start_line": 86,
"end_line": 127,
"range_sha256": "36875d3d73a96c8efc1fd370f97f0919d378dc978b3ba83f0d00af0d562eca84",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC1",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.GCC.TOOLS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 127,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Well-ordered disjointification, arbitrary strong-star assembly and largest inner central part"
},
{
"id": "OA-FLOW.L123.IC3",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic3",
"start_line": 128,
"end_line": 168,
"range_sha256": "7066f13e367e57ae6cbdc8cf2c81eda0a1af8b23caae12aac35077272efc4a35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.CF.8",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 168,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Constructed vector and general normal-functional slices, bimodule normality, state separation and positive decomposition"
},
{
"id": "OA-FLOW.L123.IC4",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic4",
"start_line": 169,
"end_line": 199,
"range_sha256": "297dbf3cd4ce32200b138b1e39e2eaa159a6ee86c9d2c3a80dd00aba63e6673e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.L123.IC1",
"OA-FLOW.L123.IC2",
"OA-FLOW.L123.IC3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 199,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full slice-family carrier exhaustion and both directions of tensor-product innerness"
},
{
"id": "OA-FLOW.L123.IC5",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.ic5",
"start_line": 200,
"end_line": 218,
"range_sha256": "1c7bbe31a691e8540c94962dce1781209e451fb7b9423941f482a49b41de9566",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC2",
"OA-FLOW.L123.IC3",
"OA-FLOW.L123.IC4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 218,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Exact state-slice cancellation and direct-sum central assembly examples"
},
{
"id": "OA-FLOW.L123.HISTORICAL.SETTING",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.historical.setting",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Statement, tensor notation and component terms; not a theorem premise"
},
{
"id": "OA-FLOW.TENSORINNER.EASY",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.tensorinner.easy",
"start_line": 234,
"end_line": 249,
"range_sha256": "9f22c7fd4a75fa904dad254cb058042da912dcad2346d111a7b22092876f8212",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 249,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained complete easy implication with full normal-density extension"
},
{
"id": "OA-FLOW.TENSORINNER.SLICE",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.tensorinner.slice",
"start_line": 250,
"end_line": 286,
"range_sha256": "c7bba8adba1053ca8e389f388583f8f41b064c7814ee1841bb4639d2fd7dd441",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.L123.IC3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 286,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained nonzero-state slice intertwiner with constructed general normal slices"
},
{
"id": "OA-FLOW.TENSORINNER.POLAR",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.tensorinner.polar",
"start_line": 287,
"end_line": 350,
"range_sha256": "5c94a48b8fedce1d7fc2037879b68fa14d634c27f04fa1fcab0d47fac18233d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TENSORINNER.SLICE",
"OA-FLOW.L123.IC1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 350,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained polar centrality and adjoint-inverse invariance alternative"
},
{
"id": "OA-FLOW.TENSORINNER.PATCH",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.tensorinner.patch",
"start_line": 351,
"end_line": 416,
"range_sha256": "26c2ab9808bbed39ee3269025909bf6e82fe340b20f2fb154e83a129562ce1f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.L123.IC1",
"OA-FLOW.L123.IC2",
"OA-FLOW.L123.IC3",
"OA-FLOW.TENSORINNER.EASY",
"OA-FLOW.TENSORINNER.SLICE",
"OA-FLOW.TENSORINNER.POLAR",
"OA-FLOW.CF.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 416,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained full maximal-family remainder proof, tensor-corner construction and arbitrary-index unitary sum"
},
{
"id": "OA-FLOW.L123.PROBLEM",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.problem",
"start_line": 417,
"end_line": 421,
"range_sha256": "acac8bc2c3197c98bc15b043b6d74e645646e575b3d60524632290b3f07921ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC3",
"OA-FLOW.L123.IC5",
"OA-FLOW.TENSORINNER.PATCH"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 421,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 219,
"end_line": 233,
"range_sha256": "622f002920ab86293bacc6e1186224a09e9ce2c83c9adbd343940f877c9eb13d",
"full_statement_context": "\n## Maximal-remainder proof: tensor-product innerness from slice intertwiners\n\nIf a tensor product of two automorphisms is inner, each factor automorphism is inner. The reverse implication is immediate; the content is extracting a nonzero intertwiner from an implementing tensor unitary, turning its polar part into a unitary on an invariant central summand, and patching all such summands without a countability assumption.\n\nLet $M_1,M_2$ be nonzero von Neumann algebras, let $\\sigma_i\\in\\operatorname{Aut}(M_i)$, and put\n\n\n$$\nM=M_1\\,\\overline\\otimes\\,M_2,\n\\qquad\n\\sigma=\\sigma_1\\,\\overline\\otimes\\,\\sigma_2.\n\\tag{T1}\n$$\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Complete retained solved problem on preselected state cancellation and fresh remainder choice"
},
{
"id": "OA-FLOW.L123.SOURCES",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.sources",
"start_line": 422,
"end_line": 426,
"range_sha256": "76c5279baaecd9a3a8d12b0d0283448b0be5191f250a8078056462a417fda697",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 1,
"end_line": 15,
"range_sha256": "7c33860bf270734c145766f401229f63ece2bb2ef1f8fdbaaaa10315ae5c55ed",
"full_statement_context": "\n# From an intertwiner family to tensor-product innerness\n\nA single normal state can send a tensor unitary to zero. A family of separating states keeps the information we need: its slice intertwiners have central carriers whose join is the identity. We first construct the actual tensor product and normal maps, then show how any family of intertwiners assembles a largest inner central part. Constructed normal slices will supply a family with full support. The maximal-remainder argument gives a second complete proof.\n\nThroughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n## Exact earlier inputs\n\nThe thirteen direct proofs are [CF1](OA-FLOW-CF.md#oa-flow.cf.1), [CF6](OA-FLOW-CF.md#oa-flow.cf.6), [CF8](OA-FLOW-CF.md#oa-flow.cf.8), [CP1](OA-FLOW-CP.md#oa-flow.cp.1), [CP2](OA-FLOW-CP.md#oa-flow.cp.2), [CP4](OA-FLOW-CP.md#oa-flow.cp.4), [CP6](OA-FLOW-CP.md#oa-flow.cp.6), [PC1](OA-FLOW-PC.md#oa-flow.pc.1), [GCC TOOLS](OA-FLOW-GCC.md#gcc-tools), [NCF1](OA-FLOW-NCF.md#ncf-1), [BD4](OA-FLOW-BD.md#oa-flow.bd.4), [BD5](OA-FLOW-BD.md#oa-flow.bd.5), and [NF4](OA-FLOW-NF.md#oa-flow.nf.4).\n\nNF4 concerns a bounded positive functional preserving bounded increasing positive suprema on an arbitrary concrete von Neumann algebra; its full source setting is included. BD4 applies to the actual nondegenerate elementary tensor algebra, and BD5 turns uniformly bounded strong nets into ultraweak nets. NCF1's matrix argument is for an arbitrary auxiliary Hilbert space and a normal homomorphism with the actual concrete preduals; no group theorem from its surrounding lesson is a tensor-map premise. CP2 supplies the algebraic quotient by bilinearity. Its projective norm is not the Hilbert tensor norm: we construct the latter next. The general commutation and support arguments in GCC TOOLS are used at their stated bounded-operator scope, without a group action assumption.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L123.md",
"source_sha256": "b2c6de7e1323b9467520dd5d6cd30211dca4a3015ab5fd6667cc33f1cde1aafa",
"start_line": 6,
"end_line": 6,
"range_sha256": "7884c90735ffc560b4885eec7c91d392050f71cffc1441e18e6068cdf11c55bc",
"full_statement_context": "Throughout, \\(M_i\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space \\(H_i\\), with identity \\(1_{H_i}\\), and \\(\\sigma_i\\) is an algebraic unital star automorphism of \\(M_i\\). Thus \\(H_i\\ne0\\). For the single-algebra statements, \\(N\\ne0\\) is a concrete von Neumann algebra on an arbitrary Hilbert space \\(K\\), with automorphism \\(\\sigma\\). Inner products are linear in the first variable. An automorphism is inner if it has the form \\(x\\mapsto v x v^*\\) for a unitary \\(v\\) in the actual algebra. A central corner \\(zN\\) has identity \\(z\\); its implementing unitary satisfies \\(v^*v=vv^*=z\\). There is no factor, separability, countability, sigma-finiteness or faithful-normal-state assumption. Arbitrary orthogonal sums below mean nets over finite subsets.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Bounded actual source and organization comparison; no proof premise"
},
{
"id": "OA-FLOW.L123.FIGURE",
"unit": "OA-FLOW-L123",
"source_file": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda",
"reader": "OA-FLOW-L123.html#oa-flow.l123.figure",
"start_line": 427,
"end_line": 436,
"range_sha256": "8d857b37db474094d9dcc7a5751d6519644b6767515edbfc6b01e9f82db76648",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC1",
"OA-FLOW.L123.IC2",
"OA-FLOW.L123.IC3",
"OA-FLOW.L123.IC4",
"OA-FLOW.L123.IC5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null
},
{
"id": "OA-FLOW.L124.SETTING",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.setting",
"start_line": 8,
"end_line": 16,
"range_sha256": "6ea5ea8b577a5383b470123b1962eae93ad315e5d4bb0723773d8c6b835c4f85",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full arbitrary-algebra setting, exact inputs and bounded-use descriptions; not a theorem premise"
},
{
"id": "OA-FLOW.L124.CE0",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce0",
"start_line": 17,
"end_line": 64,
"range_sha256": "2076b7c60e18a0409a88153f784b8a6ffb3101fd7bbeb33528c38848ae8a15ed",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.NF.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 64,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 34,
"end_line": 34,
"range_sha256": "79d2866361a4e438d29b9c2f01ca1f3a62afa2c663300892de18707cbd131404",
"full_statement_context": "Let \\(D\\) now be any von Neumann algebra and let \\(\\delta:\\mathbb T\\to\\operatorname{Aut}(D)\\) be point-ultraweakly continuous. For \\(x\\in D\\), define the Fourier mean by its evaluations:\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Automatic normality fully derived from functional theorem; scalar-pairing Fourier means and unrestricted ultraweak Fejer density"
},
{
"id": "OA-FLOW.L124.CE1",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce1",
"start_line": 65,
"end_line": 84,
"range_sha256": "fce29e567e41b0b041e226c04a9934e67e4823cc168be26a7de8e17f93b598a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 84,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 34,
"end_line": 34,
"range_sha256": "79d2866361a4e438d29b9c2f01ca1f3a62afa2c663300892de18707cbd131404",
"full_statement_context": "Let \\(D\\) now be any von Neumann algebra and let \\(\\delta:\\mathbb T\\to\\operatorname{Aut}(D)\\) be point-ultraweakly continuous. For \\(x\\in D\\), define the Fourier mean by its evaluations:\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 69,
"end_line": 69,
"range_sha256": "73907f1d8c94038fa7910ce97d844802b11ba6086d9e7c976acdad4c5ee38c71",
"full_statement_context": "Assume \\(D\\ne0\\) is abelian. Let \\(\\mathcal B\\) be any group of automorphisms commuting with \\(\\delta\\), with \\(D^{\\mathcal B}=\\mathbb C1\\). No topology on \\(\\mathcal B\\) is required. A normal element \\(x\\) in an abelian algebra has equal initial and range supports by PC1; write this common projection \\(s(x)\\). It is the least projection \\(p\\) with \\(px=x\\). An automorphism sends such least projections to such least projections, and multiplication by a nonzero scalar preserves the support. Consequently \\(s(x)\\) is fixed by \\(\\delta\\) whenever \\(x\\in D_n\\). Set\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 79,
"end_line": 79,
"range_sha256": "b9200a8701fe189595c22ee82ceb9f37edb34a3695eb6e17b903300064d0dd7b",
"full_statement_context": "Now suppose \\(\\delta\\) is faithful. \\(S=\\{0\\}\\) would make all Fourier means fixed, and CE0 would make the whole action trivial on the nonzero algebra. Otherwise choose the least positive \\(k\\in S\\); division with remainder proves \\(S=k\\mathbb Z\\). If \\(k>1\\), \\(\\delta_{2\\pi/k}\\) fixes every \\(D_n\\); normality and CE0 make it the identity on \\(D\\), contradicting faithfulness. We have proved\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Abelian carrier dichotomy, nonzero-product subgroup and faithful-circle full integer spectrum"
},
{
"id": "OA-FLOW.L124.CE2",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce2",
"start_line": 85,
"end_line": 101,
"range_sha256": "e37742c9ab2dacba2d4e6954316335ea2dcbccb10db7c261784084f8dd0e1435",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE1",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 101,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 34,
"end_line": 34,
"range_sha256": "79d2866361a4e438d29b9c2f01ca1f3a62afa2c663300892de18707cbd131404",
"full_statement_context": "Let \\(D\\) now be any von Neumann algebra and let \\(\\delta:\\mathbb T\\to\\operatorname{Aut}(D)\\) be point-ultraweakly continuous. For \\(x\\in D\\), define the Fourier mean by its evaluations:\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 69,
"end_line": 69,
"range_sha256": "73907f1d8c94038fa7910ce97d844802b11ba6086d9e7c976acdad4c5ee38c71",
"full_statement_context": "Assume \\(D\\ne0\\) is abelian. Let \\(\\mathcal B\\) be any group of automorphisms commuting with \\(\\delta\\), with \\(D^{\\mathcal B}=\\mathbb C1\\). No topology on \\(\\mathcal B\\) is required. A normal element \\(x\\) in an abelian algebra has equal initial and range supports by PC1; write this common projection \\(s(x)\\). It is the least projection \\(p\\) with \\(px=x\\). An automorphism sends such least projections to such least projections, and multiplication by a nonzero scalar preserves the support. Consequently \\(s(x)\\) is fixed by \\(\\delta\\) whenever \\(x\\in D_n\\). Set\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 89,
"end_line": 89,
"range_sha256": "b6dd7c36050dc19b26422c129a1a7a88420d042e4efd10aa79ef5665d237c624",
"full_statement_context": "For a fixed \\(n\\) with \\(c_n=1\\), consider families of nonzero \\(x_i\\in D_n\\) with pairwise orthogonal supports \\(p_i\\). Chain unions remain families, so CF1 gives a maximal family. Each \\(p_i\\) is fixed by \\(\\delta\\); normality, or the order-isomorphism property, fixes their join \\(p\\) and \\(q=1-p\\). If \\(q\\ne0\\), the support join \\(c_n=1\\) yields \\(qx\\ne0\\) for some \\(x\\in D_n\\). Its support lies below \\(q\\), so it extends the family, a contradiction. Hence \\(\\bigvee_i p_i=1\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Full carrier gives an eigenunitary by an arbitrary orthogonal finite-subset net"
},
{
"id": "OA-FLOW.L124.CE3",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce3",
"start_line": 102,
"end_line": 137,
"range_sha256": "4ff1c86a5961f6fed7d6fcb5a024b33c0f561943710abc96410fca2ad117b6b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.NCF.1",
"OA-FLOW.CONNES.STABILIZATION",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 137,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 106,
"end_line": 111,
"range_sha256": "4cc23897a1599360876e6a6949ffe441614075e09407046776c61bd30d7a6e42",
"full_statement_context": "Work on \\(H\\otimes\\ell^2(\\mathbb Z)\\), with standard vectors \\(e_m\\). Define\n\\(U\\xi(k)=\\xi(k+1)\\), \\(V_\\theta\\xi(k)=e^{ik\\theta}\\xi(k)\\),\n\\(\\widetilde M=M\\bar\\otimes B(\\ell^2(\\mathbb Z))\\),\n\\(\\widetilde\\sigma=\\sigma\\bar\\otimes\\operatorname{Ad}U\\), and\n\\(\\delta_\\theta=\\operatorname{Ad}(1\\otimes V_\\theta)\\).\nNormal amplification on the entire tensor product is constructed by NCF1 and L115 STABILIZATION. The diagonal unitaries are strongly continuous first on finite-coordinate vectors and then by their common norm one. Conjugating a fixed bounded \\(X\\) is continuous on vector coefficients; the uniform bound and CP4/6 summable tails make \\(\\delta\\) point-ultraweakly continuous.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Direct matrix extraction w=a tensor U* and sigma=Ad(a*) on the entire original algebra"
},
{
"id": "OA-FLOW.L124.CE4",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce4",
"start_line": 138,
"end_line": 171,
"range_sha256": "3f082011dee798cba7aeee40fbd5668e257318471099cfa8c35b880acfa42264",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE3",
"OA-FLOW.NCF.1",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CONNES.CENTERKERNEL",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 171,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 106,
"end_line": 111,
"range_sha256": "4cc23897a1599360876e6a6949ffe441614075e09407046776c61bd30d7a6e42",
"full_statement_context": "Work on \\(H\\otimes\\ell^2(\\mathbb Z)\\), with standard vectors \\(e_m\\). Define\n\\(U\\xi(k)=\\xi(k+1)\\), \\(V_\\theta\\xi(k)=e^{ik\\theta}\\xi(k)\\),\n\\(\\widetilde M=M\\bar\\otimes B(\\ell^2(\\mathbb Z))\\),\n\\(\\widetilde\\sigma=\\sigma\\bar\\otimes\\operatorname{Ad}U\\), and\n\\(\\delta_\\theta=\\operatorname{Ad}(1\\otimes V_\\theta)\\).\nNormal amplification on the entire tensor product is constructed by NCF1 and L115 STABILIZATION. The diagonal unitaries are strongly continuous first on finite-coordinate vectors and then by their common norm one. Conjugating a fixed bounded \\(X\\) is continuous on vector coefficients; the uniform bound and CP4/6 summable tails make \\(\\delta\\) point-ultraweakly continuous.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Constructed cyclic fixed realization, exact inverse-parameter dual convention, generation and tensor center"
},
{
"id": "OA-FLOW.L124.CE5",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.ce5",
"start_line": 172,
"end_line": 184,
"range_sha256": "0eac288e83891aa14ccf95a41cd93c5ae445bddf6af56ae9d0d4fbde0c91bb8b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE1",
"OA-FLOW.L124.CE2",
"OA-FLOW.L124.CE3",
"OA-FLOW.L124.CE4",
"OA-FLOW.NCF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 106,
"end_line": 111,
"range_sha256": "4cc23897a1599360876e6a6949ffe441614075e09407046776c61bd30d7a6e42",
"full_statement_context": "Work on \\(H\\otimes\\ell^2(\\mathbb Z)\\), with standard vectors \\(e_m\\). Define\n\\(U\\xi(k)=\\xi(k+1)\\), \\(V_\\theta\\xi(k)=e^{ik\\theta}\\xi(k)\\),\n\\(\\widetilde M=M\\bar\\otimes B(\\ell^2(\\mathbb Z))\\),\n\\(\\widetilde\\sigma=\\sigma\\bar\\otimes\\operatorname{Ad}U\\), and\n\\(\\delta_\\theta=\\operatorname{Ad}(1\\otimes V_\\theta)\\).\nNormal amplification on the entire tensor product is constructed by NCF1 and L115 STABILIZATION. The diagonal unitaries are strongly continuous first on finite-coordinate vectors and then by their common norm one. Conjugating a fixed bounded \\(X\\) is continuous on vector coefficients; the uniform bound and CP4/6 summable tails make \\(\\delta\\) point-ultraweakly continuous.\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 176,
"end_line": 179,
"range_sha256": "52e86c34a6783ac07f2a43ce3ce01c2c0840cc0dcc3d2c69b275aa89d9fecc44",
"full_statement_context": "Assume \\(\\Gamma(\\sigma)=\\{1\\}\\) and\n\\(Z(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1\\), where\n\\(\\operatorname{Aut}_\\sigma(M)=\\{\\beta:\\beta\\sigma=\\sigma\\beta\\}\\).\nLet \\(\\mathcal A\\) be the group of automorphisms of \\(\\widetilde M\\) commuting with \\(\\widetilde\\sigma\\) and every \\(\\delta_\\theta\\). All these automorphisms are normal by CE0. They preserve \\(N\\) and \\(D=Z(N)\\). The Weyl relation shows that each \\(\\delta_\\theta\\) belongs to \\(\\mathcal A\\). NCF1 constructs \\(\\beta\\bar\\otimes\\mathrm{id}\\), also in \\(\\mathcal A\\), for every \\(\\beta\\in\\operatorname{Aut}_\\sigma(M)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Centralizer ergodicity and trivial Connes spectrum imply innerness, using the tensor-innerness theorem"
},
{
"id": "OA-FLOW.L124.HISTORICAL.SETTING",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.historical.setting",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Statement, component terms and conditional alternative; not a proof premise"
},
{
"id": "OA-FLOW.CENTRALIZER.STABILIZE",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.centralizer.stabilize",
"start_line": 218,
"end_line": 300,
"range_sha256": "ca4f4e47c474e48409fd971a43a26443ae0ebd63532fcb941fe020e095ef951d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE3",
"OA-FLOW.L124.CE4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 300,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained E3–E7b full stabilization and generation with all matrix and Fourier details supplied earlier"
},
{
"id": "OA-FLOW.CENTRALIZER.ERGODIC",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.centralizer.ergodic",
"start_line": 301,
"end_line": 343,
"range_sha256": "b7e59dc0dd6d47b158814fd19f12e2b550de08d79a04536df61b784ec1d84a7a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE4",
"OA-FLOW.L124.CE5",
"OA-FLOW.CENTRALIZER.STABILIZE",
"OA-FLOW.NCF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 343,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained commuting-group ergodicity on the fixed center"
},
{
"id": "OA-FLOW.CENTRALIZER.GRADING",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.centralizer.grading",
"start_line": 344,
"end_line": 397,
"range_sha256": "9b29f19f6fe7268cbbfc6739c7423fd8cab32cbd719411735e5c032ed152881f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE1",
"OA-FLOW.L124.CE4",
"OA-FLOW.CENTRALIZER.ERGODIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 397,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained faithful circle, full spectral carriers and integer subgroup argument"
},
{
"id": "OA-FLOW.CENTRALIZER.UNITARY",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.centralizer.unitary",
"start_line": 398,
"end_line": 441,
"range_sha256": "e388c4f12c04b7dbb5c83ba279c66950526c330a801c867c675b63810e37e66d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE2",
"OA-FLOW.CENTRALIZER.GRADING",
"OA-FLOW.PC.1",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 441,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained entire arbitrary-support polar-unitary construction"
},
{
"id": "OA-FLOW.CENTRALIZER.INNER",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.centralizer.inner",
"start_line": 442,
"end_line": 498,
"range_sha256": "6a28ece4851fd18584c6d61a73879408e5323249a79b5c922e0260ab48e16c0c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE3",
"OA-FLOW.L124.CE4",
"OA-FLOW.L124.CE5",
"OA-FLOW.CENTRALIZER.STABILIZE",
"OA-FLOW.CENTRALIZER.UNITARY",
"OA-FLOW.TENSORINNER.PATCH"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 498,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Retained stabilized implementation and unconditional E25 via CE3; the accepted earlier L123 PATCH supplies the complete tensor-descent alternative."
},
{
"id": "OA-FLOW.L124.PROBLEM",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.problem",
"start_line": 499,
"end_line": 508,
"range_sha256": "cc016dfabd319b837394a20805db2af25e813e98409b7c78767ce047e3ef8879",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE1",
"OA-FLOW.CENTRALIZER.GRADING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 508,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
},
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 185,
"end_line": 217,
"range_sha256": "f02841e3ae7a6e399cff72ba96c4d57245e293525698389e924b7db881775890",
"full_statement_context": "\n\n## Stabilization and tensor-descent argument\n\nCE0–CE5 provide the Fourier, normality, matrix and arbitrary-family facts used below. The concluding tensor descent uses [From an intertwiner family to tensor-product innerness](OA-FLOW-L123.md#oa-flow.l123.ic4); CE3 also gives a direct matrix-coefficient descent.\n\n## Trivial Connes spectrum and centralizer ergodicity force innerness\n\nTrivial Connes spectrum does not by itself make an automorphism inner. The missing rigidity comes from the automorphisms commuting with it: if that centralizer acts ergodically on the center, stabilization turns the problem into a faithful circle action on an abelian center. Its degree-one spectral space contains a unitary, and that unitary implements the stabilized automorphism.\n\nLet $M$ be a nonzero von Neumann algebra and $\\sigma\\in\\operatorname{Aut}(M)$. Regard $\\sigma$ as the action $n\\mapsto\\sigma^n$ of $\\mathbb Z$, whose dual is written multiplicatively as $\\mathbb T$. Put\n\n\n\n$$\n\\operatorname{Aut}_\\sigma(M)\n=\\{\\beta\\in\\operatorname{Aut}(M):\\beta\\sigma=\\sigma\\beta\\}.\n\\tag{E1}\n$$\n\nAssume\n\n\n\n$$\n\\Gamma(\\sigma)=\\{1\\},\n\\qquad\nZ(M)^{\\operatorname{Aut}_\\sigma(M)}=\\mathbb C1.\n\\tag{E2}\n$$\n\nThere is no separability, sigma-finiteness, or countability hypothesis.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Complete retained solved problem excluding kZ for k>1"
},
{
"id": "OA-FLOW.L124.SOURCES",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.sources",
"start_line": 509,
"end_line": 515,
"range_sha256": "901aedd3de396e293f547042af96ed52dcef6f26f85ec97816e890463c94e0c2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 1,
"end_line": 16,
"range_sha256": "d1de0a5309ced9c4653cb6ee3881e26860abba009b2a0b460f79105b47fac1ed",
"full_statement_context": "# Read an inner implementer from a circle eigenunitary\n\nTrivial Connes spectrum acquires force when the commuting automorphisms are ergodic on the center. The mechanism has two parts: a faithful circle action on an abelian algebra admits a unitary of degree one, and a central degree-one unitary in the cyclic stabilization has a matrix coefficient that implements the original automorphism. We prove the circle statement first and construct the matrix descent before applying the Connes-spectrum hypothesis.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## The setting and earlier proof bodies\n\nThroughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n\nOur complete earlier inputs are the scalar circle measure and Fejér estimates of [CC0](OA-FLOW-CC.md#oa-flow.cc.0), the summable vector-series and predual-duality proofs [CP4](OA-FLOW-CP.md#oa-flow.cp.4) and [CP6](OA-FLOW-CP.md#oa-flow.cp.6), the maximal principle [CF1](OA-FLOW-CF.md#oa-flow.cf.1), the arbitrary projection joins, bounded polar decomposition and orthogonal sums [PC1](OA-FLOW-PC.md#oa-flow.projection.pc1), the bounded positive functional theorem [NF4](OA-FLOW-NF.md#oa-flow.nf.4), the entire normal amplification and matrix-entry proof [NCF1](OA-FLOW-NCF.md#ncf-1), and L115's complete [stabilization](OA-FLOW-L115.md#oa-flow.connes.stabilization) and [dual-center kernel](OA-FLOW-L115.md#oa-flow.connes.centerkernel) bodies.\n\nOnly CC0's scalar measure and finite polynomial estimates are used; its modular-weight setting is not a hypothesis here. PC1's bounded polar and arbitrary-sum proof is used without the countable-decomposability hypothesis of its later classification sections. NF4 proves a theorem about functionals; the next paragraph supplies the passage to automorphisms. The tensor-descent alternative uses the earlier L123 proof.\n\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L124.md",
"source_sha256": "846e413bee8b84cc00c2b6cbcebdf71f62ae5696d9324c59e520e2029b0908bd",
"start_line": 11,
"end_line": 11,
"range_sha256": "814deb3479dab77c1dc8a61057ef3f606fd445051e4c7a6dd6c42e4bd1f95555",
"full_statement_context": "Throughout, \\(M\\ne0\\) is an arbitrary concrete von Neumann algebra on an arbitrary Hilbert space, and \\(\\sigma\\) is a star automorphism of \\(M\\). The cyclic action is \\(j\\mapsto\\sigma^j\\), with discrete group \\(\\mathbb Z\\) and multiplicative dual \\(\\mathbb T\\); its Connes spectrum is the intersection of the spectra of all nonzero fixed corners at the exact [L115 conventions](OA-FLOW-L115.md#oa-flow.l115.conventions). We will prove algebraic automorphisms normal. Circle parameters are modulo \\(2\\pi\\), and circle integrals use \\(d\\theta/(2\\pi)\\).\n",
"usage": "Full arbitrary-algebra global setting, subject to explicit local hypotheses; context is not a proof premise."
}
],
"scope_boundary": "Bounded source reading, expression comparison and illustration terms; not a mathematical premise"
},
{
"id": "OA-FLOW.L124.FIGURE",
"unit": "OA-FLOW-L124",
"source_file": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654",
"reader": "OA-FLOW-L124.html#oa-flow.l124.figure",
"start_line": 517,
"end_line": 531,
"range_sha256": "e80b889e4d6daef156c2b7b778e82ca2f49a893f7a38626ce071a9d043ec7d0c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L124.CE0",
"OA-FLOW.L124.CE1",
"OA-FLOW.L124.CE2",
"OA-FLOW.L124.CE3",
"OA-FLOW.L124.CE4",
"OA-FLOW.L124.CE5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null
},
{
"id": "OA-FLOW.L125.SETTING",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.setting",
"start_line": 7,
"end_line": 18,
"range_sha256": "d39d3395548bc38f0d601726f44f3a73c5bc8a2ac85335002a53f6850b4a2d26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n\\[\n\\|(x_j-x_i)\\xi\\|^2\\leq C\\langle(x_j-x_i)\\xi,\\xi\\rangle.\n\\tag{CO1}\n\\]\n\nFor each \\(\\xi\\), the quadratic values have a finite scalar supremum. Once they are close to it, comparison with a common upper index makes both vector differences small by CO1 and the triangle inequality. Thus \\(x_i\\xi\\) is a Cauchy net. To obtain a limit from Hilbert completeness, choose increasing indices after which its errors are below \\(2^{-n}\\); their sequence converges, and the same errors make the entire net converge to that limit. The vector limits define a linear bounded positive operator of norm at most \\(C\\). Strong convergence puts it in the weakly closed algebra \\(M\\). Passing each quadratic order inequality to the limit shows that this operator dominates the net and is below every selfadjoint upper bound. It is its supremum. Adding a scalar reduces uniformly bounded increasing selfadjoint nets to this case.\n\nIn an abelian von Neumann algebra \\(D\\), commuting selfadjoint elements have the pointwise maximum\n\n\n\n\\[\na\\vee b=\\tfrac12(a+b+|a-b|).\n\\tag{CO2}\n\\]\n\nCF6 identifies the commutative algebra generated by \\(a,b\\) with continuous scalar functions. If another selfadjoint element of \\(D\\) bounds both, it commutes with them and the same scalar comparison in the enlarged commutative algebra makes it bound their displayed maximum. This is a least upper bound in \\(D\\), not a lattice assertion for noncommuting operators.\n\nFor any uniformly bounded selfadjoint family in \\(D\\), finite maxima form a uniformly bounded increasing net. The preceding limit constructs its supremum in \\(D\\). Addition of a fixed selfadjoint element translates all upper bounds, so commutes with this supremum. A star automorphism is an order isomorphism; applying it and its inverse to the defining upper-bound inequalities proves that it carries a supremum to the supremum of the image family. This particular assertion uses order, without a continuity assumption on the map. Infima follow by changing signs.\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "Begin with any increasing positive net \\(0\\leq x_i\\leq C1\\) in \\(M\\). For \\(j\\geq i\\), the positive difference is between zero and \\(C1\\). Continuous calculus gives \\((x_j-x_i)^2\\leq C(x_j-x_i)\\), hence\n\n\n\n\\[\n\\|(x_j-x_i)\\xi\\|^2\\leq C\\langle(x_j-x_i)\\xi,\\xi\\rangle.\n\\tag{CO1}\n\\]\n\nFor each \\(\\xi\\), the quadratic values have a finite scalar supremum. Once they are close to it, comparison with a common upper index makes both vector differences small by CO1 and the triangle inequality. Thus \\(x_i\\xi\\) is a Cauchy net. To obtain a limit from Hilbert completeness, choose increasing indices after which its errors are below \\(2^{-n}\\); their sequence converges, and the same errors make the entire net converge to that limit. The vector limits define a linear bounded positive operator of norm at most \\(C\\). Strong convergence puts it in the weakly closed algebra \\(M\\). Passing each quadratic order inequality to the limit shows that this operator dominates the net and is below every selfadjoint upper bound. It is its supremum. Adding a scalar reduces uniformly bounded increasing selfadjoint nets to this case.\n\nIn an abelian von Neumann algebra \\(D\\), commuting selfadjoint elements have the pointwise maximum\n\n\n\n\\[\na\\vee b=\\tfrac12(a+b+|a-b|).\n\\tag{CO2}\n\\]\n\nCF6 identifies the commutative algebra generated by \\(a,b\\) with continuous scalar functions. If another selfadjoint element of \\(D\\) bounds both, it commutes with them and the same scalar comparison in the enlarged commutative algebra makes it bound their displayed maximum. This is a least upper bound in \\(D\\), not a lattice assertion for noncommuting operators.\n\nFor any uniformly bounded selfadjoint family in \\(D\\), finite maxima form a uniformly bounded increasing net. The preceding limit constructs its supremum in \\(D\\). Addition of a fixed selfadjoint element translates all upper bounds, so commutes with this supremum. A star automorphism is an order isomorphism; applying it and its inverse to the defining upper-bound inequalities proves that it carries a supremum to the supremum of the image family. This particular assertion uses order, without a continuity assumption on the map. Infima follow by changing signs.\n"
}
],
"scope_boundary": "Complete increasing-net strong limit, abelian finite maxima and arbitrary bounded selfadjoint-family order suprema"
},
{
"id": "OA-FLOW.L125.CO1",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.co1",
"start_line": 47,
"end_line": 80,
"range_sha256": "ec36639a212ebca3321ce70120d1a75a0eb3d57bd39825821c3d19550e833cc4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO0"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 80,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## CO1. Solve a central bounded cocycle for any group\n\nLet \\(G\\) be any group, let \\(\\alpha\\) be an automorphism action, and let the uniformly bounded family \\(h_s\\in Z(M)_{\\rm sa}\\) satisfy\n\n\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n\nPutting \\(s=t=e\\) gives \\(h_e=0\\). Its bounded central transfer is\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## CO1. Solve a central bounded cocycle for any group\n\nLet \\(G\\) be any group, let \\(\\alpha\\) be an automorphism action, and let the uniformly bounded family \\(h_s\\in Z(M)_{\\rm sa}\\) satisfy\n\n\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n\nPutting \\(s=t=e\\) gives \\(h_e=0\\). Its bounded central transfer is\n\n"
}
],
"scope_boundary": "Central uniformly bounded selfadjoint additive cocycles are coboundaries for every group without amenability"
},
{
"id": "OA-FLOW.L125.CO2",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.co2",
"start_line": 81,
"end_line": 137,
"range_sha256": "a2bf1fe5d5609f93f93bcc5feeceada07be180b4c9d3e34ccb4f57fc87e201b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO0",
"OA-FLOW.L125.CO1",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 137,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## CO2. Correct the logarithm without increasing its norm\n\nSuppose \\(v\\) implements an automorphism \\(\\sigma\\) commuting with every \\(\\alpha_s\\), and suppose its principal argument is\n\n\n\n\\[\nv=e^{ib},\\qquad b=b^*,\\qquad B:=\\|b\\|<\\pi/2.\n\\tag{CO6}\n\\]\n\nThe equality \\(\\operatorname{Ad}(\\alpha_s(v))=\\operatorname{Ad}(v)\\) implies \\(a_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M))\\): multiply the equality of conjugations on the left by \\(v^*\\) and on the right by \\(\\alpha_s(v)\\) to get \\(a_sx=xa_s\\). Conversely a central unitary quotient makes the conjugations equal.\n\nNow \\(\\alpha_s(v)=va_s\\) commutes with \\(v\\). Their principal arguments commute because continuous calculus commutes with every commuting normal generator. A star automorphism is isometric, by CF6's contractivity applied also to its inverse, and transports continuous calculus by polynomial approximation and uniqueness. Therefore \\(\\alpha_s(b)\\) is the principal argument of \\(\\alpha_s(v)\\). Set\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## CO2. Correct the logarithm without increasing its norm\n\nSuppose \\(v\\) implements an automorphism \\(\\sigma\\) commuting with every \\(\\alpha_s\\), and suppose its principal argument is\n\n\n\n\\[\nv=e^{ib},\\qquad b=b^*,\\qquad B:=\\|b\\|<\\pi/2.\n\\tag{CO6}\n\\]\n\nThe equality \\(\\operatorname{Ad}(\\alpha_s(v))=\\operatorname{Ad}(v)\\) implies \\(a_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M))\\): multiply the equality of conjugations on the left by \\(v^*\\) and on the right by \\(\\alpha_s(v)\\) to get \\(a_sx=xa_s\\). Conversely a central unitary quotient makes the conjugations equal.\n\nNow \\(\\alpha_s(v)=va_s\\) commutes with \\(v\\). Their principal arguments commute because continuous calculus commutes with every commuting normal generator. A star automorphism is isometric, by CF6's contractivity applied also to its inverse, and transports continuous calculus by polynomial approximation and uniqueness. Therefore \\(\\alpha_s(b)\\) is the principal argument of \\(\\alpha_s(v)\\). Set\n\n"
}
],
"scope_boundary": "Principal-log branch, central additive difference and fixed implementer preserving the original B\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## CO4. Localize centrally, then apply the order correction\n\nAssume \\(\\operatorname{Sp}(S)\\subset V(\\varepsilon)\\), with \\(0<\\varepsilon<\\pi/2\\). Choose \\(0<\\delta<\\varepsilon\\), so \\(\\delta+\\varepsilon<\\pi\\). Let \\(E\\) be the closed \\(\\delta\\)-arc about \\(e^{i\\theta}\\), and \\(c=z(P(E))\\). For \\(\\eta>0\\), put\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## CO4. Localize centrally, then apply the order correction\n\nAssume \\(\\operatorname{Sp}(S)\\subset V(\\varepsilon)\\), with \\(0<\\varepsilon<\\pi/2\\). Choose \\(0<\\delta<\\varepsilon\\), so \\(\\delta+\\varepsilon<\\pi\\). Let \\(E\\) be the closed \\(\\delta\\)-arc about \\(e^{i\\theta}\\), and \\(c=z(P(E))\\). For \\(\\eta>0\\), put\n\n"
},
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 235,
"end_line": 235,
"range_sha256": "34cc4a4ca0401e0fbe1b61c5745bfa5d27896b370101d24bcceed4c299ca4fda",
"full_statement_context": "If \\(0<\\varepsilon<\\pi/4\\), its principal argument \\(b\\) is defined by continuous calculus on that closed arc and has \\(B=\\|b\\|\\leq\\delta+\\varepsilon<2\\varepsilon<\\pi/2\\). CO2 applies to any group action commuting with \\(S\\) and produces\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "If \\(0<\\varepsilon<\\pi/4\\), its principal argument \\(b\\) is defined by continuous calculus on that closed arc and has \\(B=\\|b\\|\\leq\\delta+\\varepsilon<2\\varepsilon<\\pi/2\\). CO2 applies to any group action commuting with \\(S\\) and produces\n"
}
],
"scope_boundary": "Delta+epsilon corner bound, finite disjoint central partition, actual corner inverses and arbitrary-group fixed implementer with spectrum in V(2epsilon)"
},
{
"id": "OA-FLOW.L125.CO5",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.co5",
"start_line": 244,
"end_line": 269,
"range_sha256": "453c9cfc50d61029016aee26cf2640269aa3990f74598c9f6f88a6bb5ecafeda",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CP.6",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 269,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n\\[\nh_{st}=h_s+\\alpha_s(h_t).\n\\tag{CO3}\n\\]\n"
}
],
"scope_boundary": "General noncentral norm-bounded cocycles for abelian groups: constructed weak-star compactness and finite Cesaro fixed-point proof"
},
{
"id": "OA-FLOW.L125.CO6",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.co6",
"start_line": 270,
"end_line": 297,
"range_sha256": "0cde4c7b471167458f81c1469c322574c568a661fdd4c50c0b4d0dc2e007a4f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO1",
"OA-FLOW.L125.CO2",
"OA-FLOW.L125.CO4",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 297,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## CO6. Inspect an exact central correction in two blocks\n\nLet \\(M=M_2(\\mathbb C)\\oplus M_2(\\mathbb C)\\) and let the two-element group exchange the blocks. In units of \\(\\pi/32\\), take\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## CO6. Inspect an exact central correction in two blocks\n\nLet \\(M=M_2(\\mathbb C)\\oplus M_2(\\mathbb C)\\) and let the two-element group exchange the blocks. In units of \\(\\pi/32\\), take\n\n"
}
],
"scope_boundary": "Exact two-block pi/32 correction, logarithm norm and finite-dimensional Banach spectrum example"
},
{
"id": "OA-FLOW.L125.HISTORICAL.SETTING",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.historical.setting",
"start_line": 298,
"end_line": 336,
"range_sha256": "1c48fbdb80b41278bea0df188198f8edf5d588265ae9dfaef68875de675bf89d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"scope_boundary": "Retained separated-phase central supports and compact-set separation with full earlier uniform estimate"
},
{
"id": "OA-FLOW.SMALLINNER.LOCALIZE",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.smallinner.localize",
"start_line": 435,
"end_line": 571,
"range_sha256": "f4dcc0257358ab5df04f47c88fc32d8da0e0097f471fe125b8ca64707ed8d1ea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO4",
"OA-FLOW.SMALLINNER.ORTHOGONAL",
"OA-FLOW.PC.1",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 571,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"scope_boundary": "Retained complete small implementer construction, disjointification and delta+epsilon estimate at radius below pi"
},
{
"id": "OA-FLOW.SMALLINNER.COCYCLE",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.smallinner.cocycle",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
}
],
"scope_boundary": "Exact additive cocycle and coboundary definitions for general groups; definition context, not an earlier proved theorem"
},
{
"id": "OA-FLOW.SMALLINNER.COBOUNDARY",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.smallinner.coboundary",
"start_line": 603,
"end_line": 673,
"range_sha256": "ccc26eff556f36c37416ddf93de63f21a60715f7eeded111c41eb02ca5f32c09",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 673,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
}
],
"hypothesis_context_ids": [
"OA-FLOW.SMALLINNER.COCYCLE"
],
"scope_boundary": "Retained full abelian bounded-cocycle transfer theorem, with actual finite-average fixed-point proof earlier"
},
{
"id": "OA-FLOW.SMALLINNER.LOGARITHM",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.smallinner.logarithm",
"start_line": 674,
"end_line": 783,
"range_sha256": "1d7b70cea703e2e1eebdd8f997dcf60458293c5d78e92dd45c1416f224b1b18b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO2",
"OA-FLOW.L125.CO4",
"OA-FLOW.SMALLINNER.LOCALIZE",
"OA-FLOW.CF.6",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 783,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
}
],
"hypothesis_context_ids": [
"OA-FLOW.SMALLINNER.COCYCLE"
],
"scope_boundary": "Retained full strict logarithm branch, centrality and continuous additive cocycle at original abelian-action hypotheses"
},
{
"id": "OA-FLOW.SMALLINNER.FIXED",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.smallinner.fixed",
"start_line": 784,
"end_line": 850,
"range_sha256": "a3805f58aa21114845d4757269d6f4ca2e18fb81140926eee1f48ebf2c826a6c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO2",
"OA-FLOW.L125.CO5",
"OA-FLOW.SMALLINNER.COBOUNDARY",
"OA-FLOW.SMALLINNER.LOGARITHM",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 850,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COBOUNDARY",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 603,
"end_line": 673,
"range_sha256": "ccc26eff556f36c37416ddf93de63f21a60715f7eeded111c41eb02ca5f32c09",
"full_statement_context": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.LOGARITHM",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 674,
"end_line": 783,
"range_sha256": "1d7b70cea703e2e1eebdd8f997dcf60458293c5d78e92dd45c1416f224b1b18b",
"full_statement_context": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COBOUNDARY",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 603,
"end_line": 673,
"range_sha256": "ccc26eff556f36c37416ddf93de63f21a60715f7eeded111c41eb02ca5f32c09",
"full_statement_context": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.LOGARITHM",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 674,
"end_line": 783,
"range_sha256": "1d7b70cea703e2e1eebdd8f997dcf60458293c5d78e92dd45c1416f224b1b18b",
"full_statement_context": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n"
}
],
"hypothesis_context_ids": [
"OA-FLOW.SMALLINNER.COCYCLE",
"OA-FLOW.SMALLINNER.COBOUNDARY",
"OA-FLOW.SMALLINNER.LOGARITHM"
],
"scope_boundary": "Retained complete abelian fixed implementing unitary route; general central-order strengthening kept separately"
},
{
"id": "OA-FLOW.L125.PROBLEM",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.problem",
"start_line": 851,
"end_line": 871,
"range_sha256": "5971e263aef313aea83b26cfcf68168268265bfb8207b46ca7f6f5684d67ffbf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L125.CO1",
"OA-FLOW.L125.CO2",
"OA-FLOW.L125.CO5",
"OA-FLOW.SMALLINNER.LOGARITHM",
"OA-FLOW.SMALLINNER.FIXED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 871,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COBOUNDARY",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 603,
"end_line": 673,
"range_sha256": "ccc26eff556f36c37416ddf93de63f21a60715f7eeded111c41eb02ca5f32c09",
"full_statement_context": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.LOGARITHM",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 674,
"end_line": 783,
"range_sha256": "1d7b70cea703e2e1eebdd8f997dcf60458293c5d78e92dd45c1416f224b1b18b",
"full_statement_context": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n"
}
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 11,
"end_line": 13,
"range_sha256": "6659e73d8640ce596cfa453c54bfac49d942c578c098e0ab2364323bc008e015",
"full_statement_context": "Throughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n## Phase localization and abelian fixed-point route\n\nCO3 supplies the uniform compact-set subdivision used below, CO5 supplies the complete commuting affine fixed-point proof, and CO2 proves the precise logarithm branch. In P12–P13 take \\(\\delta+\\varepsilon<\\pi\\), as in CO4; a radius at least \\(\\pi\\) gives the whole circle and needs no \\(V(r)\\) notation outside its defined range. Its group hypotheses remain those actually stated, even though the central order route works for a larger group class.\n\n## Phase localization and fixed small-spectrum implementers\n\nA small operator spectrum for an inner automorphism can be converted into a\nsmall spectrum for one of its implementing unitaries. If the automorphism also\ncommutes with an abelian action, a central logarithmic correction makes that\nimplementer fixed by the action. The proof has two delicate points: the\ncentral spectral pieces must form an actually orthogonal partition, and the\nprincipal logarithm may be made additive only after a strict branch estimate.\n\nLet $M$ be a nonzero von Neumann algebra. For $0\n\n$$\nV(r)=\\{e^{it}:|t|\n\n$$\np(\\theta,r)\n=1_{\\{z\\in\\mathbb T:d_{\\mathbb T}(z,e^{i\\theta})\\le r\\}}(u).\n\\tag{P2}\n$$\n\nThe symbol $\\operatorname{Sp}(\\operatorname{Ad}(u))$ below denotes the\nordinary Banach-operator spectrum of $\\operatorname{Ad}(u)$ on $M$; it is not\nthe dual-group spectrum of an action.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COCYCLE",
"context_role": "standing_locally_compact_covariant_system_and_continuous_additive_cocycle_definition",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 572,
"end_line": 602,
"range_sha256": "575c1301edb21e83f4c920e42b4587e9a4d81d37b7ee0d5a08e891eef7f8288b",
"full_statement_context": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Additive cocycles and coboundaries\n\nLet $G$ be a locally compact group and let\n$\\alpha:G\\to\\operatorname{Aut}(M)$ be point-ultraweakly continuous. In\nmultiplicative notation, an **additive one-cocycle** is an ultraweakly\ncontinuous map $h:G\\to M$ satisfying\n\n\n\n$$\nh(st)=h(s)+\\alpha_s(h(t))\n\\qquad(s,t\\in G).\n\\tag{P22}\n$$\n\nIt is a **coboundary** if one element $k\\in M$ satisfies\n\n\n\n$$\nh(s)=\\alpha_s(k)-k\n\\qquad(s\\in G).\n\\tag{P23}\n$$\n\nThese definitions make sense for nonabelian $G$. The bounded-cocycle result\nproved next uses the standing abelian hypothesis, because its affine maps must\ncommute.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.COBOUNDARY",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 603,
"end_line": 673,
"range_sha256": "ccc26eff556f36c37416ddf93de63f21a60715f7eeded111c41eb02ca5f32c09",
"full_statement_context": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### Bounded additive cocycles for abelian actions are coboundaries\n\nAssume throughout this section that $G$ is abelian, and suppose\n\n\n\n$$\nC:=\\sup_{s\\in G}\\lVert h(s)\\rVert<\\infty.\n\\tag{P24}\n$$\n\nFor $s\\in G$, define the ultraweakly continuous affine isometry\n\n\n\n$$\nT_s(x)=\\alpha_s(x)-h(s).\n\\tag{P25}\n$$\n\nThe cocycle identity gives\n\n\n\n$$\nT_sT_t(x)\n=\\alpha_{st}(x)-\\alpha_s(h(t))-h(s)\n=T_{st}(x).\n\\tag{P26}\n$$\n\nThus the $T_s$ commute. The orbit of any $x\\in M$ is norm bounded, since\n\n\n\n$$\n\\lVert T_s(x)\\rVert\\le\\lVert x\\rVert+C.\n\\tag{P27}\n$$\n\nLet $K$ be its ultraweakly closed convex hull. The closed ball of $M$, viewed\nas the dual of its predual, is ultraweakly compact; hence $K$ is a nonempty\ncompact convex set. Moreover\n\n\n\n$$\nT_s\\{T_t(x):t\\in G\\}=\\{T_r(x):r\\in G\\},\n\\tag{P28}\n$$\n\nso every $T_s$ preserves $K$. The Markov–Kakutani fixed-point theorem supplies\na common fixed point $k\\in K$:\n\n\n\n$$\nT_s(k)=k\n\\quad\\Longleftrightarrow\\quad\nh(s)=\\alpha_s(k)-k.\n\\tag{P29}\n$$\n\nTherefore every norm-bounded additive cocycle for the standing locally compact\nabelian action is a coboundary. If the cocycle takes selfadjoint values in an\n$\\alpha$-invariant von Neumann subalgebra $A$, the argument may be run in $A$\nand $k$ replaced by $(k+k^*)/2$. In particular, a central selfadjoint cocycle\nhas a central selfadjoint transfer element.\n\n"
},
{
"referenced_body_scope_id": "OA-FLOW.SMALLINNER.LOGARITHM",
"context_role": "full_earlier_application_conditions_and_constructed_objects",
"consumed_as_new_strict_proof_edge": false,
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 674,
"end_line": 783,
"range_sha256": "1d7b70cea703e2e1eebdd8f997dcf60458293c5d78e92dd45c1416f224b1b18b",
"full_statement_context": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n",
"usage": "Exact declared context; definition references add no proved-premise edge.",
"full_conditions_and_proof": "\n\n### A strict logarithm branch turns the central unitary cocycle additive\n\nReturn to the locally compact abelian covariant system $(M,G,\\alpha)$. Let\n$\\sigma\\in\\operatorname{Int}(M)$ commute with every $\\alpha_s$, and assume\n\n\n\n$$\n\\operatorname{Sp}(\\sigma)\\subset V(\\varepsilon),\n\\qquad 0<\\varepsilon<\\frac{\\pi}{4}.\n\\tag{P30}\n$$\n\nBy (P21), choose $v\\in\\mathcal U(M)$ such that\n\n\n\n$$\n\\sigma=\\operatorname{Ad}(v),\n\\qquad\n\\operatorname{Sp}_M(v)\\subset V(2\\varepsilon).\n\\tag{P31}\n$$\n\nSince $\\sigma$ commutes with $\\alpha_s$,\n\n\n\n$$\n\\operatorname{Ad}(v)\n=\\alpha_s\\operatorname{Ad}(v)\\alpha_s^{-1}\n=\\operatorname{Ad}(\\alpha_s(v)).\n\\tag{P32}\n$$\n\nTwo unitaries implement the same inner automorphism exactly when their quotient\nis central. Hence\n\n\n\n$$\na_s=v^*\\alpha_s(v)\\in\\mathcal U(Z(M)),\n\\qquad\na_{st}=a_s\\alpha_s(a_t).\n\\tag{P33}\n$$\n\nLet $b=\\operatorname{Arg}(v)$ on the principal branch. Compactness of\n$\\operatorname{Sp}_M(v)$ inside the open arc in (P31) gives\n\n\n\n$$\nb=b^*,\n\\qquad\nv=e^{ib},\n\\qquad\n\\lVert b\\rVert<2\\varepsilon.\n\\tag{P34}\n$$\n\nEquation (P33) says $\\alpha_s(v)=va_s$, so $v$ and $\\alpha_s(v)$ commute.\nTheir principal arguments $b$ and $\\alpha_s(b)$ therefore commute. Put\n\n\n\n$$\nd_s=\\alpha_s(b)-b.\n\\tag{P35}\n$$\n\nThen\n\n\n\n$$\n\\lVert d_s\\rVert<4\\varepsilon<\\pi,\n\\qquad\na_s=e^{-ib}e^{i\\alpha_s(b)}=e^{id_s}.\n\\tag{P36}\n$$\n\nThe strict norm bound keeps the spectrum of $d_s$ inside the injectivity\ninterval of the exponential. Consequently\n\n\n\n$$\nh(s):=\\operatorname{Arg}(a_s)=d_s=\\alpha_s(b)-b.\n\\tag{P37}\n$$\n\nBecause $a_s$ is central, functional calculus makes $h(s)$ central and\nselfadjoint. Equation (P37), rather than a false global rule that logarithms\nturn arbitrary products into sums, gives directly\n\n\n\n$$\nh(st)\n=\\alpha_{st}(b)-b\n=h(s)+\\alpha_s(h(t)).\n\\tag{P38}\n$$\n\nThus $h:G\\to Z(M)_{\\mathrm{sa}}$ is an ultraweakly continuous additive\ncocycle, bounded by $4\\varepsilon$.\n\n"
}
],
"hypothesis_context_ids": [
"OA-FLOW.SMALLINNER.COCYCLE",
"OA-FLOW.SMALLINNER.COBOUNDARY",
"OA-FLOW.SMALLINNER.LOGARITHM"
],
"scope_boundary": "Entire retained solved problem: central correction preserves the inner automorphism"
},
{
"id": "OA-FLOW.L125.SOURCES",
"unit": "OA-FLOW-L125",
"source_file": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9",
"reader": "OA-FLOW-L125.html#oa-flow.l125.sources",
"start_line": 872,
"end_line": 877,
"range_sha256": "f8d07d078ba1fdef3ab3d7cb4acab333950ed4705c05031dc1506cfa7e9879f9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"statement_context_ranges": [
{
"source": "src/OA-FLOW-L125.md",
"source_sha256": "ea3555a84d4f6a4da38273a48ea29131bd91218bcbe54d012197794b4e896fdb",
"start_line": 1,
"end_line": 18,
"range_sha256": "6c474d8ffe36293d78de10c41902846ec75b1f9c182ba001f6a991f132d4747f",
"full_statement_context": "# A central order correction fixes the implementer and retains its bound\n\nAn inner automorphism with small Banach-operator spectrum has an implementer with a small principal logarithm. A commuting group can then be removed from that implementer by a central correction. We first prove the order construction that makes this correction possible for an arbitrary group. We next construct the small logarithm by spectral separation. The independent compact convex argument for a general abelian bounded cocycle and the entire earlier P argument remain available as alternative routes.\n\n*Original exposition and illustrations: CC0-1.0 to the extent of rights held; existing component and font terms apply.*\n\n\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## Full hypotheses and earlier proofs\n\nThroughout, \\(M\\ne0\\) is a concrete von Neumann algebra on an arbitrary complex Hilbert space. The center is \\(Z(M)\\). A group action \\(\\alpha:G\\to\\operatorname{Aut}(M)\\) uses multiplicative notation. The central order results impose no topology, commutativity or amenability on \\(G\\). The separate general noncentral cocycle theorem requires \\(G\\) abelian. When continuity is mentioned it means point-ultraweak continuity, and cocycle continuity uses the ultraweak topology.\n\nFor a unitary \\(v\\), the notation \\(S=\\operatorname{Ad}(v)\\) denotes a bounded linear isometry of the Banach space \\(M\\), with isometric inverse. Its \\(\\operatorname{Sp}(S)\\) is the ordinary Banach-operator spectrum; \\(\\operatorname{Sp}_M(v)\\) is the spectrum of the unitary in its algebra. These are not action spectra. We use \\(V(r)=\\{e^{it}:|t|\n\n## LC1. A small cyclic spectrum supplies a central fixed lift\n\nFix $t\\in G$ and put $\\sigma=\\alpha_t$. Suppose there exist a nonzero fixed projection $e$ and a radius $0\n\\[\n\\operatorname{Sp}_{\\mathcal L(eMe)}(\\sigma^e)\\subset V(r).\n\\tag{LC4}\n\\]\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.COMPACT.SELECTION",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#lc2",
"start_line": 79,
"end_line": 134,
"range_sha256": "461fe2f6d363796a163bac9053a0ca9308b93aeed827b6e6cdcf927df1b91839",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.ABSORPTION",
"OA-FLOW.CONNES.SUBGROUP",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.L121.SC0",
"OA-FLOW.DIRECTED.THICKENING",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 134,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n\n## LC2. Compact saturated thickenings select a corner\n\nNow assume precisely\n\n\\[\nS=\\pi(\\operatorname{Sp}(\\alpha))\\text{ is compact},\n\\qquad \\pi:H\\longrightarrow H/\\Gamma.\n\\tag{LC9}\n\\]\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.CENTRAL.FIXED.TIMES",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#lc3",
"start_line": 135,
"end_line": 164,
"range_sha256": "90de734980a12e43c8e6541bd88b3c417078ab1f3edf694fe2013031ac82f52c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.LOCAL.RECOGNITION",
"OA-FLOW.L126.COMPACT.SELECTION",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.AT.3",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.L122.DIRECTANNIHILATOR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 164,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n\n## LC2. Compact saturated thickenings select a corner\n\nNow assume precisely\n\n\\[\nS=\\pi(\\operatorname{Sp}(\\alpha))\\text{ is compact},\n\\qquad \\pi:H\\longrightarrow H/\\Gamma.\n\\tag{LC9}\n\\]\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 135,
"end_line": 145,
"full_statement_context": "\n\n## LC3. The annihilator is exactly the centrally implemented times\n\nTake $t\\in\\Gamma^\\perp$. The continuous character $p\\mapsto(t,p)$ is constant on $\\Gamma$-cosets, so the quotient topology supplies a continuous character $\\chi_t:H/\\Gamma\\to\\mathbb T$. Choose radii\n\n\\[\n0\n\n## LC4. An exact finite example and the meaning of the chosen phase\n\nLet $M=M_3(\\mathbb C)$, $G=\\mathbb R$, and\n\n\\[\nW_s=\\operatorname{diag}(1,e^{is},e^{2is}),\\qquad\n\\alpha_s=\\operatorname{Ad}(W_s),\\qquad e=E_{22}.\n\\tag{LC18}\n\\]\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.SOURCE.COMPARISON",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#l126-source-comparison",
"start_line": 190,
"end_line": 198,
"range_sha256": "81ce2ef8cb158cdfbaf196375c9686aa9ae85d46ec2b8dbc23db901f8932de5f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 198,
"source_level_hypothesis_ranges": []
},
{
"id": "OA-FLOW.L126.HIST.SETTING",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-alternative",
"start_line": 212,
"end_line": 251,
"range_sha256": "74c065f93df48afd0fc91817ba4186d7b971d8f48fffd13c906bb9e93aeb55a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 251,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.THICKENINGS",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-1",
"start_line": 252,
"end_line": 358,
"range_sha256": "f6208529ae999e8d5bd01350b13c6211dc4574a5f370e600925d24ab7ad5de59",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.ABSORPTION",
"OA-FLOW.CONNES.SUBGROUP",
"OA-FLOW.L121.SC0",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.DIRECTED.THICKENING",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 358,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 252,
"end_line": 358,
"full_statement_context": "\n### Compact quotient thickenings shrink to the identity coset\n\nFor $0\\ne e\\in\\operatorname{Proj}(D)$, the reduced spectrum absorbs\n$\\Gamma$:\n\n\n$$\n\\operatorname{Sp}(\\alpha^e)+\\Gamma\n=\\operatorname{Sp}(\\alpha^e).\n\\tag{Q3}\n$$\n\nIndeed, every nonzero fixed projection in $eMe$ is also a nonzero fixed\nprojection in $M$, so\n\n\n$$\n\\Gamma(\\alpha)\\subseteq\\Gamma(\\alpha^e).\n\\tag{Q4}\n$$\n\nApply the Connes-spectrum absorption law of\n[The dual-center kernel through central overlap](OA-FLOW-L115.md#oa-flow.connes.absorption) to $\\alpha^e$;\n(Q3) follows from (Q4).\n\nLet $U$ range over compact neighborhoods of $0$ in $H$, and set\n\n\n$$\nA_{e,U}=\\operatorname{Sp}(\\alpha^e)+U,\n\\qquad\nK_{e,U}=\\pi(A_{e,U}).\n\\tag{Q5}\n$$\n\nThe sets $A_{e,U}$ are closed: the sum of a closed set and a compact set is\nclosed in a Hausdorff topological group. Equation (Q3) makes them\n$\\Gamma$-saturated, so\n\n\n$$\n\\pi^{-1}(K_{e,U})=A_{e,U}.\n\\tag{Q6}\n$$\n\nThe quotient topology and (Q6) make $K_{e,U}$ closed. Moreover,\n\n\n$$\nK_{e,U}\n\\subseteq S+\\pi(U),\n\\tag{Q7}\n$$\n\nand the right side is compact by (Q2). Hence every $K_{e,U}$ is compact.\n\nThe directed-thickening theorem in\n[Mutual corner approximation through a family of spectral bridges](OA-FLOW-L121.md#oa-flow.directed.thickening)\nshows that the $A_{e,U}$, and therefore the $K_{e,U}$, are downward directed\nunder inclusion. Restricting to compact neighborhoods loses nothing: inside\nany prescribed neighborhood of zero there is a compact neighborhood small\nenough for the same refinement argument.\n\nTheir intersection is the identity coset. First,\n$\\Gamma\\subseteq A_{e,U}$ for every pair $(e,U)$. Conversely, if\n$p\\notin\\Gamma$, the central-corner formula for the Connes spectrum supplies\n$0\\ne e\\in\\operatorname{Proj}(D)$ with\n$p\\notin\\operatorname{Sp}(\\alpha^e)$. Closedness of that spectrum supplies a\ncompact neighborhood $U$ of zero such that\n\n\n$$\np\\notin\\operatorname{Sp}(\\alpha^e)+U.\n\\tag{Q8}\n$$\n\nThus\n\n\n$$\n\\bigcap_{e,U}A_{e,U}=\\Gamma.\n\\tag{Q9}\n$$\n\nBecause every member is saturated, inverse images commute with this\nintersection without losing representatives:\n\n\n$$\n\\pi^{-1}\\!\\left(\\bigcap_{e,U}K_{e,U}\\right)\n=\\bigcap_{e,U}\\pi^{-1}(K_{e,U})\n=\\Gamma.\n\\tag{Q10}\n$$\n\nConsequently\n\n\n$$\n\\boxed{\\bigcap_{e,U}K_{e,U}=\\{0+\\Gamma\\}.}\n\\tag{Q11}\n$$\n\nThis saturation step is what justifies passing the infinite intersection to\nthe quotient.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.SMALL.CORNER",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-2",
"start_line": 359,
"end_line": 427,
"range_sha256": "57d5b80053be8b5703f3c3daaed06de797664b9cb18006ac4684faf9ee30e263",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.HIST.THICKENINGS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 427,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 359,
"end_line": 427,
"full_statement_context": "\n### Select a central corner with a small time spectrum\n\nFix\n\n\n$$\nt\\in\\Gamma^\\perp\n=\\{s\\in G:(s,p)=1\\text{ for every }p\\in\\Gamma\\}.\n\\tag{Q12}\n$$\n\nThe character $p\\mapsto(t,p)$ is constant on cosets of $\\Gamma$, and hence\ndescends to a continuous character $\\chi_t$ of $H/\\Gamma$. Choose\n\n\n$$\n0<\\varepsilon<1-\\frac1{\\sqrt2},\n\\tag{Q12a}\n$$\n\nand define\n\n\n$$\nO_\\varepsilon\n=\\{q\\in H/\\Gamma:\\operatorname{Re}\\chi_t(q)>1-\\varepsilon\\},\n\\qquad\nU_\\varepsilon=\\pi^{-1}(O_\\varepsilon).\n\\tag{Q13}\n$$\n\nWe use a compact directed-family lemma. If a downward-directed family of\nnonempty compact sets has intersection contained in an open set $O$, then one\nmember is contained in $O$. To see this, fix one member $K_0$. If no member\nlay in $O$, directedness would make the compact sets\n\n\n$$\nK_0\\cap K\\cap O^c\n\\tag{Q14}\n$$\n\nnonempty with the finite-intersection property. Their intersection would give\na point in the total intersection outside $O$, a contradiction.\n\nApply this lemma to (Q11) and $O_\\varepsilon$. For some\n$0\\ne e\\in\\operatorname{Proj}(D)$ and compact neighborhood $U$ of zero,\n\n\n$$\nK_{e,U}\\subset O_\\varepsilon.\n\\tag{Q15}\n$$\n\nSince $\\operatorname{Sp}(\\alpha^e)\\subset A_{e,U}$, equations\n(Q6), (Q13), and (Q15) give the desired spectral corner:\n\n\n$$\n\\boxed{\n\\operatorname{Sp}(\\alpha^e)\\subset U_\\varepsilon\n=\\{p\\in H:\\operatorname{Re}(t,p)>1-\\varepsilon\\}.}\n\\tag{Q16}\n$$\n\nOnly compactness of the image $S$ in (Q2) was used to make the quotient\nthickenings compact.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.GLOBAL.INNERNESS",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-3",
"start_line": 428,
"end_line": 508,
"range_sha256": "9387043a761d16c21e8dca15c50a7be72b5a93a4fb3d0ff6740e8e51681dfa68",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.HIST.SMALL.CORNER",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.AT.3",
"OA-FLOW.CONNES.SUBGROUP",
"OA-FLOW.MINFIX.FULLCONNES",
"OA-FLOW.PC.2",
"OA-FLOW.L124.CE5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 508,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 428,
"end_line": 508,
"full_statement_context": "\n### The small cyclic spectrum forces global innerness\n\nSet\n\n\n$$\n\\sigma=\\alpha_t.\n\\tag{Q17}\n$$\n\nThe projection $e$ in (Q16) is fixed by $\\sigma$. The individual-operator\nspectrum formula from [The spectrum of one action operator](OA-FLOW-L89.md#oa-flow.opsp.formula),\napplied inside $eMe$, gives\n\n\n$$\n\\operatorname{Sp}_{\\mathcal L(eMe)}(\\sigma^e)\n=\\overline{\\{(t,p):p\\in\\operatorname{Sp}(\\alpha^e)\\}}.\n\\tag{Q18}\n$$\n\nIt is the action spectrum $\\operatorname{Sp}(\\alpha^e)$ that belongs on the\nright side of (Q18). Equation (Q16) yields\n\n\n$$\n\\operatorname{Sp}_{\\mathcal L(eMe)}(\\sigma^e)\n\\subseteq\n\\{z\\in\\mathbb T:\\operatorname{Re}z\\ge1-\\varepsilon\\}.\n\\tag{Q19}\n$$\n\nRegard $\\sigma^e$ as the action of $\\mathbb Z$ on $eMe$. Its action spectrum\nis the operator spectrum in (Q18), and its Connes spectrum is a subgroup of\n$\\mathbb T$ contained in that spectrum. The choice (Q12a) has\n$0<\\varepsilon<1$, so the closed arc on the right of (Q19) is contained in a\nproper open semicircle. No\nnontrivial subgroup of $\\mathbb T$ can be contained in such an arc: the powers\nof any nonidentity element eventually leave it. Therefore\n\n\n$$\n\\Gamma(\\sigma^e)=\\{1\\}.\n\\tag{Q20}\n$$\n\nCentral ergodicity makes $e$ full in $M$. Indeed, its ambient central support\n$z_M(e)$ is a nonzero $\\alpha$-invariant central projection, so\n\n\n$$\nz_M(e)=1.\n\\tag{Q21}\n$$\n\nThe full fixed-corner theorem from\n[Cancel an inner subgroup, then detect the exact spectrum](OA-FLOW-L119.md#oa-flow.minfix.fullconnes) now\ngives\n\n\n$$\n\\Gamma(\\sigma)=\\Gamma(\\sigma^e)=\\{1\\}.\n\\tag{Q22}\n$$\n\nEvery $\\alpha_s$ commutes with $\\sigma$ because $G$ is abelian. Hence\n$\\alpha(G)\\subseteq\\operatorname{Aut}_\\sigma(M)$, and the action of\n$\\operatorname{Aut}_\\sigma(M)$ on $Z(M)$ is ergodic. Applying the\ncentralizer-ergodicity theorem from\n[Read an inner implementer from a circle eigenunitary](OA-FLOW-L124.md#ce5)\nto (Q22) proves\n\n\n$$\n\\boxed{\\sigma\\in\\operatorname{Int}(M).}\n\\tag{Q23}\n$$\n\nThe centralizer-ergodicity criterion applies to the whole algebra here.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.FIXED.CORRECTION",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-4",
"start_line": 509,
"end_line": 565,
"range_sha256": "96181f644a61751a80d258ae817c920b727e47955ee897ed02e7558a459b199c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.HIST.GLOBAL.INNERNESS",
"OA-FLOW.L125.CO4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 565,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 509,
"end_line": 565,
"full_statement_context": "\n### Choose consistent constants and fix the corner implementer\n\nThe tolerance chosen in (Q12a) satisfies\n\n\n$$\n0<\\varepsilon<1-\\frac1{\\sqrt2}.\n\\tag{Q24}\n$$\n\nPut\n\n\n$$\n\\theta_0=\\arccos(1-\\varepsilon).\n\\tag{Q25}\n$$\n\nThen $0<\\theta_0<\\pi/4$. Choose a strictly larger open-arc radius\n\n\n$$\n\\theta_0<\\delta<\\frac\\pi4.\n\\tag{Q26}\n$$\n\nEquations (Q19), (Q25), and (Q26) give\n\n\n$$\n\\operatorname{Sp}_{\\mathcal L(eMe)}(\\sigma^e)\n\\subseteq\\{e^{ir}:|r|\\le\\theta_0\\}\n\\subset V(\\delta).\n\\tag{Q27}\n$$\n\nThe strict enlargement in (Q26) is needed because (Q19) is a closed spectral\nbound.\n\nBy (Q23), $\\sigma^e$ is inner. It commutes with the reduced action\n$\\alpha^e$. The fixed small-spectrum implementer theorem from\n[A central order correction fixes the implementer and retains its bound](OA-FLOW-L125.md#co4),\napplied with the radius $\\delta<\\pi/4$, supplies\n\n\n$$\nv\\in\\mathcal U((eMe)^{\\alpha^e})\n=\\mathcal U(eM^\\alpha e),\n\\qquad\n\\sigma^e=\\operatorname{Ad}(v).\n\\tag{Q28}\n$$\n\nThus the corner implementer is fixed by the entire reduced action, not merely\nby $\\sigma^e$.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.UNIQUE.LIFT",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-5",
"start_line": 566,
"end_line": 620,
"range_sha256": "f61e7e5d2a473a4037306b0efa9f8f878b345e0784aefb2cbb360c051ccb448b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.HIST.FIXED.CORRECTION",
"OA-FLOW.FULLCORNER.INNER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 620,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 566,
"end_line": 620,
"full_statement_context": "\n### Full-corner uniqueness puts the lift in the fixed center\n\nEquations (Q21) and (Q28) meet the hypotheses of the unique inner-lifting\ntheorem from\n[Lifting innerness and cocycles from a full corner](OA-FLOW-L117.md#oa-flow.fullcorner.inner).\nThere is a unique $u\\in\\mathcal U(M)$ such that\n\n\n$$\n\\sigma=\\operatorname{Ad}(u),\n\\qquad\nue=eu=v.\n\\tag{Q29}\n$$\n\nFor $s\\in G$, the unitary $\\alpha_s(u)$ also implements $\\sigma$, because\n$\\alpha_s$ commutes with $\\sigma$. Since $e$ and $v$ are fixed,\n\n\n$$\n\\alpha_s(u)e\n=\\alpha_s(ue)\n=\\alpha_s(v)\n=v.\n\\tag{Q30}\n$$\n\nUniqueness in (Q29) forces\n\n\n$$\n\\alpha_s(u)=u\n\\qquad(s\\in G),\n\\tag{Q31}\n$$\n\nso $u\\in M^\\alpha$. Finally, $\\sigma=\\alpha_t$ fixes every element of\n$M^\\alpha$ pointwise. Equation (Q29) therefore gives\n\n\n$$\nuxu^*=x\n\\qquad(x\\in M^\\alpha).\n\\tag{Q32}\n$$\n\nTogether, (Q31)–(Q32) show\n\n\n$$\n\\boxed{u\\in\\mathcal U(Z(M^\\alpha)).}\n\\tag{Q33}\n$$\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.ANNIHILATOR",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-6",
"start_line": 621,
"end_line": 660,
"range_sha256": "80faf4b88d357459494da5f5a10be6e309c2aa38f96db8b092a8ac0f886b7d41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L126.HIST.UNIQUE.LIFT",
"OA-FLOW.L122.DIRECTANNIHILATOR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 660,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
},
{
"source": "src/OA-FLOW-L126.md",
"start_line": 621,
"end_line": 660,
"full_statement_context": "\n### Identify the annihilator exactly\n\nWe have proved the difficult inclusion\n\n\n$$\n\\Gamma^\\perp\n\\subseteq\n\\{s\\in G:\\alpha_s=\\operatorname{Ad}(u)\n\\text{ for some }u\\in\\mathcal U(Z(M^\\alpha))\\}.\n\\tag{Q34}\n$$\n\nFor the reverse inclusion, a central fixed implementer is in particular a\nfixed implementer. The fixed-implementer annihilator theorem from\n[Small fixed corners, inner spectra and annihilating times](OA-FLOW-L122.md#oa-flow.l122.directannihilator) gives\n\n\n$$\n\\alpha_s=\\operatorname{Ad}(u),\\quad\nu\\in\\mathcal U(Z(M^\\alpha))\n\\quad\\Longrightarrow\\quad\ns\\in\\Gamma^\\perp.\n\\tag{Q35}\n$$\n\nThis proves Theorem XI.2.9(iii):\n\n\n$$\n\\boxed{\n\\begin{aligned}\n\\Gamma(\\alpha)^\\perp\n=\\{s\\in G:\\ &\\alpha_s=\\operatorname{Ad}(u)\\\\\n&\\text{for some }u\\in\\mathcal U(Z(M^\\alpha))\\}.\n\\end{aligned}}\n\\tag{Q36}\n$$\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.COMPLETE",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-step-1",
"start_line": 252,
"end_line": 660,
"range_sha256": "3d52fbe324ded79b610af2492e6a10d71f7cf7854d6d45baece606d4b23b07ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.ABSORPTION",
"OA-FLOW.CONNES.SUBGROUP",
"OA-FLOW.L121.SC0",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.INTERSECTION",
"OA-FLOW.DIRECTED.THICKENING",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.AT.3",
"OA-FLOW.MINFIX.FULLCONNES",
"OA-FLOW.PC.2",
"OA-FLOW.FULLCORNER.INNER",
"OA-FLOW.L122.DIRECTANNIHILATOR",
"OA-FLOW.L124.CE5",
"OA-FLOW.L125.CO4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 660,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.HIST.SOLVED.PROBLEM",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#historical-problem",
"start_line": 661,
"end_line": 679,
"range_sha256": "6206fd4577648939f57644f6c503816101e2877e582e88472db67b548db7c4cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 679,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L126.md",
"start_line": 9,
"end_line": 9,
"full_statement_context": "Let $M\\ne0$ be a concrete von Neumann algebra on an arbitrary Hilbert space, let $G$ be a locally compact Hausdorff abelian group, and let $\\alpha:G\\to\\operatorname{Aut}(M)$ be a normal point-ultraweakly continuous action. Suppose $Z(M)^\\alpha=\\mathbb C1$. Write $H=\\widehat G$, $F=M^\\alpha$, $D=Z(F)$ and $\\Gamma=\\Gamma(\\alpha)$. Frequencies have the positive L115 convention. For $0\n$$\n\\Gamma=\\Gamma(\\alpha),\n\\qquad\nD=Z(M^\\alpha),\n\\qquad\n\\pi:H\\longrightarrow H/\\Gamma.\n\\tag{Q1}\n$$\n\nAssume the precise compactness hypothesis\n\n\n$$\nS:=\\pi(\\operatorname{Sp}(\\alpha))\n\\text{ is compact in }H/\\Gamma.\n\\tag{Q2}\n$$\n\nThe whole quotient $H/\\Gamma$ need not be compact. No separability,\nsigma-finiteness, second-countability, or countable-multiplicity assumption is\nmade.\n\n",
"usage": "Full hypotheses and statement context; not a separate proved premise."
}
]
},
{
"id": "OA-FLOW.L126.CURRENT.DEVELOPMENT.BOUNDARY",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#l126-current-development-boundary",
"start_line": 680,
"end_line": 684,
"range_sha256": "ba31f69c6714994e0b0eb3de91b5f50bf338bd23b0c5e18b658a298da9a00533",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 684,
"source_level_hypothesis_ranges": []
},
{
"id": "OA-FLOW.L126.FIGURE",
"unit": "OA-FLOW-L126",
"source_file": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f",
"reader": "OA-FLOW-L126.html#l126-figure",
"start_line": 199,
"end_line": 211,
"range_sha256": "0ab39178966ccc0bb500e24924f66ba5f509630787c9285dcb0ffdc878cf9260",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 211,
"source_level_hypothesis_ranges": []
},
{
"id": "OA-FLOW.L127.SETTING",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.setting",
"start_line": 8,
"end_line": 17,
"range_sha256": "e949ca7e8cfa6310c67ed3936594c3f590d319c1aca38b98baf10bca40bfdd64",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.L127.CN0",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn0",
"start_line": 20,
"end_line": 34,
"range_sha256": "1e6151de952761d2c23b08cbda5e60c93eaeca79d1bbb8504b4fc6b3d3aa7f69",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.L127.CN1",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn1",
"start_line": 37,
"end_line": 79,
"range_sha256": "7868e068a7a9b4fdc39031c80a803f8763690e876668294c19fe029aeaccf901",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN0"
]
},
{
"id": "OA-FLOW.L127.CN2.TOPOLOGY",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn2.topology",
"start_line": 82,
"end_line": 105,
"range_sha256": "323be2e8f4aaa98e8e9bfe0dfb1e23abb122bd5cbb71eb9d0a0d57130d4cca9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.SUBGROUP",
"OA-FLOW.L118.DUAL-MAP",
"OA-FLOW.XGAPS.CSTAR.H3"
]
},
{
"id": "OA-FLOW.L127.CN2.IMPLEMENTERS",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn2.implementers",
"start_line": 108,
"end_line": 123,
"range_sha256": "2f53e1c9eae1ef93fc714cfe87f5dc307feb9240a21938450156cd0a451cbeec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN0",
"OA-FLOW.L127.CN1",
"OA-FLOW.L127.CN2.TOPOLOGY",
"OA-FLOW.L126.CENTRAL.FIXED.TIMES",
"OA-FLOW.CONNES.ABSORPTION",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.L127.CN3",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn3",
"start_line": 126,
"end_line": 157,
"range_sha256": "d40e954b6d358790b9da771d74a2132f5aa4ba852b8db2b93cce407116266062",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN2.IMPLEMENTERS",
"OA-FLOW.REPEXT.GROUP"
]
},
{
"id": "OA-FLOW.L127.CONTINUITY",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.continuity",
"start_line": 160,
"end_line": 187,
"range_sha256": "70af49278df19fe0f8126ee569186dcab96861b4e6f5771c025d05bf027ff363",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CP.6"
]
},
{
"id": "OA-FLOW.L127.CN4",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn4",
"start_line": 190,
"end_line": 212,
"range_sha256": "214319c7f597e7159aad88868ffdd0f510125bf7d31d713ca774eebfc7622dbd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN3",
"OA-FLOW.L127.CONTINUITY",
"OA-FLOW.L122.DIRECTANNIHILATOR"
]
},
{
"id": "OA-FLOW.L127.KERNELALTERNATIVE",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.kernelalternative",
"start_line": 215,
"end_line": 232,
"range_sha256": "0424a8d2e039f1a45de73c83cb69b7505e644929530f6812587d771de12a80fe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CONTINUITY",
"OA-FLOW.COCMAT.ACTION",
"OA-FLOW.COCMAT.CORNERS",
"OA-FLOW.OPSP.FORMULA",
"OA-FLOW.AT.3",
"OA-FLOW.GCC.DEFINITION",
"OA-FLOW.L115.CONVENTIONS"
]
},
{
"id": "OA-FLOW.L127.CN5",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn5",
"start_line": 235,
"end_line": 265,
"range_sha256": "fca636da58ccd885c7a98ea69249ae4d626d7017e1245fcbce3aea6501a39dc5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L118.DUAL-MAP",
"OA-FLOW.L127.CN1"
]
},
{
"id": "OA-FLOW.L127.CN6",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.cn6",
"start_line": 268,
"end_line": 290,
"range_sha256": "04a82545f6bf5489a6cb4056bb6b254ed1cdb678240df2af515de3983197d177",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN1",
"OA-FLOW.GCC.SETTING",
"OA-FLOW.GCC.DEFINITION"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.SETTING",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.setting",
"start_line": 308,
"end_line": 349,
"range_sha256": "75c07936ecd9bf5893d438c00977000d6bea0902128d5557ed91a9f0317253f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.L127.HISTORICAL.CIRCLE",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.circle",
"start_line": 352,
"end_line": 417,
"range_sha256": "9895983d8cb9d8a50c4c63c27c62702e97b7673fa943f41efe88e011f5f5ab55",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN5",
"OA-FLOW.L118.DUAL-MAP"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.DISCRETE",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.discrete",
"start_line": 420,
"end_line": 452,
"range_sha256": "89b0951d1125108d87fff6a95dc946009beaa434f0ffef281234c727e036f316",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN2.TOPOLOGY",
"OA-FLOW.L127.CN2.IMPLEMENTERS"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.MULTIPLIER",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.multiplier",
"start_line": 455,
"end_line": 510,
"range_sha256": "893a9e558ddb7270a61366b8b6247dbc732e1243d753f04b2f1acfd83b732382",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN1",
"OA-FLOW.L127.HISTORICAL.DISCRETE"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.SPLIT",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.split",
"start_line": 513,
"end_line": 618,
"range_sha256": "7470bf0046d005df05cb781d44b48bdecb433ca22fc6c7f214f276b7c3684718",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN5",
"OA-FLOW.L127.HISTORICAL.MULTIPLIER"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.EXTENSION",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.extension",
"start_line": 621,
"end_line": 670,
"range_sha256": "8cfa902abbf044e2fa3d143577f90f7e16177fca6f6765f492fdcd463803bd21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN3",
"OA-FLOW.L127.CONTINUITY",
"OA-FLOW.L127.HISTORICAL.SPLIT",
"OA-FLOW.REPEXT.GROUP"
]
},
{
"id": "OA-FLOW.L127.HISTORICAL.KERNEL",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.historical.kernel",
"start_line": 673,
"end_line": 741,
"range_sha256": "71191f3ba9f1cd78d35bdbe2aba41769f08d3bfbac43f8319b0bcb5658ca7c7b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN4",
"OA-FLOW.L127.KERNELALTERNATIVE",
"OA-FLOW.L127.HISTORICAL.EXTENSION",
"OA-FLOW.GCC.DEFINITION"
]
},
{
"id": "OA-FLOW.L127.PROBLEM",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.problem",
"start_line": 744,
"end_line": 760,
"range_sha256": "021e4dcfc969ae38e7d3bffeb6cfc4a941bb041ce5ff5ee328ddd743f1aef083",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L127.CN1"
]
},
{
"id": "OA-FLOW.L127.SOURCES",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.sources",
"start_line": 763,
"end_line": 770,
"range_sha256": "2b3040096f0f934e548e333d5e2fcf4d3eb71461db706268cf02d17f79497fb4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.L127.FIGURE",
"unit": "OA-FLOW-L127",
"source_file": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9",
"reader": "OA-FLOW-L127.html#oa-flow.l127.figure",
"start_line": 291,
"end_line": 306,
"range_sha256": "ecc030cb498014e32cd3c2092ab4872fbc4d8f66b5100f7ef5ebb27c61ec2601",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": []
},
{
"id": "OA-FLOW.L128.SETTING",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-setting",
"start_line": 1,
"end_line": 16,
"range_sha256": "b3e08f4ddbf455a212655874472014d00d2e239bff3fcb1692a520f3c8024b7e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null
},
{
"id": "OA-FLOW.L128.RT0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#regular-transport",
"start_line": 18,
"end_line": 21,
"range_sha256": "1eca234b82a1f5237d82ee51049d8e117c9054b87621507216f60f75606b5a13",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NCF.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 21
},
{
"id": "OA-FLOW.L128.FC0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc0",
"start_line": 22,
"end_line": 41,
"range_sha256": "a75de6bd5bd8e799dee8c89b85a82dabf8e99527d6ce9ca74851165a9c58f050",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.RT0",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 41
},
{
"id": "OA-FLOW.L128.FC1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc1",
"start_line": 42,
"end_line": 62,
"range_sha256": "de9d6f3d9673efe33746d7e7b283048b038c07de8263d4c14752a2137a9f3843",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 62
},
{
"id": "OA-FLOW.L128.FC2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc2",
"start_line": 63,
"end_line": 81,
"range_sha256": "0c24f24e2c1ca9b7f99b354708a50afcc04e3e52c2c409e9805878cb42fdb24d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 81
},
{
"id": "OA-FLOW.L128.FC3",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc3",
"start_line": 111,
"end_line": 129,
"range_sha256": "b0a5c893ad0395b6a7f330d67954dbeebcbe9ea6a2ccc66b5424b4ff9306a4f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.CF.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 129
},
{
"id": "OA-FLOW.L128.FC4",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc4",
"start_line": 130,
"end_line": 154,
"range_sha256": "bda6a8ef00949c0b68386d1ce05a44bd31606fb36cf44515bca50a01ae52884e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 154
},
{
"id": "OA-FLOW.L128.FC5",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc5",
"start_line": 155,
"end_line": 198,
"range_sha256": "abbf4f7d0e5b839607d2b28725aa9934d82bd08c1a26f225f46ace8c4b6d5733",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC3",
"OA-FLOW.CF.6",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 198
},
{
"id": "OA-FLOW.L128.FC6",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc6",
"start_line": 199,
"end_line": 208,
"range_sha256": "94e461d7657db139ce2b6ee400895ce0b39d047c81403876ac221d97b27acfd8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 208
},
{
"id": "OA-FLOW.L128.TC0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc0",
"start_line": 209,
"end_line": 220,
"range_sha256": "685a9dddd3327ed8939fd0ece4268af7a8bbe83b969327cb911c6d3fa50183c0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.PC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 220
},
{
"id": "OA-FLOW.L128.TC1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc1",
"start_line": 221,
"end_line": 247,
"range_sha256": "fa64dc28267c93c62f2ee93237165620ed91aec6d057687c81c603b3c5f803e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.TC0",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 247
},
{
"id": "OA-FLOW.L128.TC2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc2",
"start_line": 248,
"end_line": 266,
"range_sha256": "3b4481a1b046ff85f984cb2b01b81ee89004a51331eb19e5a5b46db9bef22e7c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.TC0",
"OA-FLOW.L128.TC1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 266
},
{
"id": "OA-FLOW.L128.TC3",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc3",
"start_line": 267,
"end_line": 281,
"range_sha256": "cfed6f94c058a8fc304a99c243939e201ec28c921f27e17bb427772e77c24678",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.TC0",
"OA-FLOW.L128.TC1",
"OA-FLOW.L128.TC2",
"OA-FLOW.PC.1",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 281
},
{
"id": "OA-FLOW.L128.TC4",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc4",
"start_line": 316,
"end_line": 340,
"range_sha256": "b6afe05bff214345ad23009b838ac097b6f079334a8e6ac1305aec9816874528",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 340
},
{
"id": "OA-FLOW.L128.TC5",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-tc5",
"start_line": 341,
"end_line": 369,
"range_sha256": "6dd4dd283d270807514d5111a422b5c0326f929c8bf7e78889f89be766cae9c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.TC4",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.NF.4",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 369
},
{
"id": "OA-FLOW.L128.OR0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-or0",
"start_line": 370,
"end_line": 381,
"range_sha256": "8cadf2d5b6124775298077e1db0d8699ac9aa7a7d63e06bddad46a21682f04d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.HR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 381
},
{
"id": "OA-FLOW.L128.OR1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-or1",
"start_line": 468,
"end_line": 483,
"range_sha256": "5f11d4853ac2528275261f3403827e5e471b0d8f4b3e1452ff5971d2a822eadb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OR0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.HR.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 483
},
{
"id": "OA-FLOW.L128.MD0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-md0",
"start_line": 484,
"end_line": 495,
"range_sha256": "8df08d5ecb3bd2e21b804e4aac8855a459260c286488686819f5b9919f13976b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 495
},
{
"id": "OA-FLOW.L128.MD1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-md1",
"start_line": 496,
"end_line": 525,
"range_sha256": "f193c7a7a7cb9f99d0ec886136cf4350a010404f643bb29ba6c11689d85c2957",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.MD0",
"OA-FLOW.CF.8",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 525
},
{
"id": "OA-FLOW.L128.OR2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-or2",
"start_line": 526,
"end_line": 553,
"range_sha256": "81da1297999356c4fa922e61e235530835100262186523c973722fa10e3c12b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.OR1",
"OA-FLOW.L128.MD0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.CF.1",
"OA-FLOW.CC.0",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 553
},
{
"id": "OA-FLOW.L128.HT0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ht0",
"start_line": 554,
"end_line": 575,
"range_sha256": "d4ce48f13b020ef16f7c8a9646c9feae993dbc2abf8e3ca4a2d24999209db241",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CF.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 575
},
{
"id": "OA-FLOW.L128.HT1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ht1",
"start_line": 576,
"end_line": 585,
"range_sha256": "d0d2cd31456dd1d50cc697d12c8a24d2804659bcab8f1732905439f8a4403f16",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 585
},
{
"id": "OA-FLOW.L128.TY0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ty0",
"start_line": 586,
"end_line": 655,
"range_sha256": "6d54854b502dbb34cfdd44eedd12f6577a872e09338d5dc9b22a249c4780704f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC4",
"OA-FLOW.L128.TC0",
"OA-FLOW.L128.TC1",
"OA-FLOW.L128.TC2",
"OA-FLOW.L128.TC3",
"OA-FLOW.L128.TC4",
"OA-FLOW.L128.TC5",
"OA-FLOW.L128.HT0",
"OA-FLOW.L128.HT1",
"OA-FLOW.L128.OR0",
"OA-FLOW.CC.0",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 655
},
{
"id": "OA-FLOW.L128.TY1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ty1",
"start_line": 656,
"end_line": 697,
"range_sha256": "15b6aa54e6b7b528168f46b10726e82cd9f5a1c8af7eb589987e913cfb8e0bdb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC5",
"OA-FLOW.L128.TY0",
"OA-FLOW.L128.OR2",
"OA-FLOW.L128.RT0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.CC.0",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 697
},
{
"id": "OA-FLOW.L128.TY2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ty2",
"start_line": 698,
"end_line": 778,
"range_sha256": "3c0b261f21b34174e1b60b9d5f381fe8053a8566d7d0d3add5816f195eac67bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC5",
"OA-FLOW.L128.TY0",
"OA-FLOW.L128.RT0",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 778
},
{
"id": "OA-FLOW.L128.MA0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ma0",
"start_line": 779,
"end_line": 817,
"range_sha256": "a8a9beee64d271fa67e0b84a6a73cbd9bb5cb6c85b9b3b31c5b53a616da9b925",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.OR1",
"OA-FLOW.L128.OR2",
"OA-FLOW.PC.1",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 817
},
{
"id": "OA-FLOW.L128.FG0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fg0",
"start_line": 818,
"end_line": 920,
"range_sha256": "ecd57ed87c9d00a85cf15dba25f5ae5a9b8311d4381fc66c4cf6b55979fbc1cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.TC3",
"OA-FLOW.CC.0",
"OA-FLOW.CF.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 920
},
{
"id": "OA-FLOW.L128.MA1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-ma1",
"start_line": 921,
"end_line": 939,
"range_sha256": "1b7a0b44c7e2622beefdef5c9322e4d4b7c7d8e21a3fe523e2b43c53f72cae11",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC4",
"OA-FLOW.CC.0",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 939
},
{
"id": "OA-FLOW.L128.MD2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-md2",
"start_line": 940,
"end_line": 963,
"range_sha256": "32e3d3990b43c3d1883a6be8557bc2ac77497b03303437dc43de9994a33dadcc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.MD0",
"OA-FLOW.L128.MD1",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 963
},
{
"id": "OA-FLOW.L128.OS0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-os0",
"start_line": 964,
"end_line": 1026,
"range_sha256": "a24320ca9bfd413008954e8303940eabd413220348d1a1e136252c9344a9a0b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.OR0",
"OA-FLOW.CF.1",
"OA-FLOW.HR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1026
},
{
"id": "OA-FLOW.L128.OS1",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-os1",
"start_line": 1027,
"end_line": 1068,
"range_sha256": "007c89b129ca5a9d677e8222510e9bc4b0a2c45bcf2a8f78a4f305fe2e351abe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OS0",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1068
},
{
"id": "OA-FLOW.L128.OR3",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-or3",
"start_line": 1069,
"end_line": 1083,
"range_sha256": "cc4aa2ab1e44a7ac47ef3c3597c16671e72e550d7768bd80568d0b2b3c2962df",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.OS0",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1083
},
{
"id": "OA-FLOW.L128.OS2",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-os2",
"start_line": 1084,
"end_line": 1151,
"range_sha256": "789cb49fab172fc59b82e6e7081cd06f6def7e28053e56358ee5fa3aa10acda8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.OS0",
"OA-FLOW.L128.OS1",
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.OR3",
"OA-FLOW.CF.10",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1151
},
{
"id": "OA-FLOW.L128.MD3",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-md3",
"start_line": 1152,
"end_line": 1204,
"range_sha256": "3da459c35f010da39189bddb47c220d44e0aa40c6467ca81e3bc49b49c9c8daf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC4",
"OA-FLOW.L128.MD2",
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.OR3",
"OA-FLOW.L128.OS0",
"OA-FLOW.L128.OS1",
"OA-FLOW.L128.OS2",
"OA-FLOW.CCM.7",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1204
},
{
"id": "OA-FLOW.L128.HX0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-hx0",
"start_line": 1205,
"end_line": 1480,
"range_sha256": "0adf5a89147555ea0ddee20734ef14affd30b7ba6d1ca7ee8dadf09e5b0b3612",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.FC3",
"OA-FLOW.L128.FC4",
"OA-FLOW.L128.FC5",
"OA-FLOW.PC.1",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1480
},
{
"id": "OA-FLOW.L128.WK0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-wk0",
"start_line": 1481,
"end_line": 1493,
"range_sha256": "73c13ec428f2f6e819bdba3e308c06d7991f9e94a094dc79978dc7b8ff2d1083",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.FC3",
"OA-FLOW.L128.FC4",
"OA-FLOW.L128.FC6",
"OA-FLOW.L128.TY0",
"OA-FLOW.L128.OS0",
"OA-FLOW.L128.OS1",
"OA-FLOW.L128.OR3",
"OA-FLOW.L128.OS2",
"OA-FLOW.L128.MD3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1493
},
{
"id": "OA-FLOW.L129.SETTING",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-setting",
"start_line": 7,
"end_line": 16,
"range_sha256": "fb1edb8d904b7daa2cb97768a8e90c1415f22d93f01f5322460ac4f917e9118e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 16,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP0",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp0",
"start_line": 17,
"end_line": 63,
"range_sha256": "33dc5182773f7525ade229c3358482795b71e5e3878dec35bdc0a2edc899353e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC4",
"OA-FLOW.PC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 63,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP1",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp1",
"start_line": 64,
"end_line": 93,
"range_sha256": "669e6b2fc5b4282256b2e1ebe579c1cea9b5591a97bcda864e6529771ff7f9ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 93,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP2",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp2",
"start_line": 94,
"end_line": 132,
"range_sha256": "cad163fce7a30d149f4c392adb53997748cf318ecd7c642f86860b8ade3e0fc1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 132,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP3",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp3",
"start_line": 133,
"end_line": 178,
"range_sha256": "9f275985121ff5cd40e08886d67c5627a93147e386070f77c38af5cce46cd2a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP0",
"OA-FLOW.L129.FP1",
"OA-FLOW.L129.FP2",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.NF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 178,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP4",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp4",
"start_line": 179,
"end_line": 205,
"range_sha256": "4a0fd53bda0940ef66c94dac4094520661d94a62fadef7e8a54b008982dedab3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP0",
"OA-FLOW.L129.FP3",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 205,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP5",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp5",
"start_line": 206,
"end_line": 230,
"range_sha256": "10ad701c777fb5bfef217c822a6a3c105c66a4aa5b3876955bf2dffd58618ad2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.3",
"OA-FLOW.PC.7",
"OA-FLOW.L120.CENTRAL-CARDINAL-PARTITION",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.L120.CARDINAL-ARITHMETIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 230,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FP6",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#fp6",
"start_line": 231,
"end_line": 264,
"range_sha256": "7f759e7587bf40991cc83835a64d588a7fb62a8cfa219dc3f9ba0d5305eb99b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP0",
"OA-FLOW.L129.FP4",
"OA-FLOW.L129.FP5",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 264,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.FIGURE",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-figure",
"start_line": 265,
"end_line": 279,
"range_sha256": "472bd9dae5c7ecb5cd629f1116546037acfe80c7d79ba305cf7c9aacb506faef",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 279,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.SETTING",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-standard-form-alternative",
"start_line": 280,
"end_line": 308,
"range_sha256": "7ae1b6b7dae93343e52e87ab1c62b4ac91a8d67861eb2b024e6a1b447175ecda",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 308,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.MATRIX.REDUCTION",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-matrix-reduction",
"start_line": 309,
"end_line": 388,
"range_sha256": "dee23cc468ee85541b39318945d8423f49c8d2c0372a69a6978a5e9c003f0881",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 388,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.AMPLIFIED.FREENESS",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-amplified-freeness",
"start_line": 389,
"end_line": 435,
"range_sha256": "6dcee0fcfd9608acbddf20d8e1d5b9852c3eaec206b327c5a9da05f47797ac98",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.ALT.MATRIX.REDUCTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 435,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.STANDARD.CENTER",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-standard-center",
"start_line": 436,
"end_line": 537,
"range_sha256": "18505949101498736b5c445ca8277eb268c08bb15f508f72a74355b883a1e6a7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.9",
"OA-FLOW.MC.1",
"OA-FLOW.NR.1",
"OA-FLOW.ST.2",
"OA-FLOW.CF.6",
"OA-FLOW.PC.1",
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC4",
"OA-FLOW.L129.ALT.AMPLIFIED.FREENESS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 537,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.ALTERNATIVE.PROJECTIONS",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-alternative-projections",
"start_line": 538,
"end_line": 542,
"range_sha256": "d50e3a3ef659b41cae3a7d5b8669f9455bafb8f0b98d2516f176242dcc6fde76",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CP.6",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.GW.3",
"OA-FLOW.GW.4",
"OA-FLOW.NF.1",
"OA-FLOW.NF.5",
"OA-FLOW.NF.6",
"OA-FLOW.ST.2",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.L120.CENTRAL-CARDINAL-PARTITION",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.L120.CARDINAL-ARITHMETIC"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 542,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.ADJOINT.TEST",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-adjoint-test",
"start_line": 543,
"end_line": 590,
"range_sha256": "50a4861619a81195bdf6606ecf55894e0ee06973b4f05ccb327180f49bd6db17",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 590,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.FINITENESS.TRANSFER",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-finiteness-transfer",
"start_line": 591,
"end_line": 696,
"range_sha256": "5266f230429cf87e4ca0a58a30fc17ba24794b48f09c55705c818acf51bb2fda",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP0",
"OA-FLOW.L129.FP1",
"OA-FLOW.L129.FP2",
"OA-FLOW.L129.FP4",
"OA-FLOW.FCT.7",
"OA-FLOW.NR.1",
"OA-FLOW.L129.ALT.STANDARD.CENTER",
"OA-FLOW.L129.ALT.ADJOINT.TEST"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 696,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.HOMOGENEOUS.FAMILIES",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-homogeneous-families",
"start_line": 697,
"end_line": 763,
"range_sha256": "09579ec422004fb44cb7d82131a9f3c931fd5aff760c680d10d123cd8f725e5c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7",
"OA-FLOW.L120.RESIDUAL-ABSORPTION",
"OA-FLOW.L120.HOMOGENEOUS-FILLING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 763,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.CARDINAL.COMPARISON",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-cardinal-comparison",
"start_line": 764,
"end_line": 836,
"range_sha256": "716ce6a4421384ffcb1e14c1edd097665865e734b1b844e13bc207b54c2d2ffe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.ALT.HOMOGENEOUS.FAMILIES",
"OA-FLOW.L120.COVERING-BOUND",
"OA-FLOW.L120.CARDINAL-ARITHMETIC",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4",
"OA-FLOW.PC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 836,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.INFINITE.CORNERS",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-infinite-corners",
"start_line": 837,
"end_line": 861,
"range_sha256": "490143b38f42c56ffed8401b173356518230052582792cdc73a5a4d87f53cef0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.FP6",
"OA-FLOW.L129.ALT.CARDINAL.COMPARISON",
"OA-FLOW.L129.ALT.FINITENESS.TRANSFER",
"OA-FLOW.L129.ALT.STANDARD.CENTER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 861,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.FINITE.CORNERS",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-finite-corners",
"start_line": 862,
"end_line": 907,
"range_sha256": "29625da3bfa2e08f336eeabc63df4b2d43e14259945b70d429df9dc77aba2e3f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.L129.ALT.STANDARD.CENTER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 907,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.ALT.COBOUNDARY",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-coboundary",
"start_line": 908,
"end_line": 949,
"range_sha256": "09a3e24a4d4d1e9e3fc5243e645bd3fc5ade47372f690d3d172a377f53e282b1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L129.ALT.INFINITE.CORNERS",
"OA-FLOW.L129.ALT.FINITE.CORNERS",
"OA-FLOW.L129.ALT.MATRIX.REDUCTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 949,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.L129.PROBLEM",
"unit": "OA-FLOW-L129",
"source_file": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e",
"reader": "OA-FLOW-L129.html#l129-problem",
"start_line": 950,
"end_line": 972,
"range_sha256": "7d02b30cd4d72c6ca01f0639d60e8b67938eb9f707ae36b74644d3fb1a41b9ea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 972,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L129.md",
"start_line": 7,
"end_line": 16,
"full_statement_context": "\n\n## Setting and earlier proofs\n\nThroughout, finite groups act by normal star automorphisms on nonzero concrete von Neumann algebras of arbitrary Hilbert dimension. An action is **free** if, for every nonidentity group element $s$, the relation $cx=\\alpha_s(x)c$ for every $x$ forces $c=0$. A **unitary cocycle** is a family $u_s\\in\\mathcal U(M)$ satisfying $u_{st}=u_s\\alpha_s(u_t)$ and $u_e=1$. We impose no separability or countable-decomposability hypothesis. Sums of nonnegative numbers mean suprema of finite subsums; orthogonal operator sums mean strong limits of finite sums.\n\nThe regular crossed-product coefficients, their uniqueness, the faithful normal completely positive identity-coefficient expectation, and the free relative commutant are proved in [Discrete Fourier columns, finite corners and orbit MASAs](OA-FLOW-L128.md#l128-fc0), FC0–FC4. Its [regular-model transport theorem](OA-FLOW-L128.md#regular-transport) identifies the model in every faithful normal representation. [Projection comparison and the countably decomposable type III case](OA-FLOW-PC.md#oa-flow.pc.0), PC0–PC7, supplies projection joins, polar decomposition, central supports, comparison, Cantor–Bernstein, full-corner centers, finite joins, proper infiniteness and the exact faithful-normal-state criterion for countable decomposability. The cardinal tools are HF1–HF2, LC2–LC3 and CA1 of [Transport a fixed corner, then assemble its local cardinal](OA-FLOW-L120.md#oa-flow.l120.central-cardinal-partition). Those local projection lemmas apply to arbitrary algebras, without the action hypotheses used elsewhere in that lesson.\n\nThe trace input is the general finite-algebra theorem in [The normalized trace with its full center retained](OA-FLOW-FCT.md#oa-flow.fct.6), FCT6–FCT8. The trace Hilbert space and its representation are constructed in [Finite ideals and GNS spaces for arbitrary weights](OA-FLOW-GW.md#oa-flow.gw.3), GW3–GW4; their normality follows from [Order-normal positive functionals and ultraweak continuity](OA-FLOW-NF.md#oa-flow.nf.5), NF5, and the faithful normal inverse and ultrastrong topology are proved in [Concrete predual balls and faithful ultraweak representations](OA-FLOW-ST12.md#oa-flow.st.2), ST2. Normal functional supports use NF1, and the normality criterion for positive functionals uses NF1–NF4. The standard-form alternative uses [All projection corners and canonical positive-functional vectors](OA-FLOW-CR.md#oa-flow.cr.9), CR9 and its natural-cone construction, [Canonical standard-form transport and finite balanced matrix cones](OA-FLOW-MC.md#oa-flow.mc.1), MC1, and [A normal regular construction on arbitrary Hilbert spaces](OA-FLOW-NR.md#oa-flow.nr.1), NR1. A finite discrete group automatically satisfies the continuity hypotheses of NR1.\n\n"
}
]
},
{
"id": "OA-FLOW.INTSPACE.DEFINITION",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.definition",
"start_line": 25,
"end_line": 73,
"range_sha256": "dce9967dc28249f573c2fdd0adbadc4a9d01c0239326c5524cd8c0c8aa3cf9dc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 73,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.BASIS",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.basis",
"start_line": 75,
"end_line": 180,
"range_sha256": "9456126c47fd5010a05fe2be2cc0eac962571e91781351d25e6bf4aea65558a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.PC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 180,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.CLOSED",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.closed",
"start_line": 182,
"end_line": 247,
"range_sha256": "8e16e81b9d35370c82459fd22fa6101e1f8db8d4e7cfc125ed9edb8c4648ef61",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 247,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.TENSOR",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.tensor",
"start_line": 249,
"end_line": 332,
"range_sha256": "a36c0c4fcf7e7ad0c0daa68bc975403e17c73d76392bf1c9b16865c75765ba8b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 332,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.SUM",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.sum",
"start_line": 334,
"end_line": 432,
"range_sha256": "a2265da2198d4bac7227941d981efd178a154b4e79d22ed5c5984b245456aabb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 432,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.ADJOINT",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.adjoint",
"start_line": 434,
"end_line": 497,
"range_sha256": "af71f1046b7014c92e4fd9b75287b3d8f0ab1cef2eb82e803a9cc15b9148b330",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.TENSOR",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 497,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.DIMONE",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.dimone",
"start_line": 499,
"end_line": 528,
"range_sha256": "de066887d898e9010159a17b17cc3552f076086599b87df81dea3ba8986a6098",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 528,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.DIMTWO",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.dimtwo",
"start_line": 530,
"end_line": 562,
"range_sha256": "1db5221bb5091697ad0b457b86ba971464940f32642d61b08477f17411937b97",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.INTSPACE.ADJOINT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 562,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTSPACE.PROBLEM",
"unit": "OA-FLOW-L130",
"source_file": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f",
"reader": "OA-FLOW-L130.html#oa-flow.intspace.problem",
"start_line": 564,
"end_line": 579,
"range_sha256": "0205e4428365fa64fe759640b7a125d1b4f2a3b2b20916b261fba3e3c390ad78",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.INTSPACE.CLOSED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 579,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L130.md",
"line": 13,
"through_line": 16,
"description": "Nonzero concrete unital von Neumann algebra on arbitrary Hilbert space; linear-first inner products; arbitrary-index finite-subset nets."
}
]
},
{
"id": "OA-FLOW.INTOPS.COORDINATES",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.coordinates",
"start_line": 16,
"end_line": 37,
"range_sha256": "3b8299fcc9f6a4dff7ea69bd0d6fad16f3c6b3af527c1650611da3d2d8c797ab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 37,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.FINITE",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.finite",
"start_line": 39,
"end_line": 98,
"range_sha256": "72da394a048d5e945f1aa5dd1eea3e9d226a51c5e8f95742b7a5dfc1c8b348d0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 98,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.CLOSURES",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.closures",
"start_line": 100,
"end_line": 175,
"range_sha256": "35b66711af7fee30fbade279ac0349b0215e42a1bdfa9c6710c5280e46c1c6ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTOPS.FINITE",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 175,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.ENDOMORPHISM",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.endomorphism",
"start_line": 177,
"end_line": 210,
"range_sha256": "f9bde4001a47207a8ac6f173abebf624e30a4e39969be2eb5f3dd3cb962268ba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTOPS.CLOSURES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 210,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.SPLITTING",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.splitting",
"start_line": 212,
"end_line": 268,
"range_sha256": "b64f8d3a8e3b21a6c7abe717bb294801b16baed3161c1d1212c578952cff28e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.CLOSURES",
"OA-FLOW.INTOPS.ENDOMORPHISM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 268,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.BASIS",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.basis",
"start_line": 270,
"end_line": 291,
"range_sha256": "37acf3034fcc2802fc52818f575ac09755c56cf392933cf9200b86f41da42c00",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTOPS.ENDOMORPHISM",
"OA-FLOW.INTSPACE.BASIS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 291,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.INVARIANT",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.invariant",
"start_line": 293,
"end_line": 339,
"range_sha256": "792f46b9b8bf4ccabd65e2474de315bc208e12728ab35513add799827af80063",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.ENDOMORPHISM",
"OA-FLOW.INTOPS.BASIS",
"OA-FLOW.NF.6",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 339,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.FULLNESS",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.fullness",
"start_line": 341,
"end_line": 360,
"range_sha256": "9bd72dd8091a0a6f137e73cf36ec57e11422ed6dcf2de7aa4b7ac507e4663fc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTOPS.FINITE",
"OA-FLOW.INTSPACE.DEFINITION",
"OA-FLOW.INTSPACE.BASIS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 360,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.INTOPS.EXAMPLE",
"unit": "OA-FLOW-L131",
"source_file": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4",
"reader": "OA-FLOW-L131.html#oa-flow.intops.example",
"start_line": 362,
"end_line": 396,
"range_sha256": "eb64922f75dbd98207eaf73b287584d7082569ad9fe57bbfe442e618899177f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INTOPS.COORDINATES",
"OA-FLOW.INTOPS.CLOSURES",
"OA-FLOW.INTOPS.ENDOMORPHISM",
"OA-FLOW.INTOPS.SPLITTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 396,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L131.md",
"line": 5,
"through_line": 13,
"description": "Nonzero concrete unital von Neumann algebra; full norm-closed scalar Hilbert subspaces; linear-first inner products; arbitrary cardinalities. Section 8 explicitly relaxes fullness."
}
]
},
{
"id": "OA-FLOW.L132.SETTING",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.l132.setting",
"start_line": 1,
"end_line": 29,
"range_sha256": "8816bea1097e64fd9c2a428ece0ebd4b46af6717e38e752ba80b096dcb0a2618",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0"
],
"range_bytes": 1253
},
{
"id": "OA-FLOW.L132.INPUTS",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.l132.inputs",
"start_line": 30,
"end_line": 47,
"range_sha256": "4293ce1191ab598dc7094b87811fa397d29ced81925f2b30d250d5776f6e6fd9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L123.IC0",
"OA-FLOW.L123.IC3",
"OA-FLOW.CP.6",
"OA-FLOW.PC.7",
"OA-FLOW.PC.5"
],
"range_bytes": 3963
},
{
"id": "OA-FLOW.HOMREP.FINITEORBIT",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.finiteorbit",
"start_line": 48,
"end_line": 105,
"range_sha256": "322372a686fb0d636e0c691330087d568fb8cf14fd8a860fc022f2d284074bc6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7"
],
"range_bytes": 2228
},
{
"id": "OA-FLOW.HOMREP.AMPLIFY",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.amplify",
"start_line": 106,
"end_line": 167,
"range_sha256": "ebe433dc3cb2e19d0bfa8db1e5ba70c7dc4e019df17a32795ab85ea1bed7fdd2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L132.INPUTS",
"OA-FLOW.L123.IC0",
"OA-FLOW.L123.IC3",
"OA-FLOW.CP.6"
],
"range_bytes": 3392
},
{
"id": "OA-FLOW.HOMREP.TRANSPORT",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.transport",
"start_line": 168,
"end_line": 223,
"range_sha256": "e49c31fa9ab9f49c5b90e5c020ffc22b44627adee21146da1d7f43e6ef957010",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.AMPLIFY",
"OA-FLOW.PC.5",
"OA-FLOW.INTSPACE.BASIS",
"OA-FLOW.INTOPS.CLOSURES",
"OA-FLOW.INTOPS.INVARIANT"
],
"range_bytes": 1866
},
{
"id": "OA-FLOW.HOMREP.MULTIPLIER",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.multiplier",
"start_line": 224,
"end_line": 263,
"range_sha256": "7818c6992f30901acec3b6f98a27c3338d383a6e5435184b05670d0f5f03d630",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.TRANSPORT",
"OA-FLOW.INTOPS.ENDOMORPHISM",
"OA-FLOW.INTOPS.SPLITTING"
],
"range_bytes": 883
},
{
"id": "OA-FLOW.HOMREP.UNITARY",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.unitary",
"start_line": 264,
"end_line": 387,
"range_sha256": "2c25692e425936237f5a58ec36cf613fcf37b3478de823b8fe29767d0e79b4b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.MULTIPLIER",
"OA-FLOW.L132.INPUTS",
"OA-FLOW.PC.2",
"OA-FLOW.PC.7",
"OA-FLOW.INTOPS.SPLITTING"
],
"range_bytes": 4596
},
{
"id": "OA-FLOW.HOMREP.EXTRACT",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.extract",
"start_line": 388,
"end_line": 434,
"range_sha256": "f6bd1ebe62c9227580fbcae7231776cc3b1368b425fcc7f4eaac443adc69ea3e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.UNITARY",
"OA-FLOW.INTOPS.ENDOMORPHISM"
],
"range_bytes": 1186
},
{
"id": "OA-FLOW.HOMREP.CONJUGATE",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.homrep.conjugate",
"start_line": 435,
"end_line": 488,
"range_sha256": "53777c7524c8fae3274143f374007b0acc74f1f9b692d8bfe2b5cb7f363a6eb4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.EXTRACT",
"OA-FLOW.HOMREP.TRANSPORT",
"OA-FLOW.INTSPACE.ADJOINT"
],
"range_bytes": 2356
},
{
"id": "OA-FLOW.L132.EXAMPLE",
"unit": "OA-FLOW-L132",
"source_file": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96",
"reader": "OA-FLOW-L132.html#oa-flow.l132.example",
"start_line": 489,
"end_line": 524,
"range_sha256": "327a8cf3c5444ace6ddced69bf82c9f93e375767d80a2182dfdeffca0e62f528",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.MULTIPLIER",
"OA-FLOW.HOMREP.TRANSPORT"
],
"range_bytes": 3617
},
{
"id": "OA-FLOW.L133.SETTING",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.l133.setting",
"start_line": 1,
"end_line": 35,
"range_sha256": "49a5158da7463b068336c4e0a3c3dc589058ca5608722ed3dcf8ac186ad70ff9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.AMPLIFY",
"OA-FLOW.L132.INPUTS",
"OA-FLOW.PC.5"
],
"range_bytes": 2884
},
{
"id": "OA-FLOW.TANNAKA.PRESERVE",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.preserve",
"start_line": 36,
"end_line": 123,
"range_sha256": "26111215dcee64486db1c0620b89fb126fbe4f7e4a23dd968f8e706ea593e1a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.AMPLIFY",
"OA-FLOW.L132.INPUTS",
"OA-FLOW.PC.5",
"OA-FLOW.INTOPS.CLOSURES",
"OA-FLOW.INTOPS.SPLITTING",
"OA-FLOW.INTOPS.ENDOMORPHISM"
],
"range_bytes": 2477
},
{
"id": "OA-FLOW.L133.RECONSTRUCTION",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.l133.reconstruction",
"start_line": 124,
"end_line": 153,
"range_sha256": "3bdf8f6417236e5546dd731a234645d3f7ec1085ee75d53390a2f274560d136c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7"
],
"range_bytes": 3059
},
{
"id": "OA-FLOW.TANNAKA.RING",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.ring",
"start_line": 154,
"end_line": 206,
"range_sha256": "00f00e3d6f3e68283dba72139e2baad05c2a341249ed6042ce655808a0c9871f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HOMREP.EXTRACT",
"OA-FLOW.HOMREP.UNITARY",
"OA-FLOW.INTSPACE.SUM",
"OA-FLOW.INTSPACE.TENSOR",
"OA-FLOW.PC.5"
],
"range_bytes": 2236
},
{
"id": "OA-FLOW.TANNAKA.DETERMINANT",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.determinant",
"start_line": 207,
"end_line": 263,
"range_sha256": "d1f2c7766482c4d7e2dce33eef7056503327a0e265435159e1279ae2f07156ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TANNAKA.RING",
"OA-FLOW.INTSPACE.DIMONE"
],
"range_bytes": 2138
},
{
"id": "OA-FLOW.TANNAKA.ALLIRREPS",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.allirreps",
"start_line": 264,
"end_line": 312,
"range_sha256": "2333558628ae8f0a139c13d1e484c021083af95f6999006db117d005ecbefabc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TANNAKA.RING",
"OA-FLOW.TANNAKA.DETERMINANT",
"OA-FLOW.HOMREP.FINITEORBIT",
"OA-FLOW.HOMREP.EXTRACT",
"OA-FLOW.CF.5"
],
"range_bytes": 1791
},
{
"id": "OA-FLOW.TANNAKA.NATURAL",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.natural",
"start_line": 313,
"end_line": 366,
"range_sha256": "219b24dab1e93dbedcccdfecce8c02db1792abb1b466215b6dc73846958ec3e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TANNAKA.PRESERVE",
"OA-FLOW.L133.RECONSTRUCTION",
"OA-FLOW.TANNAKA.ALLIRREPS",
"OA-FLOW.INTOPS.FINITE"
],
"range_bytes": 1812
},
{
"id": "OA-FLOW.TANNAKA.RIGIDITY",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.rigidity",
"start_line": 367,
"end_line": 428,
"range_sha256": "8e9c9de867cee9f8fbb0fe249ab2e27bd3cd8d60fbc4bf4c01003400d303b4c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TANNAKA.NATURAL",
"OA-FLOW.HOMREP.EXTRACT",
"OA-FLOW.HOMREP.UNITARY",
"OA-FLOW.HOMREP.FINITEORBIT",
"OA-FLOW.HOMREP.AMPLIFY"
],
"range_bytes": 2126
},
{
"id": "OA-FLOW.TANNAKA.REGULAR",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.tannaka.regular",
"start_line": 429,
"end_line": 467,
"range_sha256": "3b3ff18c2809f764bf1c1919c95406ad50c7bc03c98e30bae89a50aa4f39bc1b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TANNAKA.ALLIRREPS",
"OA-FLOW.L132.INPUTS",
"OA-FLOW.INTSPACE.SUM",
"OA-FLOW.PC.5"
],
"range_bytes": 1975
},
{
"id": "OA-FLOW.L133.EXAMPLE",
"unit": "OA-FLOW-L133",
"source_file": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115",
"reader": "OA-FLOW-L133.html#oa-flow.l133.example",
"start_line": 468,
"end_line": 496,
"range_sha256": "e5d6c4a89815614297b8458ee4a713236abee6c780097f2f7a7095e163231d35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L133.RECONSTRUCTION",
"OA-FLOW.L132.EXAMPLE"
],
"range_bytes": 3436
},
{
"id": "OA-FLOW.WEYLPLANE.ACTION",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-action",
"start_line": 1,
"end_line": 90,
"range_sha256": "41db2ee23e7b5a28acdc76ba25b696685d03558db478ce6c2b14876b8a8af3ba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.2",
"OA-FLOW.SC.4",
"OA-FLOW.CP.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 90
},
{
"id": "OA-FLOW.WEYLPLANE.INTEGRABLE",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-integrable",
"start_line": 91,
"end_line": 143,
"range_sha256": "10e129896e1e9d0fe893a65f29e0dc59ce00176869c5250477aa5939ceaca9a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.FF.1",
"OA-FLOW.FF.3",
"OA-FLOW.SF.SB3",
"OA-FLOW.AT.5",
"OA-FLOW.L115.GENERAL-AVERAGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 143
},
{
"id": "OA-FLOW.WEYLPLANE.CONNES",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-connes",
"start_line": 144,
"end_line": 177,
"range_sha256": "7c3eaa9ac0c1b3025ab33d1028c4ce66cd843b5577016efef825be2d119ec39b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.AT.5",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CONNES.CENTERKERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 177
},
{
"id": "OA-FLOW.WEYLPLANE.CENTRALIZER",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-centralizer",
"start_line": 178,
"end_line": 233,
"range_sha256": "14dfc37e199afb34ccf132e3a1010716bc7c92348d810cd69dc9c8e9b3f571f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.WEYLPLANE.CONNES",
"OA-FLOW.L123.IC0",
"OA-FLOW.CP.4",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 233
},
{
"id": "OA-FLOW.WEYLPLANE.TRACE",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-subgroups",
"start_line": 234,
"end_line": 291,
"range_sha256": "e246d90396e969fcaf17258e3ea52d31f29ba5cb50405f181f29b88123a70512",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 291
},
{
"id": "OA-FLOW.WEYLPLANE.DIVISION",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-fourier",
"start_line": 292,
"end_line": 362,
"range_sha256": "4df484e53076982e4baeaf451ac83c75d75b966fdfcb87e9544d79c848bded62",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.GI.FOURIER",
"OA-FLOW.RF.1",
"OA-FLOW.CF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 362
},
{
"id": "OA-FLOW.WEYLPLANE.SUBGROUPS",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-synthesis",
"start_line": 363,
"end_line": 790,
"range_sha256": "25c2fbdb0ddeb38c1d2b7550d0146c8e8820f1e974d10ddc01fa7019991965c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.WEYLPLANE.INTEGRABLE",
"OA-FLOW.WEYLPLANE.TRACE",
"OA-FLOW.WEYLPLANE.DIVISION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 790
},
{
"id": "OA-FLOW.WEYLPLANE.DEFECT",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-continuous",
"start_line": 791,
"end_line": 822,
"range_sha256": "1dbf6b4efc45d8d7babf497fa74e0baecbd03b45e1c7562427b7e636ae220818",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 822
},
{
"id": "OA-FLOW.WEYLPLANE.CONTINUOUS",
"unit": "OA-FLOW-L134",
"source_file": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f",
"reader": "OA-FLOW-L134.html#weylplane-convolution",
"start_line": 823,
"end_line": 916,
"range_sha256": "29c2829c72be1de2a104ad0e9380f66edabf6f2e687ae03b44f1d45dd5981855",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WEYLPLANE.ACTION",
"OA-FLOW.WEYLPLANE.DEFECT",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.FF.2",
"OA-FLOW.INTOPS.CLOSURES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 916
},
{
"id": "OA-FLOW.SMALLAUT.BRANCH",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.branch",
"start_line": 1,
"end_line": 56,
"range_sha256": "3e55f38da2eb53f53346228a7ae5f8d42f69e6df0d14611b30cbf8a7bba20124",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.2",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 56
},
{
"id": "OA-FLOW.SMALLAUT.GROUPS",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.groups",
"start_line": 57,
"end_line": 134,
"range_sha256": "c67ed0c40c5aa97b3718dd21248df2af3978fb9ed670d8570a61867311bef32e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 134
},
{
"id": "OA-FLOW.SMALLAUT.ARC",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.arc",
"start_line": 135,
"end_line": 174,
"range_sha256": "e561c574387e3037f997913b4bf91460aa538f8dde698b1168166dd842cda1e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERSP.FACTOR",
"OA-FLOW.CF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 174
},
{
"id": "OA-FLOW.SMALLAUT.NUMERICAL",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.numerical",
"start_line": 175,
"end_line": 266,
"range_sha256": "b8f1e1ac0df1c5acefbabde6dbda769afda1a315480545c47b70e2695cb8498e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.2",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 266
},
{
"id": "OA-FLOW.SMALLAUT.CENTER",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.center",
"start_line": 267,
"end_line": 321,
"range_sha256": "60739646e69d395238e9c9e91cbc0e6a151f276aa9665675e158448f277abe82",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.NUMERICAL",
"OA-FLOW.CF.1",
"OA-FLOW.CF.6",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.ISOM.RIGIDITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 321
},
{
"id": "OA-FLOW.SMALLAUT.ATOMIC",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.atomic",
"start_line": 322,
"end_line": 387,
"range_sha256": "5fedc666322ff6ee28e63ee79c13ba63c93452e82eaef86c852fddd5176098ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.CENTER",
"OA-FLOW.SMALLAUT.ARC",
"OA-FLOW.SMALLAUT.NUMERICAL",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 387
},
{
"id": "OA-FLOW.SMALLAUT.PRODUCTLOG",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.productlog",
"start_line": 388,
"end_line": 438,
"range_sha256": "f1e0c8d9de3154d18e757587ed782e67c7691608aa7bca42be81f5da9fc1912b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.ATOMIC",
"OA-FLOW.SMALLAUT.BRANCH",
"OA-FLOW.CF.1",
"OA-FLOW.INNERSP.MULTIPLIERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 438
},
{
"id": "OA-FLOW.SMALLAUT.DESCENT",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.descent",
"start_line": 439,
"end_line": 504,
"range_sha256": "77f95e6e523b13b7d05f45b64e6490bbbe8cd243d074d913e9229ec4e52105eb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.ATOMIC",
"OA-FLOW.SMALLAUT.PRODUCTLOG",
"OA-FLOW.SMALLAUT.BRANCH",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 504
},
{
"id": "OA-FLOW.SMALLAUT.INNER",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.inner",
"start_line": 505,
"end_line": 546,
"range_sha256": "ec800cebf01de94616971970d9e775faea111940fce3ba0d2a51a9f6e5b91b01",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.DESCENT",
"OA-FLOW.SMALLAUT.GROUPS",
"OA-FLOW.GI.INNERNESS",
"OA-FLOW.CF.6",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 546
},
{
"id": "OA-FLOW.SMALLAUT.CORNERS",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.corners",
"start_line": 547,
"end_line": 583,
"range_sha256": "e28a75fe2aeebbc3181b15ad25f0ee9590af87e6361bffbf2564b7ad77f7e828",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.INNER",
"OA-FLOW.L34.8a",
"OA-FLOW.L34.9",
"OA-FLOW.CF.6"
],
"external_proof_links": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 583,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L135.md",
"line": 547,
"through_line": 583,
"description": "Arbitrary nonzero von Neumann algebra and normal automorphism; projection fixed by the automorphism and having ambient central support one. The zero-algebra convention and strict-threshold/nonfull examples are explicit."
}
]
},
{
"id": "OA-FLOW.SMALLAUT.SHARP",
"unit": "OA-FLOW-L135",
"source_file": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5",
"reader": "OA-FLOW-L135.html#smallaut.sharp",
"start_line": 584,
"end_line": 621,
"range_sha256": "52843b05e789d9e748e21f8562c4fbdcafb600a2b939e7e75e15ebb3a972184c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SMALLAUT.INNER",
"OA-FLOW.CF.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 621
},
{
"id": "OA-FLOW.INNERFLOW.SETTING",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.setting",
"start_line": 1,
"end_line": 41,
"range_sha256": "9f102f0dcf16b87223aeb043a7a6bb9d19ec3bdb0f2eb93f94ccaebb84dafdd7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.DA.ACTION",
"OA-FLOW.AT.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 41
},
{
"id": "OA-FLOW.INNERFLOW.POLISH",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.polish",
"start_line": 42,
"end_line": 93,
"range_sha256": "4757084d734c3a19b37fbc67fca403c4ab35f2ed8b0b05b0d7508db55a286c62",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GNS.5.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.ST.2",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 93
},
{
"id": "OA-FLOW.INNERFLOW.AUTOMORPHISMS",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.automorphisms",
"start_line": 94,
"end_line": 142,
"range_sha256": "26bbbf95e9820b5b14cb863d5a91d1533fcb6dfb7622334eb3439ee4693143a5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.POLISH",
"OA-FLOW.CP.6",
"OA-FLOW.CF.8",
"OA-FLOW.AT.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 142
},
{
"id": "OA-FLOW.INNERFLOW.SELECTION",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.selection",
"start_line": 143,
"end_line": 205,
"range_sha256": "b9c37955d1aecfa5e632376289de517291951f426520a736291b297b3c009ddf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.POLISH",
"OA-FLOW.INNERFLOW.AUTOMORPHISMS",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 205
},
{
"id": "OA-FLOW.INNERFLOW.DEFECT",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.defect",
"start_line": 206,
"end_line": 250,
"range_sha256": "090fc6afb25f1413976ced18ce552679cd6113c6de6ec454eda432dde2de6e1c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.SELECTION",
"OA-FLOW.AT.4",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 250
},
{
"id": "OA-FLOW.INNERFLOW.TRIVIAL",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.trivial",
"start_line": 251,
"end_line": 280,
"range_sha256": "410ffad43c781a91a3f584423445ba0ce0407d475c13d9ffee5b9eb609ca9bbe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.DEFECT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 280
},
{
"id": "OA-FLOW.INNERFLOW.ABELIAN",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.abelian",
"start_line": 281,
"end_line": 316,
"range_sha256": "aa3bd76c3cf0225653271cebd0f4c5b3dc7d1dbd4071ff9b2250ce9fe8335d42",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.TRIVIAL",
"OA-FLOW.NR.3",
"OA-FLOW.CP.6",
"OA-FLOW.DA.ACTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 316
},
{
"id": "OA-FLOW.INNERFLOW.TWIST",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.twist",
"start_line": 317,
"end_line": 368,
"range_sha256": "60717b736cfe748c42bb2c9f96079af60c2601b34bfd95243fdc810ae08e05d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.ABELIAN",
"OA-FLOW.INNERFLOW.POLISH",
"OA-FLOW.ST.2",
"OA-FLOW.CP.6",
"OA-FLOW.TW.1",
"OA-FLOW.SC.5",
"OA-FLOW.DS.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 368
},
{
"id": "OA-FLOW.INNERFLOW.INTEGRABILITY",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.integrability",
"start_line": 369,
"end_line": 407,
"range_sha256": "d1ed0489bd4f0ba97462678cf26d9a3ad5d5966b8c1f2fec343795f61e498e0b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.TWIST",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.FF.3",
"OA-FLOW.CONNES.REGULARINT",
"OA-FLOW.L115.CORNER-INTEGRABILITY",
"OA-FLOW.TW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 407
},
{
"id": "OA-FLOW.INNERFLOW.ORBITS",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.orbits",
"start_line": 408,
"end_line": 473,
"range_sha256": "a04eb139b3aa1e5ae2784c0c82df88bbd201ec4ae7f0fb0fc8580ea9cfe08f59",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.INTEGRABILITY",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.ORBITS.HOMOGENEOUS",
"OA-FLOW.ORBITS.ERGODIC",
"OA-FLOW.NF.6",
"OA-FLOW.FF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 473
},
{
"id": "OA-FLOW.INNERFLOW.EIGENGROUP",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.eigengroup",
"start_line": 474,
"end_line": 492,
"range_sha256": "d066f90cac255bdeb4e44447189835f9fe846edabad083fc2b4686488b7b1ab4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.ORBITS",
"OA-FLOW.SC.5",
"OA-FLOW.ST.2",
"OA-FLOW.SF.SF4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 492
},
{
"id": "OA-FLOW.INNERFLOW.CANCELLATION",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.cancellation",
"start_line": 493,
"end_line": 521,
"range_sha256": "17007bb6966160c907022652cbf473ee5d5d0eb0f28bdc0b87b0adcf0c9f566f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.EIGENGROUP",
"OA-FLOW.INNERFLOW.ABELIAN",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NR.3",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 521
},
{
"id": "OA-FLOW.INNERFLOW.THEOREM",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.theorem",
"start_line": 522,
"end_line": 542,
"range_sha256": "1184dfb971c3f24f4514455674d88b3b68d66467dde78f485737d4646589ff71",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.CANCELLATION",
"OA-FLOW.INNERFLOW.SELECTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 542
},
{
"id": "OA-FLOW.INNERFLOW.DIAGNOSTICS",
"unit": "OA-FLOW-L136",
"source_file": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6",
"reader": "OA-FLOW-L136.html#innerflow.diagnostics",
"start_line": 543,
"end_line": 555,
"range_sha256": "4c4755c6c24026206637ae39de4b3e1418bd508f1b11898fb592d1027ffe6cde",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.INNERFLOW.THEOREM",
"OA-FLOW.INNERFLOW.ABELIAN",
"OA-FLOW.INNERFLOW.EIGENGROUP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 555
},
{
"id": "OA-FLOW.BOUNDARIES.SETTING",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.setting",
"start_line": 1,
"end_line": 34,
"range_sha256": "be56892b0a129cf99cad90475ebe26f9a68d3ee41d8ee15f5fd876848ba6d996",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.9",
"OA-FLOW.NF.6",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 34
},
{
"id": "OA-FLOW.BOUNDARIES.PRIMITIVE",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.primitive",
"start_line": 35,
"end_line": 203,
"range_sha256": "d69f7d0476c25e5df1dae434188291d25bc96796e7ed306e2e7712b355917643",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4",
"OA-FLOW.GI.INNERNESS",
"OA-FLOW.GI.BIDUAL",
"OA-FLOW.GI.REPRESENTATIONS",
"OA-FLOW.CF.1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 203
},
{
"id": "OA-FLOW.BOUNDARIES.EMBEDDING",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.embedding",
"start_line": 204,
"end_line": 253,
"range_sha256": "6c2bb73746ed560a8be857058f242e62af28c4e7790d31354e5924cbbcd17c0b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.PRIMITIVE",
"OA-FLOW.CF.6",
"OA-FLOW.CF.8",
"OA-FLOW.GNS.5.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 253
},
{
"id": "OA-FLOW.BOUNDARIES.MULTIPLIER",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.multiplier",
"start_line": 254,
"end_line": 268,
"range_sha256": "868e7f28355ea005970c13c2546215bdd2517730110f89cb4badef74a9d6868d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.PRIMITIVE",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 268
},
{
"id": "OA-FLOW.BOUNDARIES.UNIFORMITY",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.uniformity",
"start_line": 269,
"end_line": 442,
"range_sha256": "974cbade2594f169356a00d5d01e101154b644995b62f0fefcc5dda6d66d66d1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.SF.SB4",
"OA-FLOW.SF.SB6",
"OA-FLOW.PC.1",
"OA-FLOW.PC.5",
"OA-FLOW.SMALLAUT.BRANCH",
"OA-FLOW.SMALLAUT.DESCENT",
"OA-FLOW.SMALLAUT.INNER",
"OA-FLOW.GI.INNERNESS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 442
},
{
"id": "OA-FLOW.BOUNDARIES.GENERATOR",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.generator",
"start_line": 443,
"end_line": 471,
"range_sha256": "7b168708037bed732143d071e0e7e93a5e2a00b1fc88ef43b8fed5019bf5509b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.UNIFORMITY",
"OA-FLOW.L34.5",
"OA-FLOW.L34.5a",
"OA-FLOW.GI.INNERNESS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 471
},
{
"id": "OA-FLOW.BOUNDARIES.ANALYTIC",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.analytic",
"start_line": 472,
"end_line": 515,
"range_sha256": "ff4aba26923cab9282b25fb88a1c431e0f3416e69ec5e0b74f93faff25363486",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.2",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 515
},
{
"id": "OA-FLOW.BOUNDARIES.RADIAL",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.radial",
"start_line": 516,
"end_line": 576,
"range_sha256": "f4c372391d719ada80adfcc9238cd44fe419694d62ee75b5898e88cade83ea13",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.ANALYTIC",
"OA-FLOW.SC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 576
},
{
"id": "OA-FLOW.BOUNDARIES.RIESZ",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.riesz",
"start_line": 577,
"end_line": 630,
"range_sha256": "a929cd8b2aeeaa07d9e0772603fe5ed4b5237a0f2694882e79056167893e1dd8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.RADIAL",
"OA-FLOW.SC.7",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.FF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 630
},
{
"id": "OA-FLOW.BOUNDARIES.EXTERNAL",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.external",
"start_line": 631,
"end_line": 666,
"range_sha256": "60ebc1a2f835382f27d92373349c0f1bd5dca60a58a0589536c2236c015f9890",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.POSFLOW.SETTING",
"OA-FLOW.POSFLOW.MODELS",
"OA-FLOW.POSFLOW.NECESSITY",
"OA-FLOW.SUM.CLAIM",
"OA-FLOW.AT.1",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 666
},
{
"id": "OA-FLOW.BOUNDARIES.INTERNAL",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.internal",
"start_line": 667,
"end_line": 689,
"range_sha256": "9893eb13017fa1cf9ff9c653bae038a31f1eb16eee2d3d3484c20b5d7a4fb86d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.EXTERNAL",
"OA-FLOW.POSFLOW.MEASURE",
"OA-FLOW.POSFLOW.RECOVERY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 689
},
{
"id": "OA-FLOW.BOUNDARIES.POSITIVE",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.positive",
"start_line": 690,
"end_line": 707,
"range_sha256": "442502c30a5eff426981ecf14724690fdaa69e974fcfb5d3dfc757425b5d9fde",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.INTERNAL",
"OA-FLOW.BOUNDARIES.EXTERNAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 707
},
{
"id": "OA-FLOW.BOUNDARIES.UNIFORM-IMPLEMENTERS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.uniform-implementers",
"start_line": 709,
"end_line": 762,
"range_sha256": "0d5c6eebe16727da53d80b20896d7e4e33643d183cf6b36a6ef524a46c4caf2a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4",
"OA-FLOW.CF.1",
"OA-FLOW.GI.BIDUAL",
"OA-FLOW.GI.INNERNESS",
"OA-FLOW.BOUNDARIES.PRIMITIVE",
"OA-FLOW.BOUNDARIES.EMBEDDING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 762,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 709,
"through_line": 762,
"description": "The first proposition concerns an arbitrary nonzero simple C*-algebra and an everywhere-defined complex or star derivation. The direct sum has an arbitrary index set; its innerness theorem assumes simple coordinate algebras. The multiplier-product and coordinate-derivation constructions do not require simplicity."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 709,
"through_line": 762,
"description": "Complete local proof of uniform implementer bounds, double-centralizer product, ideal invariance and bounded assembly; exact scalar-choice counterexample."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.DIRECT-SUMS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.direct-sums",
"start_line": 763,
"end_line": 838,
"range_sha256": "188ac135921ebb56d33636712f856969b24356746dcb08e26fbb745c0165c9f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.UNIFORM-IMPLEMENTERS",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.BOUNDARIES.MULTIPLIER",
"OA-FLOW.L34.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 838,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 763,
"through_line": 838,
"description": "The first proposition concerns an arbitrary nonzero simple C*-algebra and an everywhere-defined complex or star derivation. The direct sum has an arbitrary index set; its innerness theorem assumes simple coordinate algebras. The multiplier-product and coordinate-derivation constructions do not require simplicity."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 763,
"through_line": 838,
"description": "Complete local proof of uniform implementer bounds, double-centralizer product, ideal invariance and bounded assembly; exact scalar-choice counterexample."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.MULTIPLIER-EXTENSION",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.multiplier-extension",
"start_line": 840,
"end_line": 883,
"range_sha256": "b227e816198fd08ad5bfc3fd56cf960f951d2b5a8273634e81887b83221b3d59",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.BIDUAL",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.CF.1",
"OA-FLOW.CF.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 883,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 840,
"through_line": 883,
"description": "The extension concerns an arbitrary C*-algebra. The local implementation assumes a full projection in its multiplier algebra and a commutative corner. Assembly assumes the stated positive central partition with strict sum one, ideal support, and uniform local implementer bound. No general continuous-trace cover theorem or characterization is asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 840,
"through_line": 883,
"description": "Complete proof of multiplier extension and central vanishing; full-corner module construction and norm estimates; supported weighted multiplier extension and strict assembly."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.ABELIAN-CORNERS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.abelian-corners",
"start_line": 884,
"end_line": 1003,
"range_sha256": "8820ddc73da8f16f13602d134c56d79993a653dc2411c6896d6e77b877a6937b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.MULTIPLIER-EXTENSION",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.BOUNDARIES.UNIFORM-IMPLEMENTERS",
"OA-FLOW.L34.4",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1003,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 884,
"through_line": 1003,
"description": "The extension concerns an arbitrary C*-algebra. The local implementation assumes a full projection in its multiplier algebra and a commutative corner. Assembly assumes the stated positive central partition with strict sum one, ideal support, and uniform local implementer bound. No general continuous-trace cover theorem or characterization is asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 884,
"through_line": 1003,
"description": "Complete proof of multiplier extension and central vanishing; full-corner module construction and norm estimates; supported weighted multiplier extension and strict assembly."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.CENTRAL-GLUING",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.central-gluing",
"start_line": 1004,
"end_line": 1082,
"range_sha256": "962b8c22a5034c8bfb92d3031744a9d26deba228ed441f4ec534d46e6872a8bb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.MULTIPLIER-EXTENSION",
"OA-FLOW.BOUNDARIES.ABELIAN-CORNERS",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.CF.7",
"OA-FLOW.L34.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1082,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1004,
"through_line": 1082,
"description": "The extension concerns an arbitrary C*-algebra. The local implementation assumes a full projection in its multiplier algebra and a commutative corner. Assembly assumes the stated positive central partition with strict sum one, ideal support, and uniform local implementer bound. No general continuous-trace cover theorem or characterization is asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1004,
"through_line": 1082,
"description": "Complete proof of multiplier extension and central vanishing; full-corner module construction and norm estimates; supported weighted multiplier extension and strict assembly."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.TOPOLOGICAL-PARTITIONS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.topological-partitions",
"start_line": 1084,
"end_line": 1143,
"range_sha256": "d912c3c67c57530b3d4491228a4e09e6abab4d74b285495c7e9df8b4f74a58c2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1143,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1084,
"through_line": 1143,
"description": "The partition theorem assumes a second-countable locally compact Hausdorff space. Continuous trace is defined by the Hausdorff hull-kernel spectrum and Fell local rank-one condition; the central action and local compactness are derived. Separability is used to obtain a countable base. The final direct-sum condition is sufficient; structural necessity is not asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1084,
"through_line": 1143,
"description": "Complete cutoff and partition construction; ideal and local-corner spectrum arguments; actual central multiplier construction and nondegeneracy; strict assembly for continuous-trace algebras, including a changing-fiber example."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.LOCAL-RANK-ONE",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.local-rank-one",
"start_line": 1144,
"end_line": 1241,
"range_sha256": "de44f957cc361c4f4ecbbb7d36cf408590d72abbb8b4d155a418875e48e57295",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.TOPOLOGICAL-PARTITIONS",
"OA-FLOW.BOUNDARIES.ABELIAN-CORNERS",
"OA-FLOW.BOUNDARIES.EMBEDDING",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1241,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1144,
"through_line": 1241,
"description": "The partition theorem assumes a second-countable locally compact Hausdorff space. Continuous trace is defined by the Hausdorff hull-kernel spectrum and Fell local rank-one condition; the central action and local compactness are derived. Separability is used to obtain a countable base. The final direct-sum condition is sufficient; structural necessity is not asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1144,
"through_line": 1241,
"description": "Complete cutoff and partition construction; ideal and local-corner spectrum arguments; actual central multiplier construction and nondegeneracy; strict assembly for continuous-trace algebras, including a changing-fiber example."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.CENTRAL-FUNCTIONS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.central-functions",
"start_line": 1242,
"end_line": 1312,
"range_sha256": "76c307a15878aff51e91738176e4a85bfee93b8c03864ac4ab5b613f3ae0dcfb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.LOCAL-RANK-ONE",
"OA-FLOW.BOUNDARIES.ABELIAN-CORNERS",
"OA-FLOW.BOUNDARIES.TOPOLOGICAL-PARTITIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1312,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1242,
"through_line": 1312,
"description": "The partition theorem assumes a second-countable locally compact Hausdorff space. Continuous trace is defined by the Hausdorff hull-kernel spectrum and Fell local rank-one condition; the central action and local compactness are derived. Separability is used to obtain a countable base. The final direct-sum condition is sufficient; structural necessity is not asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1242,
"through_line": 1312,
"description": "Complete cutoff and partition construction; ideal and local-corner spectrum arguments; actual central multiplier construction and nondegeneracy; strict assembly for continuous-trace algebras, including a changing-fiber example."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.CONTINUOUS-TRACE",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.continuous-trace",
"start_line": 1313,
"end_line": 1374,
"range_sha256": "9ace9b4958bafc2a10f3a332deae2bbea2b43334fc31dd50f906f7109e139244",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.CENTRAL-FUNCTIONS",
"OA-FLOW.BOUNDARIES.TOPOLOGICAL-PARTITIONS",
"OA-FLOW.BOUNDARIES.LOCAL-RANK-ONE",
"OA-FLOW.BOUNDARIES.CENTRAL-GLUING",
"OA-FLOW.BOUNDARIES.DIRECT-SUMS",
"OA-FLOW.L34.4",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1374,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1313,
"through_line": 1374,
"description": "The partition theorem assumes a second-countable locally compact Hausdorff space. Continuous trace is defined by the Hausdorff hull-kernel spectrum and Fell local rank-one condition; the central action and local compactness are derived. Separability is used to obtain a countable base. The final direct-sum condition is sufficient; structural necessity is not asserted."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1313,
"through_line": 1374,
"description": "Complete cutoff and partition construction; ideal and local-corner spectrum arguments; actual central multiplier construction and nondegeneracy; strict assembly for continuous-trace algebras, including a changing-fiber example."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.COMPACT-UNITIZATION",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.compact-unitization",
"start_line": 1376,
"end_line": 1461,
"range_sha256": "4a615715a81f746f916c9d399cd2abd98d8b2fbb553fb834676bcb1c5a50bf4b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1461,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1376,
"through_line": 1461,
"description": "The Hilbert space is ell-two of the positive integers. A is the compact-operator unitization inside B(H). The intermediate spatial implementation applies to any concrete C*-algebra containing the compact operators. Derivations are everywhere-defined complex-linear maps; star assertions and numerical bounds specify selfadjoint implementers."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1376,
"through_line": 1461,
"description": "Complete rank-one spatial implementation, scalar ambiguity and innerness classification; exact commutator norm; bounded outer generator and exact inner times of its automorphism group; full diagonal-family exercise and solution."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.COMPACT-DERIVATION-NORM",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.compact-derivation-norm",
"start_line": 1462,
"end_line": 1498,
"range_sha256": "739a11580ae1c8cc62114a59643879c20d9a5c842ad84d68397560dab414066e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.COMPACT-UNITIZATION",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1498,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1462,
"through_line": 1498,
"description": "The Hilbert space is ell-two of the positive integers. A is the compact-operator unitization inside B(H). The intermediate spatial implementation applies to any concrete C*-algebra containing the compact operators. Derivations are everywhere-defined complex-linear maps; star assertions and numerical bounds specify selfadjoint implementers."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1462,
"through_line": 1498,
"description": "Complete rank-one spatial implementation, scalar ambiguity and innerness classification; exact commutator norm; bounded outer generator and exact inner times of its automorphism group; full diagonal-family exercise and solution."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.OUTER-COMPACT-FLOW",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.outer-compact-flow",
"start_line": 1499,
"end_line": 1568,
"range_sha256": "7292c79800ac347b8c9c60d6c9d8ef93485bf0e034ee639c2ecd8d3d989aa62e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.COMPACT-UNITIZATION",
"OA-FLOW.BOUNDARIES.COMPACT-DERIVATION-NORM",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1568,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1499,
"through_line": 1568,
"description": "The Hilbert space is ell-two of the positive integers. A is the compact-operator unitization inside B(H). The intermediate spatial implementation applies to any concrete C*-algebra containing the compact operators. Derivations are everywhere-defined complex-linear maps; star assertions and numerical bounds specify selfadjoint implementers."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1499,
"through_line": 1568,
"description": "Complete rank-one spatial implementation, scalar ambiguity and innerness classification; exact commutator norm; bounded outer generator and exact inner times of its automorphism group; full diagonal-family exercise and solution."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.ORTHOGONAL-SUMS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.orthogonal-sums",
"start_line": 1570,
"end_line": 1627,
"range_sha256": "c26dffc3aef86eb701dabc046710f9a8272bdb83268e6af7b931f9f6096e4f66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.8",
"OA-FLOW.CF.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1627,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1570,
"through_line": 1627,
"description": "The general construction uses a separable C*-algebra faithfully and nondegenerately represented on a Hilbert space of arbitrary dimension, orthogonal positive contractions, and summability of commutator norms on the stated dense sequence. Bounded real coefficients give star derivations and flows; complex coefficients give complex derivations. Orthogonality is an assumption retained by subsequence selection. The final outer-distance and inner-time formulas concern the explicit even-coordinate compact-unitization family."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1570,
"through_line": 1627,
"description": "Strong sum and exact norm-tail estimates; dense-test extension of commutators to all of A; central-sequence thinning and bounded coordinatewise continuity; integrated flows and explicit c0-difference cocycles; exact outer-distance and inner-time calculations, including a non-absolutely summable commutator exercise."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.SUMMABLE-COMMUTATORS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.summable-commutators",
"start_line": 1628,
"end_line": 1681,
"range_sha256": "ad8a29b47e863e5f6c712cdcf8872f9ec829df4a4c5a43c1e897eb00fe7cc4f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.ORTHOGONAL-SUMS",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1681,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1628,
"through_line": 1681,
"description": "The general construction uses a separable C*-algebra faithfully and nondegenerately represented on a Hilbert space of arbitrary dimension, orthogonal positive contractions, and summability of commutator norms on the stated dense sequence. Bounded real coefficients give star derivations and flows; complex coefficients give complex derivations. Orthogonality is an assumption retained by subsequence selection. The final outer-distance and inner-time formulas concern the explicit even-coordinate compact-unitization family."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1628,
"through_line": 1681,
"description": "Strong sum and exact norm-tail estimates; dense-test extension of commutators to all of A; central-sequence thinning and bounded coordinatewise continuity; integrated flows and explicit c0-difference cocycles; exact outer-distance and inner-time calculations, including a non-absolutely summable commutator exercise."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.CENTRAL-THINNING",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.central-thinning",
"start_line": 1682,
"end_line": 1715,
"range_sha256": "117eacaa72b3f73c59239dbb3ca4ab4fdcdf197229309bd767e5f09590823f8b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.SUMMABLE-COMMUTATORS",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1715,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1682,
"through_line": 1715,
"description": "The general construction uses a separable C*-algebra faithfully and nondegenerately represented on a Hilbert space of arbitrary dimension, orthogonal positive contractions, and summability of commutator norms on the stated dense sequence. Bounded real coefficients give star derivations and flows; complex coefficients give complex derivations. Orthogonality is an assumption retained by subsequence selection. The final outer-distance and inner-time formulas concern the explicit even-coordinate compact-unitization family."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1682,
"through_line": 1715,
"description": "Strong sum and exact norm-tail estimates; dense-test extension of commutators to all of A; central-sequence thinning and bounded coordinatewise continuity; integrated flows and explicit c0-difference cocycles; exact outer-distance and inner-time calculations, including a non-absolutely summable commutator exercise."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.SERIES-FLOWS",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.series-flows",
"start_line": 1716,
"end_line": 1771,
"range_sha256": "3bd609ddc4a7e5970e39b407a259621b77e785d9823a42113be3cdc76218221d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.ORTHOGONAL-SUMS",
"OA-FLOW.BOUNDARIES.SUMMABLE-COMMUTATORS",
"OA-FLOW.BOUNDARIES.CENTRAL-THINNING",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1771,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1716,
"through_line": 1771,
"description": "The general construction uses a separable C*-algebra faithfully and nondegenerately represented on a Hilbert space of arbitrary dimension, orthogonal positive contractions, and summability of commutator norms on the stated dense sequence. Bounded real coefficients give star derivations and flows; complex coefficients give complex derivations. Orthogonality is an assumption retained by subsequence selection. The final outer-distance and inner-time formulas concern the explicit even-coordinate compact-unitization family."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1716,
"through_line": 1771,
"description": "Strong sum and exact norm-tail estimates; dense-test extension of commutators to all of A; central-sequence thinning and bounded coordinatewise continuity; integrated flows and explicit c0-difference cocycles; exact outer-distance and inner-time calculations, including a non-absolutely summable commutator exercise."
}
]
},
{
"id": "OA-FLOW.BOUNDARIES.OUTER-SERIES-CLASSES",
"unit": "OA-FLOW-L137",
"source_file": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5",
"reader": "OA-FLOW-L137.html#boundaries.outer-series-classes",
"start_line": 1772,
"end_line": 1855,
"range_sha256": "04c3375a29b405a45fc3d5f901774632ef3c1a3fd744d2bc3ed160cc8312f92a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BOUNDARIES.COMPACT-UNITIZATION",
"OA-FLOW.BOUNDARIES.COMPACT-DERIVATION-NORM",
"OA-FLOW.BOUNDARIES.OUTER-COMPACT-FLOW",
"OA-FLOW.BOUNDARIES.SUMMABLE-COMMUTATORS",
"OA-FLOW.BOUNDARIES.SERIES-FLOWS",
"OA-FLOW.CF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1855,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L137.md",
"line": 1772,
"through_line": 1855,
"description": "The general construction uses a separable C*-algebra faithfully and nondegenerately represented on a Hilbert space of arbitrary dimension, orthogonal positive contractions, and summability of commutator norms on the stated dense sequence. Bounded real coefficients give star derivations and flows; complex coefficients give complex derivations. Orthogonality is an assumption retained by subsequence selection. The final outer-distance and inner-time formulas concern the explicit even-coordinate compact-unitization family."
},
{
"source": "src/OA-FLOW-L137.md",
"line": 1772,
"through_line": 1855,
"description": "Strong sum and exact norm-tail estimates; dense-test extension of commutators to all of A; central-sequence thinning and bounded coordinatewise continuity; integrated flows and explicit c0-difference cocycles; exact outer-distance and inner-time calculations, including a non-absolutely summable commutator exercise."
}
]
},
{
"id": "OA-FLOW.IW.0",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-0",
"start_line": 1,
"end_line": 29,
"range_sha256": "b8a5352c7ccf462713da4bb52620c59ff87f7f041610d90a3caee279edf9fff3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.1",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-1",
"start_line": 30,
"end_line": 102,
"range_sha256": "543c2caeebbff76144837d0959d33d149fe68f9d191c43b139ec8f2ba226d878",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 102,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.2",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-2",
"start_line": 103,
"end_line": 181,
"range_sha256": "5fe47036cde0e1dcff168abddfb58f2329fb83fb43114e26643825da107106e7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 181,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.3",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-3",
"start_line": 182,
"end_line": 292,
"range_sha256": "b05ba1403ff83ea77d6e4cdc93d893cc05a4e26ad6f8e254521708d2f37b4c30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1",
"OA-FLOW.IW.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 292,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.4",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-4",
"start_line": 293,
"end_line": 331,
"range_sha256": "5939ab439983f2626c2ecfdfe361f9c66a7df055c1ef9c8c3d896ecd0bfb573e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1",
"OA-FLOW.IW.2",
"OA-FLOW.IW.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 331,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.5",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-5",
"start_line": 332,
"end_line": 395,
"range_sha256": "ae1460f473c9a4a50fb1c1f0e1dad690304ae372ea53d06b4a33a0d3d8deaed4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1",
"OA-FLOW.IW.2",
"OA-FLOW.IW.3",
"OA-FLOW.IW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 395,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.6",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-6",
"start_line": 396,
"end_line": 415,
"range_sha256": "eb3a2156f455b4ed7c6fb163c1f97fe074263add2411df2a11ac19e90ad98e79",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1",
"OA-FLOW.IW.2",
"OA-FLOW.IW.3",
"OA-FLOW.IW.4",
"OA-FLOW.IW.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 415,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.IW.FIGURE",
"unit": "OA-FLOW-IW",
"source_file": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792",
"reader": "OA-FLOW-IW.html#iw-figure",
"start_line": 416,
"end_line": 423,
"range_sha256": "421a35ac6fd736f370f12c1ca2f70acb20074c820eefb9629c33463f32d96580",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.HR.9",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IMPR.SETTING",
"OA-FLOW.IINT.SMOOTH",
"OA-FLOW.QF.1",
"OA-FLOW.CF.8",
"OA-FLOW.CP.4",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.NCF.1",
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.QF.3",
"OA-FLOW.QF.7",
"OA-FLOW.QF.6",
"OA-FLOW.CP.6",
"OA-FLOW.IINT.RECOVERY",
"OA-FLOW.CF.6",
"OA-FLOW.IW.1",
"OA-FLOW.IW.2",
"OA-FLOW.IW.3",
"OA-FLOW.IW.4",
"OA-FLOW.IW.5",
"OA-FLOW.IW.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 423,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-IW.md",
"line": 1,
"through_line": 29,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.ISYS.STATEMENT",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.isys.statement",
"start_line": 16,
"end_line": 44,
"range_sha256": "cfd2f14d27552a129d82512ba25fc406caf5ef6ba4e267b0ce53968db16431a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.ISYS.FORWARD",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.isys.forward",
"start_line": 45,
"end_line": 78,
"range_sha256": "e8cb53270e6757197aebda8823385e4214ea3093a15a966e96aec5cae6ae7957",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 78,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.ISYS.RECONSTRUCTION",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.isys.reconstruction",
"start_line": 79,
"end_line": 113,
"range_sha256": "01520b99a41b0b031a56f7349a8b70421fc168253138c0894a3353af6df7d36e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 113,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.ISYS.RELATIVE.UNIQUENESS",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.relative-uniqueness",
"start_line": 114,
"end_line": 133,
"range_sha256": "b145a222427acdc3061686d3c71c884a1d8fef195079d20377c6f37424e522cb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.continuous-fields",
"start_line": 134,
"end_line": 177,
"range_sha256": "806498e6fd7b4e1d0aa4f3065ee3acebeb4509e02028d8c36ffbf3b73a29b599",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 177,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.evaluation-localization",
"start_line": 178,
"end_line": 190,
"range_sha256": "e280eec37356b6031037743a1841573fd33651f35e080d70e461a9ab8da4bf0d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 190,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.elementary-sections",
"start_line": 191,
"end_line": 238,
"range_sha256": "1d273324068bbc25f8cc5077de27a63eb6fb1df53d6dff1dc605e36adec877c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 238,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.EVALUATION.RANGE",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.evaluation-range",
"start_line": 239,
"end_line": 253,
"range_sha256": "57284c64112b8e1b693322aa4f02f4a806d1db4b1d34fbfc78937b5000140afe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 253,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.CORE.ISOMORPHISM",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.core-isomorphism",
"start_line": 254,
"end_line": 273,
"range_sha256": "ddf58811f5fd8d7460d06ad136474897909bc3656851793d13d1bb65364e8962",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 273,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.EVALUATION.NORMALITY",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.evaluation-normality",
"start_line": 274,
"end_line": 306,
"range_sha256": "ed93163ff387dcfb39b121e5efdde79f7509c0aaeb6f27e78f8384eacc1a91f7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 306,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.NORMAL.EXTENSION",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.indrec.normal-extension",
"start_line": 307,
"end_line": 337,
"range_sha256": "1fd14afa6028552963c6e88b6f6f3c0cb77fe7bf21439e068081cde1c2aa1fc0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 337,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 114,
"through_line": 133,
"description": "Two normal induced systems and a normal G-equivariant isomorphism preserving their specified quotient embeddings."
}
]
},
{
"id": "OA-FLOW.ISYS.TWO.FIBERS",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.isys.two-fibers",
"start_line": 338,
"end_line": 404,
"range_sha256": "26025b025576ee6880fdb90b223317b3b788d2ad3fc71f10ec1d81a0e4fb5040",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 404,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.SETTING",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-setting",
"start_line": 405,
"end_line": 423,
"range_sha256": "ac5ef19c42da6626f77795101f28da7a678dceeed842512549ae6e12c4ffea63",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 423,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-coordinates",
"start_line": 424,
"end_line": 451,
"range_sha256": "5accb737f8cecc64a78a86bdcc6d5027d3925c4da98710ec14dfdbdffb272500",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 451,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.MATRICES",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-matrices",
"start_line": 452,
"end_line": 485,
"range_sha256": "3c8ac615bab9bdb552323d88aa0667ce10188ca61368fd7342d4522a26a66b2a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 485,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.EVALUATION",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-evaluation",
"start_line": 486,
"end_line": 531,
"range_sha256": "0c5057dbf067f9fc3de2958c3615f09b6cc55ed28046bbcfa4c9c5ce9076162c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"OA-FLOW.ISYS.SEPARABLE.MATRICES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 531,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.CONTRACTIONS",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-contractions",
"start_line": 532,
"end_line": 565,
"range_sha256": "935ad49ec36f6a949789a9d78aff76e5099f2585df2e013a3eae410efc05bab2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"OA-FLOW.ISYS.SEPARABLE.MATRICES",
"OA-FLOW.ISYS.SEPARABLE.EVALUATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 565,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.DESCENT",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-descent",
"start_line": 566,
"end_line": 599,
"range_sha256": "19d92cec8a6d3e378fff4f2e1b01dba32c7d29798e7d49c423e0752d7a89ca66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"OA-FLOW.ISYS.SEPARABLE.MATRICES",
"OA-FLOW.ISYS.SEPARABLE.EVALUATION",
"OA-FLOW.ISYS.SEPARABLE.CONTRACTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 599,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.SEPARABLE.PASTING",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#isys-separable-pasting",
"start_line": 600,
"end_line": 641,
"range_sha256": "c5dfaa949b4811750d29ca1ae483df5a3c8dc35763055c88c8f1172943bbb9a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"OA-FLOW.ISYS.SEPARABLE.MATRICES",
"OA-FLOW.ISYS.SEPARABLE.EVALUATION",
"OA-FLOW.ISYS.SEPARABLE.CONTRACTIONS",
"OA-FLOW.ISYS.SEPARABLE.DESCENT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 641,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
},
{
"source": "src/OA-FLOW-L47.md",
"line": 405,
"through_line": 423,
"description": "This alternative alone assumes second-countable G and a separable faithful spatial representation; its subgroup Hilbert space is proved separable."
}
]
},
{
"id": "OA-FLOW.ISYS.EXERCISES",
"unit": "OA-FLOW-L47",
"source_file": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765",
"reader": "OA-FLOW-L47.html#oa-flow.isys.exercises",
"start_line": 642,
"end_line": 653,
"range_sha256": "33b8d9503f4a186560220ceb9cae901ce3f1297d7af9eac47e6f64ce7e1f0d02",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.CLASS",
"OA-FLOW.NR.1",
"OA-FLOW.NCF.1",
"OA-FLOW.IW.5",
"OA-FLOW.CP.4",
"OA-FLOW.QM.RHO",
"OA-FLOW.IMPR.UNITARY",
"OA-FLOW.AT.3",
"OA-FLOW.AT.6",
"OA-FLOW.CP.6",
"OA-FLOW.QF.1",
"OA-FLOW.QF.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.2",
"OA-FLOW.IS.2",
"OA-FLOW.IMPR.SMOOTH",
"OA-FLOW.IS.1",
"OA-FLOW.IND.SETTING",
"OA-FLOW.IND.ELEMENTARY",
"OA-FLOW.IS.4",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.NR.3",
"OA-FLOW.ISYS.FORWARD",
"OA-FLOW.ISYS.RECONSTRUCTION",
"OA-FLOW.ISYS.CONTINUOUS.FIELDS",
"OA-FLOW.ISYS.EVALUATION.LOCALIZATION",
"OA-FLOW.ISYS.ELEMENTARY.SECTIONS",
"OA-FLOW.ISYS.EVALUATION.RANGE",
"OA-FLOW.ISYS.CORE.ISOMORPHISM",
"OA-FLOW.ISYS.EVALUATION.NORMALITY",
"OA-FLOW.ISYS.NORMAL.EXTENSION",
"OA-FLOW.ISYS.TWO.FIBERS",
"OA-FLOW.ISYS.SEPARABLE.SETTING",
"OA-FLOW.ISYS.SEPARABLE.COORDINATES",
"OA-FLOW.ISYS.SEPARABLE.MATRICES",
"OA-FLOW.ISYS.SEPARABLE.EVALUATION",
"OA-FLOW.ISYS.SEPARABLE.CONTRACTIONS",
"OA-FLOW.ISYS.SEPARABLE.DESCENT",
"OA-FLOW.ISYS.SEPARABLE.PASTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 653,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L47.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary locally compact Hausdorff group, closed subgroup, canonical local Haar/quotient classes, continuous normal action and specified equivariant central quotient embedding."
}
]
},
{
"id": "OA-FLOW.55.OPEN.QUOTIENT",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-quotient",
"start_line": 1,
"end_line": 70,
"range_sha256": "f068d00a2130ece5282ccff6b92603046c0f051f5f079340be82da9edf70229f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 70,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.MATRIX",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-matrix",
"start_line": 71,
"end_line": 99,
"range_sha256": "32f1b1efd119f07847af88e6420d24af2e280d374006bb1e401eaf860e02f559",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 99,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.CORNER.GENERATORS",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-corner-generators",
"start_line": 100,
"end_line": 119,
"range_sha256": "6672e80565ea9881c66a92a2fd69f5ebabfc6ba67fb1db0cd493ade61c67d3ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT",
"OA-FLOW.55.OPEN.MATRIX"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 119,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.CORNER.UNITARY",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-corner-unitary",
"start_line": 120,
"end_line": 219,
"range_sha256": "d26fcd53b4ec255b338623c92c59dfd6bbdc4a1e3086e5f69c019ca4bbfa4d15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT",
"OA-FLOW.55.OPEN.MATRIX",
"OA-FLOW.55.OPEN.CORNER.GENERATORS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 219,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.THEOREM",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-theorem",
"start_line": 220,
"end_line": 325,
"range_sha256": "b1de733b7714223ad8fa5978f8b8e6ccd4dfd0d4f6f001d24d194dd56b9bf3cb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT",
"OA-FLOW.55.OPEN.MATRIX",
"OA-FLOW.55.OPEN.CORNER.GENERATORS",
"OA-FLOW.55.OPEN.CORNER.UNITARY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 325,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.FIGURE",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-figure",
"start_line": 326,
"end_line": 350,
"range_sha256": "12cef098004946f246e60d830ff21de856fbef2a0a6d48caae2a036acc0a3d8c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT",
"OA-FLOW.55.OPEN.MATRIX",
"OA-FLOW.55.OPEN.CORNER.GENERATORS",
"OA-FLOW.55.OPEN.CORNER.UNITARY",
"OA-FLOW.55.OPEN.THEOREM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 350,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.55.OPEN.EXERCISES",
"unit": "OA-FLOW-L55",
"source_file": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06",
"reader": "OA-FLOW-L55.html#open-exercises",
"start_line": 351,
"end_line": 366,
"range_sha256": "6ecada5de16c7904f4bac09f7a6d2aefd40eceb057dfc22b532edcded1b03655",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.NCF.1",
"OA-FLOW.ISYS.EXERCISES",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IS.6",
"OA-FLOW.55.OPEN.QUOTIENT",
"OA-FLOW.55.OPEN.MATRIX",
"OA-FLOW.55.OPEN.CORNER.GENERATORS",
"OA-FLOW.55.OPEN.CORNER.UNITARY",
"OA-FLOW.55.OPEN.THEOREM",
"OA-FLOW.55.OPEN.FIGURE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 366,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L55.md",
"line": 1,
"through_line": 15,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.DISCRETE.SETTING",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.setting",
"start_line": 11,
"end_line": 47,
"range_sha256": "539a6294c46d5ac3520efac9e7f48c2c593e44105fd6adbc25795180e8b85f05",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.CP.6",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 47,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.EXPECTATION",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.expectation",
"start_line": 49,
"end_line": 88,
"range_sha256": "123bc1f5e06c196de7f8ef5b308e33855788b8e0c76913bf39e22eaecff598e1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.SETTING",
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 88,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.CORNER",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.corner",
"start_line": 90,
"end_line": 148,
"range_sha256": "abf39f0cd89b91a6258a0bffc095dab7e2a02d8c12293347ad134dc18b82dbe6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.SETTING",
"OA-FLOW.DISCRETE.EXPECTATION",
"OA-FLOW.NR.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 148,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.STATE.FAMILY",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.state-family",
"start_line": 150,
"end_line": 197,
"range_sha256": "43fe012f838cefae348b4c6787bbaa762e4b64ab6e1560ba68cfb1e70b87d407",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.EXPECTATION",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.10",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.NR.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 197,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.AMPLIFICATION",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.matrix-units",
"start_line": 199,
"end_line": 322,
"range_sha256": "90b00ec38a44d3f1cb91555e178ed8bddef5cbfc1be46df98ae5924bbacf416b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.SETTING",
"OA-FLOW.DISCRETE.CORNER",
"OA-FLOW.NCF.1",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 322,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.THREE.COSETS",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.example",
"start_line": 324,
"end_line": 384,
"range_sha256": "7f93a9f17c0532854a42d436aa2094669f65cdc01a57e03e93ad4f165629f6d5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.AMPLIFICATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 384,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.DISCRETE.REPRESENTATIVES",
"unit": "OA-FLOW-L51",
"source_file": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621",
"reader": "OA-FLOW-L51.html#oa-flow.discrete.representatives",
"start_line": 386,
"end_line": 417,
"range_sha256": "7d22cb56b6603da2f18da583eddc679a8ca498125d14055898c51107dd8ce6ed",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DISCRETE.AMPLIFICATION",
"OA-FLOW.DISCRETE.CORNER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 417,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L51.md",
"line": 1,
"through_line": 11,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTCOMM.GENERATORS",
"unit": "OA-FLOW-L52",
"source_file": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca",
"reader": "OA-FLOW-L52.html#oa-flow.compactcomm.generators",
"start_line": 7,
"end_line": 52,
"range_sha256": "879a8d947acb1c31409a1c31246f0645b83f4ce1780b2f2c30b397a103da3cc4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 52,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L52.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTCOMM.EXPECTATION",
"unit": "OA-FLOW-L52",
"source_file": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca",
"reader": "OA-FLOW-L52.html#oa-flow.compactcomm.expectation",
"start_line": 54,
"end_line": 84,
"range_sha256": "9f20426fdb2425edfc782f318503fab0caf03200afcd5bf61ad83b88b6a7425e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTCOMM.GENERATORS",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.QF.6",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 84,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L52.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTCOMM.AVERAGE",
"unit": "OA-FLOW-L52",
"source_file": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca",
"reader": "OA-FLOW-L52.html#oa-flow.compactcomm.average",
"start_line": 86,
"end_line": 146,
"range_sha256": "d7d8a7f8b3c72913c3e3b1260d16606724ede66139b4831bf481ebc6c8f6f74a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTCOMM.EXPECTATION",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.QF.6",
"OA-FLOW.NR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 146,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L52.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTCOMM.EQUALITY",
"unit": "OA-FLOW-L52",
"source_file": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca",
"reader": "OA-FLOW-L52.html#oa-flow.compactcomm.equality",
"start_line": 148,
"end_line": 165,
"range_sha256": "b8aa1c92af622530dc809db3a350c2b1d3d146045263af1bc7bf7c35ddac2eda",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTCOMM.GENERATORS",
"OA-FLOW.COMPACTCOMM.AVERAGE",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.HR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 165,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L52.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTCOMM.EXAMPLE",
"unit": "OA-FLOW-L52",
"source_file": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca",
"reader": "OA-FLOW-L52.html#oa-flow.compactcomm.example",
"start_line": 167,
"end_line": 195,
"range_sha256": "f5ef3ca495e57bc62a141f4485f289463a481568878f8253225533c3be91749f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTCOMM.AVERAGE",
"OA-FLOW.COMPACTCOMM.EQUALITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 195,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L52.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTSTAB.INTERCHANGE",
"unit": "OA-FLOW-L53",
"source_file": "src/OA-FLOW-L53.md",
"source_sha256": "036657f25d6e5a9e96855f78d8d03926391f5892c69c7b3307871604d6a67713",
"reader": "OA-FLOW-L53.html#oa-flow.compactstab.interchange",
"start_line": 7,
"end_line": 62,
"range_sha256": "670c115ffa7f21a00e8d9d204f622cf6ed7a112c11632639996588362856f8d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.CP.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.COMPACTCOMM.EXPECTATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 62,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L53.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTSTAB.COORDINATES",
"unit": "OA-FLOW-L53",
"source_file": "src/OA-FLOW-L53.md",
"source_sha256": "036657f25d6e5a9e96855f78d8d03926391f5892c69c7b3307871604d6a67713",
"reader": "OA-FLOW-L53.html#oa-flow.compactstab.coordinates",
"start_line": 64,
"end_line": 173,
"range_sha256": "6e79507b6f0df9db8507b652c2e245f588a6a202560982e6f0b1e6135a8382c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTSTAB.INTERCHANGE",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IS.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.NCF.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 173,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L53.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTSTAB.THEOREM",
"unit": "OA-FLOW-L53",
"source_file": "src/OA-FLOW-L53.md",
"source_sha256": "036657f25d6e5a9e96855f78d8d03926391f5892c69c7b3307871604d6a67713",
"reader": "OA-FLOW-L53.html#oa-flow.compactstab.theorem",
"start_line": 175,
"end_line": 191,
"range_sha256": "817976f0a673fc13ae0c0e5fb2386ea6b2737fd9aa4db687f5b0b244f845d620",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.NR.4",
"OA-FLOW.IS.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 191,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L53.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.COMPACTSTAB.EXAMPLE",
"unit": "OA-FLOW-L53",
"source_file": "src/OA-FLOW-L53.md",
"source_sha256": "036657f25d6e5a9e96855f78d8d03926391f5892c69c7b3307871604d6a67713",
"reader": "OA-FLOW-L53.html#oa-flow.compactstab.example",
"start_line": 193,
"end_line": 238,
"range_sha256": "4ffd6f04814a4098b56909bb331231063b738bf64fb1c9c1d0a0397ea4571633",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.COMPACTSTAB.THEOREM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 238,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L53.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.MODEL.DOMAIN",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.domain",
"start_line": 1,
"end_line": 71,
"range_sha256": "1da680ffc7c2c901e1a918156b57e78f3e88ce992f02892851e73ce15403462d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 71,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.GENERATORS",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.generators",
"start_line": 72,
"end_line": 98,
"range_sha256": "b8693b1198e30cfc774c850aef6a9589b87654dee0c63ddb16a8c50314bb402c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 98,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.SMOOTHING",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.smoothing",
"start_line": 99,
"end_line": 170,
"range_sha256": "7480adbbfdcd36e71e7d68b096600577c3eb07e4e34dea4a39ca2a808835d14e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.GENERATORS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 170,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.SPECTRUM",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.spectrum",
"start_line": 171,
"end_line": 215,
"range_sha256": "3a5b32fae506f9ba602001fad0d6031567febb48e3ccd673b0c0fc60b124e5e1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.GENERATORS",
"OA-FLOW.MODEL.SMOOTHING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 215,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.NORMAL",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.normal",
"start_line": 216,
"end_line": 249,
"range_sha256": "8f8542dbb045d633bb9e4f538a297b4a2dda1ed1c0863cd4af4a3df192e57ebc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.GENERATORS",
"OA-FLOW.MODEL.SMOOTHING",
"OA-FLOW.MODEL.SPECTRUM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 249,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.TRANSPORT",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.transport",
"start_line": 250,
"end_line": 301,
"range_sha256": "19426312c2876564a63b9514087521375080bd99b1130e73173e0dad707c19c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.GENERATORS",
"OA-FLOW.MODEL.SMOOTHING",
"OA-FLOW.MODEL.SPECTRUM",
"OA-FLOW.MODEL.NORMAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 301,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.MODEL.EXAMPLE",
"unit": "OA-FLOW-L72",
"source_file": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850",
"reader": "OA-FLOW-L72.html#oa-flow.model.example",
"start_line": 302,
"end_line": 506,
"range_sha256": "bc87c2ce9d243792131c3fc29a21a06df5da2eb56f07422163f823328f84089e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BD.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CF.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.CF.9",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.SC.5",
"OA-FLOW.SC.8",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.GENERATORS",
"OA-FLOW.MODEL.SMOOTHING",
"OA-FLOW.MODEL.SPECTRUM",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 506,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L72.md",
"line": 1,
"through_line": 18,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.ORBITS.SETTING",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.setting",
"start_line": 7,
"end_line": 21,
"range_sha256": "17a08591f5296b2d4937e3fe4ad4650ebbb7085556a1bd88ba4016232ae7648e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 21,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.INTEGRATION",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.integration",
"start_line": 23,
"end_line": 26,
"range_sha256": "0f9cbf1f257724308c77693e1ddfaa0b7d7de86d7d51962a45389ba62e666e16",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 26,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.WITNESS",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.witness",
"start_line": 28,
"end_line": 48,
"range_sha256": "9bd729093f70e7fdc0197108f3d05233a156f846f417d92027e97e439315cb1e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.IW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 48,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.SELECTOR",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.selector",
"start_line": 50,
"end_line": 93,
"range_sha256": "69ac924e5b3db404167c867b433e99bdea9b070d308e781eb48b4775e3f366c5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.WITNESS",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.QF.6",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 93,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.REFERENCE",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.reference",
"start_line": 95,
"end_line": 133,
"range_sha256": "da5eb2181801d62fd221873eaf351fe291fcc9f1d136733dd1c796ba541e806e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 133,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.DENSITY",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.density",
"start_line": 135,
"end_line": 155,
"range_sha256": "00b13d045426c44f714b93161a5154cbcc9bc8db61db0150e6dfdaa65cb2ba2a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.CF.8",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 155,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.CONDITIONAL",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.conditional",
"start_line": 157,
"end_line": 196,
"range_sha256": "3849ce4efda60cc1366331e41064886518639981f15d15cfa97d9f27977f3d3d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.GRP.TRANSLATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 196,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.STABILIZERS",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.stabilizers",
"start_line": 198,
"end_line": 206,
"range_sha256": "a2a006b575563a2e0a7fb171d8b2601ecbc11d59b16993af0c5526bcafcbf6c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.WITNESS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 206,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.HOMOGENEOUS",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.homogeneous",
"start_line": 208,
"end_line": 234,
"range_sha256": "c5c807cb2dbabe140ad54395b1f984337dfa852e8ce7b02056f9065742acb2ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.QF.1",
"OA-FLOW.QM.RHO",
"OA-FLOW.QM.CLASS",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 234,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.STABILIZER-FIELD",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.stabilizer-field",
"start_line": 236,
"end_line": 256,
"range_sha256": "09c627db5bf2ae1c9031b89c8ee1de90eeb8fc3d142f9cbb6c5bcfac325ad243",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 256,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.ORBITS.ERGODIC",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.ergodic",
"start_line": 258,
"end_line": 280,
"range_sha256": "2f982d5e83d88218ef2080801c5f34584eab0c3e7e1c8a830c7d6f594e1b42fa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.HOMOGENEOUS",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.EOR.6",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.CP.4",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 280,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 7,
"through_line": 21,
"description": "The complete lesson setting and retained common conull base apply."
}
]
},
{
"id": "OA-FLOW.ORBITS.EXAMPLE",
"unit": "OA-FLOW-L69",
"source_file": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0",
"reader": "OA-FLOW-L69.html#oa-flow.orbits.example",
"start_line": 281,
"end_line": 313,
"range_sha256": "a45ab779ce13429aa40518b10c3845fa5fc066dc06101e7b6fecc13e933c22e5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.ORBITS.CONDITIONAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 313,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L69.md",
"line": 1,
"through_line": 7,
"description": "Initial global setting together with the explicit hypotheses in this section and its prerequisite proofs."
}
]
},
{
"id": "OA-FLOW.PROPER.THEOREM",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.theorem",
"start_line": 1,
"end_line": 70,
"range_sha256": "f99e22d531f920f5c82f6bd5979449f16576793825930be639158a8752f3ac28",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.SELECTION",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.selection",
"start_line": 71,
"end_line": 107,
"range_sha256": "de9f8923afb6a3f81fd35a905ea499b979ab7c5212e29cfdaa2e8812a6da90c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 107,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.STABILIZERS",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.stabilizers",
"start_line": 108,
"end_line": 156,
"range_sha256": "5cee85ce2843fc3618ac4e2c0562024b442c7109de90159514071be8625bbcb5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 156,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.BASE",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.base",
"start_line": 157,
"end_line": 190,
"range_sha256": "47d086e2483dc639311645cf0ba02d6b9a185cf32251644f698ccd7042d18c23",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 190,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.BUNDLE",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.bundle",
"start_line": 191,
"end_line": 231,
"range_sha256": "5cf40a820ba28fb76d9103fb87745374f48b381d5894ea9c7b4e1eaf32aad6bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 231,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.CONJUGACY",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.conjugacy",
"start_line": 232,
"end_line": 274,
"range_sha256": "6c85105a4e91023f7774c46ab8388b22443bf0b5c60190b959c5784d782ee9b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 274,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.MEASURE",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.measure",
"start_line": 275,
"end_line": 305,
"range_sha256": "330bb4d1893608a278541b3a6044b93bbd38729af750ccdd406a0e36c18a7f8c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 305,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.AVERAGES",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.averages",
"start_line": 306,
"end_line": 343,
"range_sha256": "8e30c9df01ab2c9a65936ed4cd1d84528c8a4494483f3922277380a4987092cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.PROPER.MEASURE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 343,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.ABSTRACT",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.abstract",
"start_line": 344,
"end_line": 355,
"range_sha256": "a9394023905bbfe7c53a26b773d906849c60abd190f0d58ee1e94aea7eaf1c51",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.PROPER.MEASURE",
"OA-FLOW.PROPER.AVERAGES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 355,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.EXAMPLE",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.example",
"start_line": 356,
"end_line": 400,
"range_sha256": "21b4a70e1b28b2b603288ff031a01f16ac7763a8f40c562cf3314630372bbaa3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.PROPER.MEASURE",
"OA-FLOW.PROPER.AVERAGES",
"OA-FLOW.PROPER.ABSTRACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 400,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.FIGURE",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.figure",
"start_line": 401,
"end_line": 411,
"range_sha256": "e858feffa17656471e2314f66ecb92ac0a8afeaaa22a6f06f80732f5680d2abe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.PROPER.MEASURE",
"OA-FLOW.PROPER.AVERAGES",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 411,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.PROPER.EXERCISES",
"unit": "OA-FLOW-L71",
"source_file": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95",
"reader": "OA-FLOW-L71.html#oa-flow.proper.exercises",
"start_line": 412,
"end_line": 427,
"range_sha256": "843d101a37169ef7656f09a9b07e044619d9c3d242895b43eed4e6253141cd62",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.ORBITS.STABILIZERS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.6",
"OA-FLOW.IS.1",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.CF.5",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.PROPER.STABILIZERS",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.PROPER.MEASURE",
"OA-FLOW.PROPER.AVERAGES",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.EXAMPLE",
"OA-FLOW.PROPER.FIGURE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 427,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L71.md",
"line": 1,
"through_line": 20,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.TOPOLOGY.WARNING.SPACE",
"unit": "OA-FLOW-L73",
"source_file": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1",
"reader": "OA-FLOW-L73.html#oa-flow.topology-warning.space",
"start_line": 1,
"end_line": 47,
"range_sha256": "d1a568d52488299e0eb4d8e858181156712285f2c9f6d2b0e0dd5735b440bf88",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IW.1",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.PROPER.ABSTRACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 47,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L73.md",
"line": 1,
"through_line": 7,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.TOPOLOGY.WARNING.INTEGRABLE",
"unit": "OA-FLOW-L73",
"source_file": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1",
"reader": "OA-FLOW-L73.html#oa-flow.topology-warning.integrable",
"start_line": 48,
"end_line": 75,
"range_sha256": "8e190a149196d633be20000c0ea67ea488d3ec0cb32a722308bd86471ef04695",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IW.1",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.TOPOLOGY.WARNING.SPACE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 75,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L73.md",
"line": 1,
"through_line": 7,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.TOPOLOGY.WARNING.JUMP",
"unit": "OA-FLOW-L73",
"source_file": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1",
"reader": "OA-FLOW-L73.html#oa-flow.topology-warning.jump",
"start_line": 76,
"end_line": 106,
"range_sha256": "5d0dcd7e838a75d0987f7cefb74b7bac360e60e1dedf274858c89310b38bc691",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IW.1",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.TOPOLOGY.WARNING.SPACE",
"OA-FLOW.TOPOLOGY.WARNING.INTEGRABLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 106,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L73.md",
"line": 1,
"through_line": 7,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.TOPOLOGY.WARNING.PROPER",
"unit": "OA-FLOW-L73",
"source_file": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1",
"reader": "OA-FLOW-L73.html#oa-flow.topology-warning.proper",
"start_line": 107,
"end_line": 123,
"range_sha256": "02fd50c3e783d21bec28686017dc7fefa59ba99b7d3efbeca8c29fb051ca9593",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IW.1",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.TOPOLOGY.WARNING.SPACE",
"OA-FLOW.TOPOLOGY.WARNING.INTEGRABLE",
"OA-FLOW.TOPOLOGY.WARNING.JUMP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 123,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L73.md",
"line": 1,
"through_line": 7,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.TOPOLOGY.WARNING.EXERCISES",
"unit": "OA-FLOW-L73",
"source_file": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1",
"reader": "OA-FLOW-L73.html#oa-flow.topology-warning.exercises",
"start_line": 124,
"end_line": 135,
"range_sha256": "aff38282deee2c484f287e7135950eaebf5704fee9effeb22200fd167233d57b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IW.1",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.TOPOLOGY.WARNING.SPACE",
"OA-FLOW.TOPOLOGY.WARNING.INTEGRABLE",
"OA-FLOW.TOPOLOGY.WARNING.JUMP",
"OA-FLOW.TOPOLOGY.WARNING.PROPER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 135,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L73.md",
"line": 1,
"through_line": 7,
"description": "Read the complete initial setting together with the local section hypotheses."
}
]
},
{
"id": "OA-FLOW.FIELD.SETTING",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.setting",
"start_line": 1,
"end_line": 73,
"range_sha256": "b56c22d374691de03b40b58c0cfde49a7fbd659321d317edd66844c9a29f4737",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 73,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.HAAR",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.haar",
"start_line": 74,
"end_line": 133,
"range_sha256": "b8a04d05f09c5426a661b44a34f0bf964c65e00b68afb0e88bbc99a6fb41fb0c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 133,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.MATRICES",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.matrices",
"start_line": 134,
"end_line": 203,
"range_sha256": "79f1cd6849a40529cd3a5ba753bd6717549ee00bb9a1b8600b1fac8c732560a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 203,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.AVERAGE",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.average",
"start_line": 204,
"end_line": 282,
"range_sha256": "957cdb2c90432f956297ea504ef22c4b6096b1cc1032ee943f9e81c182fc9f2e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 282,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.COEFFICIENTS",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.coefficients",
"start_line": 283,
"end_line": 351,
"range_sha256": "fd344b55f795d28afcf0a5fb533c41d39f9ac562d74f850c8345f7dc6095327c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 351,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.CROSSED",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.crossed",
"start_line": 352,
"end_line": 516,
"range_sha256": "f929682e45e36b2ab168120ec0d37155c03848304655f780254deb674ad83117",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 516,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.ABSTRACT",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.abstract",
"start_line": 517,
"end_line": 528,
"range_sha256": "dc4d2c701aa7ab1b5f1a67cc52a7de3e328e3ca9c74949f103c9ab8ba158bad9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 528,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.QUOTIENT",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.quotient",
"start_line": 529,
"end_line": 609,
"range_sha256": "8af93ac002ad5c0b51a62ffee85eca2c7bd6c916886b46d37aa1c5578363d0d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 609,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.UNITARY",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.unitary",
"start_line": 610,
"end_line": 739,
"range_sha256": "05c89eaab12ad3a89d28df92d7f002de568f1df6e98f637f06ab80a033843bc7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 739,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.FIBERS",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.fibers",
"start_line": 740,
"end_line": 847,
"range_sha256": "ef23b3dac43c539f4fba3c6935d6d4a57903bb70e49e7e2de61cbf0ff5c0d131",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.UNITARY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 847,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.DENSITY",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.density",
"start_line": 848,
"end_line": 916,
"range_sha256": "5821b429e5cd89b1e8a1d7c2d2d132ab6b2b88f1f213ca9ff1903cefa219e54b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.UNITARY",
"OA-FLOW.FIELD.FIBERS",
"OA-FLOW.ORBITS.ERGODIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 916,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.ORBITWEIGHT",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.orbit-weight",
"start_line": 917,
"end_line": 962,
"range_sha256": "9b0b584b7bb4829e2a98e79cf6dca0061129a879dff5556b6749f7a086e41d6c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.DENSITY",
"OA-FLOW.ORBITS.ERGODIC",
"OA-FLOW.PROPER.AVERAGES",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.IS.2",
"OA-FLOW.IW.1",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.QF.7",
"OA-FLOW.CF.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 962,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "The complete lesson setting and retained common conull base apply."
}
]
},
{
"id": "OA-FLOW.FIELD.THEOREM",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.theorem",
"start_line": 963,
"end_line": 979,
"range_sha256": "2aa445b357f99108ea1b7ffef6ee45c7dc0ea66b57afcff7ccf79b9f32358523",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.UNITARY",
"OA-FLOW.FIELD.FIBERS",
"OA-FLOW.FIELD.DENSITY",
"OA-FLOW.ORBITS.ERGODIC",
"OA-FLOW.FIELD.ORBITWEIGHT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 979,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.EXAMPLE",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.example",
"start_line": 980,
"end_line": 1081,
"range_sha256": "8bf7f41f09c8079bf4f2b113028f63eaa5c35f530647a500b3acb0fc01bc75e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.UNITARY",
"OA-FLOW.FIELD.FIBERS",
"OA-FLOW.FIELD.DENSITY",
"OA-FLOW.FIELD.THEOREM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1081,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.FIELD.EXERCISES",
"unit": "OA-FLOW-L74",
"source_file": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01",
"reader": "OA-FLOW-L74.html#oa-flow.field.exercises",
"start_line": 1082,
"end_line": 1099,
"range_sha256": "4290b7a9530c189812f9e174aef332c3a46084481899eeca4ed216ecfb2a6783",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.MODEL.NORMAL",
"OA-FLOW.MODEL.TRANSPORT",
"OA-FLOW.PROPER.ABSTRACT",
"OA-FLOW.PROPER.BASE",
"OA-FLOW.PROPER.BUNDLE",
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.PROPER.CONJUGACY",
"OA-FLOW.ORBITS.STABILIZER-FIELD",
"OA-FLOW.IS.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.PROPER.SELECTION",
"OA-FLOW.CF.5",
"OA-FLOW.HR.2",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ORBITS.SELECTOR",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.IW.1",
"OA-FLOW.CF.8",
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.NCF.1",
"OA-FLOW.HR.5",
"OA-FLOW.ORBITS.REFERENCE",
"OA-FLOW.ORBITS.DENSITY",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.IS.4",
"OA-FLOW.IS.5",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.COMPACTSTAB.COORDINATES",
"OA-FLOW.COMPACTCOMM.EQUALITY",
"OA-FLOW.QF.7",
"OA-FLOW.FIELD.HAAR",
"OA-FLOW.FIELD.MATRICES",
"OA-FLOW.FIELD.AVERAGE",
"OA-FLOW.FIELD.COEFFICIENTS",
"OA-FLOW.FIELD.CROSSED",
"OA-FLOW.FIELD.ABSTRACT",
"OA-FLOW.FIELD.QUOTIENT",
"OA-FLOW.FIELD.UNITARY",
"OA-FLOW.FIELD.FIBERS",
"OA-FLOW.FIELD.DENSITY",
"OA-FLOW.FIELD.THEOREM",
"OA-FLOW.FIELD.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1099,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L74.md",
"line": 1,
"through_line": 73,
"description": "Full initial group, action and proper-model hypotheses, with each section's explicit conditions."
}
]
},
{
"id": "OA-FLOW.KERNEL.SETTING",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.setting",
"start_line": 1,
"end_line": 39,
"range_sha256": "f0fe05120fa18037c9091a53179c76da526eb7994549eae93fb7af10bbea206d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.INTEGRATION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.integration",
"start_line": 40,
"end_line": 69,
"range_sha256": "c7b9992f50de8d04d0450f906f068a96db250d51da1e26559c89417dd4927ed5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 69,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 40,
"through_line": 69,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.CODING",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.coding",
"start_line": 70,
"end_line": 139,
"range_sha256": "06f3dd2e9994b672ad045148053c360270ed9858eaf77efe89675b60ab0643ea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 139,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 70,
"through_line": 139,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.QUANTILE",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.quantile",
"start_line": 140,
"end_line": 238,
"range_sha256": "2cc737f66ed23dfacde3026573516d68821e147e8ee1c2da7f47ff47ed3c3028",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 238,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 140,
"through_line": 238,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.CONCENTRATION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.concentration",
"start_line": 239,
"end_line": 276,
"range_sha256": "26286b4ccf9908188552bfa53e05aee0ee141b31300a2f05eeb52484d0bba1c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 276,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 239,
"through_line": 276,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.UNIQUENESS",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.uniqueness",
"start_line": 277,
"end_line": 305,
"range_sha256": "ec77193f98fba31bd3bc1e13444affe1f43b350cd89aa34f4a263ee791af572a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 305,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 277,
"through_line": 305,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.COMPLETION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.completion",
"start_line": 306,
"end_line": 330,
"range_sha256": "e38a07e088d617e08c0795047a9daab3680a93dda9f6aa484fe0545d9e406918",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 330,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 1,
"through_line": 39,
"description": "Initial standard Borel probability setting."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 306,
"through_line": 330,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.PROBABILITY",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.probability",
"start_line": 331,
"end_line": 337,
"range_sha256": "402e2d12e66b1d3c4ef6c534faaefe906806ce81434e2ea097612985f667bf7d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 337,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 331,
"through_line": 337,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.SIGMAFINITE",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.sigmafinite",
"start_line": 338,
"end_line": 396,
"range_sha256": "11c6699a00e3e9647ba1eb008918600c5ad1ded84636bb5799efdb88cf2169c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 396,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 338,
"through_line": 396,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.NORMALIZATION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.normalization",
"start_line": 397,
"end_line": 462,
"range_sha256": "970a7ae6c6eb08a942ee2a48f2157d2327059edc12055f59b4b92f2d11e62603",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 462,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 338,
"through_line": 356,
"description": "Nonzero sigma-finite setting and chosen positive finite normalization; zero cases are stated separately."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 397,
"through_line": 462,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.PUSHFORWARD",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.pushforward",
"start_line": 463,
"end_line": 485,
"range_sha256": "19d266c4839404aeae4894f3b556a2719b24c9f5a2d74f29441b5f458c5abd3f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 485,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 338,
"through_line": 356,
"description": "Nonzero sigma-finite setting and chosen positive finite normalization; zero cases are stated separately."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 463,
"through_line": 485,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.EXAMPLE",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.example",
"start_line": 486,
"end_line": 588,
"range_sha256": "51a1a785b4382a70998fe8f655cf7120cbe0dffbdec7109537e779d24231a6b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.PUSHFORWARD"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 588,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 486,
"through_line": 588,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.COMPOSITION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.composition",
"start_line": 589,
"end_line": 623,
"range_sha256": "852d03c8732d78a51b19b29409b94b242fadc00b370108d05371de2d9223b7d4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.PUSHFORWARD",
"OA-FLOW.KERNEL.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 623,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 589,
"through_line": 623,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.EXPECTATION",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.expectation",
"start_line": 624,
"end_line": 666,
"range_sha256": "0e6dd31591197f4dcfa9ba4cc7473405d788ac3564a0fd64dfa919e0f6da93bf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.PUSHFORWARD",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.KERNEL.COMPOSITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 666,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 589,
"through_line": 593,
"description": "Probability measure and two Borel maps of standard Borel spaces."
},
{
"source": "src/OA-FLOW-L75.md",
"line": 624,
"through_line": 666,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.KERNEL.EXERCISES",
"unit": "OA-FLOW-L75",
"source_file": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f",
"reader": "OA-FLOW-L75.html#oa-flow.kernel.exercises",
"start_line": 667,
"end_line": 700,
"range_sha256": "eea936a18cf4777111d9d4df62f51de245fa35035910a13de691441e4c1fcdcc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ORBITS.CONDITIONAL",
"OA-FLOW.SC.2",
"OA-FLOW.SC.7",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.CF.8",
"OA-FLOW.ORBITS.INTEGRATION",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.QUANTILE",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.PUSHFORWARD",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.KERNEL.COMPOSITION",
"OA-FLOW.KERNEL.EXPECTATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 700,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L75.md",
"line": 667,
"through_line": 700,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.SETTING",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.setting",
"start_line": 1,
"end_line": 39,
"range_sha256": "36b39c2e1986bbc69bd9b944c3994736921ae2bbcbed1268d68d507e98dc7d26",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.SETUP",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.setup",
"start_line": 40,
"end_line": 173,
"range_sha256": "e4e3f7e1a8876cf0cdf80604566c87c456b64213548e320db3233ac47290ca97",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 173,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 40,
"through_line": 173,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.ALMOST",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.almost",
"start_line": 174,
"end_line": 307,
"range_sha256": "b3388514f60e99ee76ef791aeb31f7a2d62b06fa2f84f0bb48b851806706da27",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 307,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 174,
"through_line": 307,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.NONSEPARABLE",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.nonseparable",
"start_line": 308,
"end_line": 329,
"range_sha256": "d8424df1d9e1caa71370381d5134a8f15ebd9a1e3a55c608392ee832e0ba990f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 329,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 308,
"through_line": 329,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.JOINT",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.joint",
"start_line": 330,
"end_line": 405,
"range_sha256": "d8dc8de66693db995bda7ddf6d833648786f6194d01219ec95716de0ee622a49",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 405,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 330,
"through_line": 405,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.RECOVERY",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.recovery",
"start_line": 406,
"end_line": 518,
"range_sha256": "20aa188d59edc25086749367ac3c490fdb8dba97750eb78a927b3bffbff85319",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 518,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 406,
"through_line": 518,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.STRICT",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.strict",
"start_line": 519,
"end_line": 654,
"range_sha256": "bb33f6cf7cd9db3d9a30de8067a72f87a19c789ece0af957ba5b6a8495c8973d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 654,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 519,
"through_line": 654,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.COVARIANCE",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.covariance",
"start_line": 655,
"end_line": 739,
"range_sha256": "505f77a991fe09d2d5736e283f4d6ffc14a608cadfc9beb405738d75fe82daa6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 739,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 655,
"through_line": 739,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.FIXED",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.fixed",
"start_line": 740,
"end_line": 785,
"range_sha256": "f2b69b90625766af32915500d08224d64dada2d769116682d4cd84461775d37e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT",
"OA-FLOW.CENTERG.COVARIANCE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 785,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 740,
"through_line": 785,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.SCALAR",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.scalar",
"start_line": 786,
"end_line": 820,
"range_sha256": "3a680d7057220e7cbac1df629aaffaa75aaf0bde4453ccd0a9cb6c936c825899",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.FIXED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 820,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 786,
"through_line": 820,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.EXAMPLE",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.example",
"start_line": 821,
"end_line": 885,
"range_sha256": "badeb64fef2831d16b1cd1b0c8f5348b63d0b2dfe5b13562c13e9fa7a413f6ea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.SCALAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 885,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 821,
"through_line": 885,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.CONCLUSION",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.conclusion",
"start_line": 886,
"end_line": 902,
"range_sha256": "41912221b311a523e2c76242edc80670df24f77b819d789d91d2dbedcc567cea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.CENTERG.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 902,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 886,
"through_line": 902,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CENTERG.EXERCISES",
"unit": "OA-FLOW-L42",
"source_file": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e",
"reader": "OA-FLOW-L42.html#oa-flow.centerg.exercises",
"start_line": 903,
"end_line": 914,
"range_sha256": "98db152e9a751ca07f37732a940df2024eff82f44f26376991d4c9de3235cddb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.CP.6",
"OA-FLOW.NC.5",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.GRP.RECOVERY",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.INTEGRATION",
"OA-FLOW.CCOV.GROUPRECOVERY",
"OA-FLOW.CP.4",
"OA-FLOW.MC.1",
"OA-FLOW.CENTERG.SETUP",
"OA-FLOW.CENTERG.ALMOST",
"OA-FLOW.CENTERG.NONSEPARABLE",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.STRICT",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.CENTERG.EXAMPLE",
"OA-FLOW.CENTERG.CONCLUSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 914,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L42.md",
"line": 1,
"through_line": 39,
"description": "Original nonzero separable-predual system and separable locally compact Hausdorff group. Individual lemmas explicitly state their own scope; the nonseparable example is a counterexample, not an input to the main theorem."
},
{
"source": "src/OA-FLOW-L42.md",
"line": 903,
"through_line": 914,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.SETTING",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.setting",
"start_line": 1,
"end_line": 48,
"range_sha256": "b9f2ab3bbc6c66adb4974025aff431a58bc3f32094edbd71b217fea07d48d6f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.COMPACT",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.compact",
"start_line": 49,
"end_line": 145,
"range_sha256": "9b0a318a694e44dd52901367c60c1888abe23c1091eb3a946fe657ab7c4e08e4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 145,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 49,
"through_line": 145,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.SPATIAL",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.spatial",
"start_line": 146,
"end_line": 275,
"range_sha256": "46f7aec0dfa5a2159eb4bf807298bd72de8c42f557e8538b85fc6e33b37cdf4b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 275,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 146,
"through_line": 275,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.COORDINATES",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.coordinates",
"start_line": 276,
"end_line": 297,
"range_sha256": "b192a6b341aaa5a7b0365e6627671ac9ec5f0298c2501f32d4ac0fb7cb0418a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 297,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 276,
"through_line": 297,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.HILBERT",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.hilbert",
"start_line": 298,
"end_line": 472,
"range_sha256": "8b31b242f385f5af872d0722558cc4105ff7579a49bc522d9edbb919f494b3aa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 472,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 298,
"through_line": 472,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.MULTIPLICATION",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.multiplication",
"start_line": 473,
"end_line": 609,
"range_sha256": "0fc7492758f7b4471e0c516bba29d0c814e9a9db5cc08cfb435a7d6a9572b2f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 609,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 473,
"through_line": 609,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.STATE.ROUTE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.state_route",
"start_line": 610,
"end_line": 684,
"range_sha256": "e08ba39d0be9cb946ac901769ba3fbd531a4f0c8855269e9a5d06eca624f8828",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 684,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 610,
"through_line": 684,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.COVARIANCE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.covariance",
"start_line": 685,
"end_line": 769,
"range_sha256": "b46e6da5355d7edb7242d0f25421caa1fe1802382fda0be27e509b4c55a6dfc1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 769,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 685,
"through_line": 769,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.QUASI",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.quasi",
"start_line": 770,
"end_line": 918,
"range_sha256": "a325251ccf7c98f31633dabe2c5b9d5b2df8913ad601ead8cffa00b6f9ee9d1a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 918,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 770,
"through_line": 918,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.ERGODIC",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.ergodic",
"start_line": 919,
"end_line": 956,
"range_sha256": "618a10347beceacb1062751023b558a350eb8a14125dbc217d1492eeef07978b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 956,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 919,
"through_line": 956,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.TRANSPORT",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.transport",
"start_line": 957,
"end_line": 995,
"range_sha256": "3029cc22296842a794066f15e48b79d835f6de1a6c284dbadf940194f2d353b1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 995,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 957,
"through_line": 995,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.CONCLUSION",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.conclusion",
"start_line": 996,
"end_line": 1004,
"range_sha256": "4a0f515eb79a4f37051d63ff829c7ffdf343a07a035b9999df3fc6ae077da7ab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1004,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 996,
"through_line": 1004,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.BASE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.base",
"start_line": 1005,
"end_line": 1045,
"range_sha256": "425c64d04860d6746b817ef5a595b919a60efb89f12929012b27bf39bac8290d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT",
"OA-FLOW.ERGDEC.CONCLUSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1045,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 1005,
"through_line": 1045,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.SIGMAFINITE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.sigmafinite",
"start_line": 1046,
"end_line": 1151,
"range_sha256": "771a4d625640978d598598c17ba690a5e1c5288896fba4a896a842fe69c15520",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT",
"OA-FLOW.ERGDEC.CONCLUSION",
"OA-FLOW.ERGDEC.BASE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1151,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 1046,
"through_line": 1151,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.EXAMPLE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.example",
"start_line": 1152,
"end_line": 1194,
"range_sha256": "44543d4e422ea791a24655ee8813078535295ae19af6205a7f961680f593b225",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT",
"OA-FLOW.ERGDEC.CONCLUSION",
"OA-FLOW.ERGDEC.BASE",
"OA-FLOW.ERGDEC.SIGMAFINITE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1194,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 1152,
"through_line": 1194,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.INFINITE.EXAMPLE",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.infinite_example",
"start_line": 1195,
"end_line": 1227,
"range_sha256": "eac72d3fef1f02196647dff8bcd6e76545bf530f4c3c2e75b7d8f0cefd4fd399",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT",
"OA-FLOW.ERGDEC.CONCLUSION",
"OA-FLOW.ERGDEC.BASE",
"OA-FLOW.ERGDEC.SIGMAFINITE",
"OA-FLOW.ERGDEC.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1227,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 1195,
"through_line": 1227,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.ERGDEC.EXERCISES",
"unit": "OA-FLOW-ED",
"source_file": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282",
"reader": "OA-FLOW-ED.html#oa-flow.ergdec.exercises",
"start_line": 1228,
"end_line": 1243,
"range_sha256": "0577a51e4ad75c224bc382ce6d6153c80ae66e913dfdb4d5f325e83dfc91e665",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.PROBABILITY",
"OA-FLOW.CENTERG.CONCLUSION",
"OA-FLOW.KERNEL.UNIQUENESS",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.ST.2",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.COMPLETION",
"OA-FLOW.CF.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.KERNEL.CONCENTRATION",
"OA-FLOW.IS.2",
"OA-FLOW.CP.6",
"OA-FLOW.CENTERG.JOINT",
"OA-FLOW.CENTERG.FIXED",
"OA-FLOW.CENTERG.RECOVERY",
"OA-FLOW.CENTERG.COVARIANCE",
"OA-FLOW.CENTERG.SCALAR",
"OA-FLOW.KERNEL.SIGMAFINITE",
"OA-FLOW.KERNEL.NORMALIZATION",
"OA-FLOW.KERNEL.EXAMPLE",
"OA-FLOW.ERGDEC.COMPACT",
"OA-FLOW.ERGDEC.SPATIAL",
"OA-FLOW.ERGDEC.COORDINATES",
"OA-FLOW.ERGDEC.HILBERT",
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.ERGDEC.STATE.ROUTE",
"OA-FLOW.ERGDEC.COVARIANCE",
"OA-FLOW.ERGDEC.QUASI",
"OA-FLOW.ERGDEC.ERGODIC",
"OA-FLOW.ERGDEC.TRANSPORT",
"OA-FLOW.ERGDEC.CONCLUSION",
"OA-FLOW.ERGDEC.BASE",
"OA-FLOW.ERGDEC.SIGMAFINITE",
"OA-FLOW.ERGDEC.EXAMPLE",
"OA-FLOW.ERGDEC.INFINITE.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1243,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ED.md",
"line": 1,
"through_line": 48,
"description": "Original strict jointly Borel nonsingular lcsc action on a standard probability space. Individual field and compact-model lemmas state their own hypotheses; the sigma-finite section explicitly replaces probability by a nonzero sigma-finite measure and supplies its normalization."
},
{
"source": "src/OA-FLOW-ED.md",
"line": 1228,
"through_line": 1243,
"description": "The explicit setting and proof of this section; incoming results retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.CSTD.SETTING",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.setting",
"start_line": 1,
"end_line": 72,
"range_sha256": "343a6479efa4322e18fefc80952d0eeedc1bce9dd67761115245bc4346be282f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
}
],
"external_proof_links": []
},
{
"id": "OA-FLOW.CSTD.SETUP",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.setup",
"start_line": 73,
"end_line": 158,
"range_sha256": "006a03fc05336aab7fa86769030509cf39d721cefcc10a2d5e3f7fdb213cce2f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.NR.1",
"OA-FLOW.CR.8",
"OA-FLOW.CR.6",
"OA-FLOW.CR.1",
"OA-FLOW.CR.4",
"OA-FLOW.ST.1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.NR.2",
"OA-FLOW.CF.6",
"OA-FLOW.HR.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 158,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 73,
"through_line": 158,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-ERGODIC/reader/measurable-actions-and-compact-models.html#4-the-compact-space-and-its-nonsingular-measure",
"../../OA-MOD/OA-MOD-DC.html#a-finite-radon-nikodym-density-from-hilbert-representation"
]
},
{
"id": "OA-FLOW.CSTD.STATIC",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.static",
"start_line": 159,
"end_line": 295,
"range_sha256": "c44e86458e4fa3db40ed53e90a8c373e336a9d81177c0ce861a4ff5684281f6d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BD.4",
"OA-FLOW.CP.6",
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 295,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 159,
"through_line": 295,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_current_proof_inputs": [
{
"id": "RD4",
"source": "docs/courses/OA-MOD/src/right-hilbert-algebra-density.md",
"reader": "docs/courses/OA-MOD/OA-MOD-RD.html",
"anchor": "oa-mod-rd-04",
"line": 260,
"through_line": 322,
"source_sha256": "8e7794931584c555b5495153ad8293f57cfea0ce62761cf49e9532642ce35e3e",
"reader_sha256": "38af4cd29d749343a64b753288798ebdcb5bbced7341f07938e00dc4470e35fe",
"physical_range_sha256": "3b72b58d19c7db64eef51273ecb769b87364a3dd83a3585b5e36a5f1c465e735",
"physical_range_bytes": 2298,
"normalized_LF_range_sha256": "3b72b58d19c7db64eef51273ecb769b87364a3dd83a3585b5e36a5f1c465e735",
"normalized_LF_range_bytes": 2298
},
{
"id": "WH4",
"source": "docs/courses/OA-MOD/src/weight-hilbert-algebra.md",
"reader": "docs/courses/OA-MOD/OA-MOD-WH.html",
"anchor": "oa-mod-wh-04",
"line": 132,
"through_line": 163,
"source_sha256": "aa03fcf989777896d5201fd76a08f2d313308c13e8cabed90ffa8137024de462",
"reader_sha256": "733fb5b8e06cea2b35068729e8535760b8a56c6c8af5d58cb1d8dd22f5e7e52a",
"physical_range_sha256": "a1e8f053d4d03064b498dd477d29669dfb7d64392047ab18297a24cb05a9eabe",
"physical_range_bytes": 1854,
"normalized_LF_range_sha256": "a1e8f053d4d03064b498dd477d29669dfb7d64392047ab18297a24cb05a9eabe",
"normalized_LF_range_bytes": 1854
},
{
"id": "HAP8",
"source": "docs/courses/OA-MOD/src/hilbert-algebra-approximation.md",
"reader": "docs/courses/OA-MOD/OA-MOD-HAP.html",
"anchor": "central-elements-preserve-both-involution-domains",
"line": 446,
"through_line": 506,
"source_sha256": "fda0941739ef12fced73cd90f7ae8698baec1879735f62a4d544895810a74154",
"reader_sha256": "c9a08d7ee7a9c99b86396cccbc88ee23e9e0222ece4e61c17310dcd704de4712",
"physical_range_sha256": "e636e4efd1fa0ee2dc5eade3d68b8c9c8b19a32081b54c4e6ffc65d7af8e845a",
"physical_range_bytes": 2726,
"normalized_LF_range_sha256": "e636e4efd1fa0ee2dc5eade3d68b8c9c8b19a32081b54c4e6ffc65d7af8e845a",
"normalized_LF_range_bytes": 2726
},
{
"id": "DI8",
"source": "docs/courses/OA-MOD/src/direct-integral-graphs-multipliers.md",
"reader": "docs/courses/OA-MOD/OA-MOD-DI.html",
"anchor": "oa-mod-di-07",
"line": 256,
"through_line": 327,
"source_sha256": "97966de372bf4843d071bef35f3546cef4d8ef1b234fc4d8046d7ddbdee2ab62",
"reader_sha256": "885a9037089279aadb82efd98dc14007e85c2452917774158c7fdfa48b260ce8",
"physical_range_sha256": "d25f0cd311476b3f45bfdab081d4ea4ef8b52cf6963cb4c797fe7cbc4e577a16",
"physical_range_bytes": 5488,
"normalized_LF_range_sha256": "d25f0cd311476b3f45bfdab081d4ea4ef8b52cf6963cb4c797fe7cbc4e577a16",
"normalized_LF_range_bytes": 5488
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-DC.html#existence-over-any-specified-central-abelian-subalgebra",
"../../OA-MOD/OA-MOD-DC.html#realizing-an-abelian-algebra-as-the-diagonal-algebra",
"../../OA-MOD/OA-MOD-MW.html#central-disintegration-of-a-separable-weighted-algebra",
"../../OA-MOD/OA-MOD-DI.html#reverse-assembly-over-a-specified-central-diagonal",
"../../OA-MOD/OA-MOD-GFR.html#the-reverse-comparison-and-the-realization-obstruction",
"../../OA-MOD/OA-MOD-MW.html#every-positive-value-and-the-exact-finite-left-ideal",
"../../OA-MOD/OA-MOD-DC.html#change-of-base-and-the-square-root-density-in-general-uniqueness",
"../../OA-MOD/OA-MOD-MW.html#constructing-the-integral-weight-and-its-full-operators",
"../../OA-MOD/OA-MOD-RD.html#oa-mod-rd-04",
"../../OA-MOD/OA-MOD-WH.html#oa-mod-wh-04",
"../../OA-MOD/OA-MOD-HAP.html#central-elements-preserve-both-involution-domains",
"../../OA-MOD/OA-MOD-DI.html#oa-mod-di-07"
]
},
{
"id": "OA-FLOW.CSTD.CONE",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.cone",
"start_line": 296,
"end_line": 351,
"range_sha256": "3396255a9d6b60034e8e1119a6ca578b58cbc364ad576d063aeed8f365778696",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.3",
"OA-FLOW.NC.5",
"OA-FLOW.CR.4",
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 351,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 296,
"through_line": 351,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-GFR.html#a-common-countable-strong-dense-contraction-family",
"../../OA-MOD/OA-MOD-DF.html#the-hilbert-direct-integral-localization-and-subsequences"
]
},
{
"id": "OA-FLOW.CSTD.NORMALIZER",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.normalizer",
"start_line": 352,
"end_line": 398,
"range_sha256": "d5545d3d588d427851ae9e4162864f20122c250c294265c67bf821737ec1ec16",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 398,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 352,
"through_line": 398,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-DC.html#uniqueness-over-the-same-diagonal-structure",
"../../OA-MOD/OA-MOD-DF.html#the-diagonal-commutant-is-exactly-the-decomposable-algebra"
]
},
{
"id": "OA-FLOW.CSTD.TRANSPORT",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.transport",
"start_line": 399,
"end_line": 444,
"range_sha256": "8c39500d2688d8667e731209687d9d8b15f5df42d9cb76b52b08897d3941c060",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.1",
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 444,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 399,
"through_line": 444,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": []
},
{
"id": "OA-FLOW.CSTD.VECTORS",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.vectors",
"start_line": 445,
"end_line": 568,
"range_sha256": "84fdfc3a1726db2afe18d60d496292dfacad252fd40380c7b76ad0cece9c467e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.8",
"OA-FLOW.NR.1",
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 568,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 445,
"through_line": 568,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-DC.html#a-finite-radon-nikodym-density-from-hilbert-representation"
]
},
{
"id": "OA-FLOW.CSTD.STRICT",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.strict",
"start_line": 569,
"end_line": 771,
"range_sha256": "edff5a787eba62064760c1320f144685420d29c388f0c89e943dcf4d72f7b645",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.MC.1",
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CSTD.VECTORS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 771,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 569,
"through_line": 771,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-GFR.html#a-common-countable-strong-dense-contraction-family",
"../../OA-MOD/OA-MOD-DF.html#compact-orthonormal-frames-and-measurable-dimension",
"../../OA-ERGODIC/reader/localizing-factor-actions-and-uniform-cocycles.html#1-repairing-an-equivariant-section",
"../../OA-ERGODIC/reader/localizing-factor-actions-and-uniform-cocycles.html#3-making-a-measurable-cocycle-strict",
"../../OA-ERGODIC/reader/localizing-factor-actions-and-uniform-cocycles.html#4-the-full-separable-locally-compact-scope",
"../../OA-ERGODIC/reader/measurable-actions-and-compact-models.html#1-a-jointly-measurable-derivative",
"../../OA-ERGODIC/reader/measurable-actions-and-compact-models.html#2-why-measurable-unitary-representations-are-continuous",
"../../OA-MOD/OA-MOD-DF.html#the-diagonal-commutant-is-exactly-the-decomposable-algebra",
"../../OA-ERGODIC/reader/localizing-factor-actions-and-uniform-cocycles.html#5-localizing-the-automorphism-action",
"../../OA-ERGODIC/reader/localizing-factor-actions-and-uniform-cocycles.html#2-polish-paths-and-closed-constant-orbits",
"../../OA-ERGODIC/reader/strict-variable-fields-and-ancillary-conjugacy.html#2-coding-the-fibre-data"
]
},
{
"id": "OA-FLOW.CSTD.BASECHANGE",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.basechange",
"start_line": 772,
"end_line": 819,
"range_sha256": "b588f360abdc9e158ac021addecf1552473118693aa4897d1f48b5d6afb43e56",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CSTD.VECTORS",
"OA-FLOW.CSTD.STRICT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 819,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 772,
"through_line": 819,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-DC.html#a-finite-radon-nikodym-density-from-hilbert-representation"
]
},
{
"id": "OA-FLOW.CSTD.CONCLUSION",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.conclusion",
"start_line": 820,
"end_line": 835,
"range_sha256": "b21903803e8cf84e34d731a61944a94e5e3bae636b58e6d61e8da1806d5951f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CSTD.VECTORS",
"OA-FLOW.CSTD.STRICT",
"OA-FLOW.CSTD.BASECHANGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 835,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 820,
"through_line": 835,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-DC.html#centers-and-the-factor-decomposition"
]
},
{
"id": "OA-FLOW.CSTD.EXAMPLE",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.example",
"start_line": 836,
"end_line": 954,
"range_sha256": "30b76f6db8386da6b891961631e9231d54e35a1a455e2c0cb4f4af83059ea6c3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CSTD.VECTORS",
"OA-FLOW.CSTD.STRICT",
"OA-FLOW.CSTD.BASECHANGE",
"OA-FLOW.CSTD.CONCLUSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 954,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 836,
"through_line": 954,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": []
},
{
"id": "OA-FLOW.CSTD.EXERCISES",
"unit": "OA-FLOW-L41",
"source_file": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde",
"reader": "OA-FLOW-L41.html#oa-flow.cstd.exercises",
"start_line": 955,
"end_line": 1022,
"range_sha256": "026811a31e44a0b29cf87b9994a511db250901fe79c4a5933cd75bc74a4bee03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CSTD.VECTORS",
"OA-FLOW.CSTD.STRICT",
"OA-FLOW.CSTD.BASECHANGE",
"OA-FLOW.CSTD.CONCLUSION",
"OA-FLOW.CSTD.EXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1022,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L41.md",
"line": 1,
"through_line": 72,
"description": "Nonzero von Neumann algebra with separable predual; continuous normal action of a separable locally compact Hausdorff group; specified invariant unital central algebra; no invariant-state or factoriality assumption."
},
{
"source": "src/OA-FLOW-L41.md",
"line": 955,
"through_line": 1022,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": []
},
{
"id": "OA-FLOW.MSF.INPUTS",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.inputs",
"start_line": 7,
"end_line": 17,
"range_sha256": "ae03dd5943b0ed44da10ac9fbf7a6d124bc5a1fb513b0a9a6644a3df4ab4cbc4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 7,
"through_line": 17,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.STATIC",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.static",
"start_line": 18,
"end_line": 61,
"range_sha256": "37a021b0839fd53c755e68dc0a50fcea88ac4dabfee640cd3fedfab04226ff1d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.CONE",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 58,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 18,
"through_line": 61,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.ASSEMBLY",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.assembly",
"start_line": 62,
"end_line": 84,
"range_sha256": "e65c15d2060dcb754f0ae9589232425aab5fd865b41a0c821a55057cbdaa7d46",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 83,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 62,
"through_line": 84,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.CONVENTIONS",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.conventions",
"start_line": 85,
"end_line": 105,
"range_sha256": "3fc4d4fcdc9ca992deb482aca3eabf16fe44dd12b127bd6180064f89e0f454e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 85,
"through_line": 105,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.TRANSPORT",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.transport",
"start_line": 106,
"end_line": 232,
"range_sha256": "a4917a949c9b2f2c4eb42033c91edcc7c495e523e51df232da6b725b94bffd7c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.1",
"OA-FLOW.CR.1",
"OA-FLOW.CR.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 228,
"definition_context": [
"OA-FLOW.MSF.CONVENTIONS"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 106,
"through_line": 232,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
},
{
"source": "src/OA-FLOW-MSF.md",
"line": 85,
"through_line": 105,
"description": "Earlier same-lesson definitions and specified measurable data; not a proved premise."
}
]
},
{
"id": "OA-FLOW.MSF.COMPOSITION",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.composition",
"start_line": 233,
"end_line": 236,
"range_sha256": "12a72c5cacd22951549846ef5a08d105419f32f3445214cbd209587ab2cb42bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MSF.TRANSPORT",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 235,
"definition_context": [
"OA-FLOW.MSF.CONVENTIONS"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 233,
"through_line": 236,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
},
{
"source": "src/OA-FLOW-MSF.md",
"line": 85,
"through_line": 105,
"description": "Earlier same-lesson definitions and specified measurable data; not a proved premise."
}
]
},
{
"id": "OA-FLOW.MSF.PREDUAL",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.predual",
"start_line": 237,
"end_line": 252,
"range_sha256": "518367ac8c0e3cca765f703de0841221bc567f9b69840d2cc7f83f0263db08af",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CR.1",
"OA-FLOW.CR.6",
"OA-FLOW.CR.8",
"OA-FLOW.MC.1",
"OA-FLOW.MSF.TRANSPORT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 249,
"definition_context": [
"OA-FLOW.MSF.CONVENTIONS"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 237,
"through_line": 252,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
},
{
"source": "src/OA-FLOW-MSF.md",
"line": 85,
"through_line": 105,
"description": "Earlier same-lesson definitions and specified measurable data; not a proved premise."
}
]
},
{
"id": "OA-FLOW.MSF.RECONSTRUCTION",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.reconstruction",
"start_line": 253,
"end_line": 397,
"range_sha256": "09c9ca0fb53730698bef2cbb436015642e36c04cade9a85c2b1ae8f0b5dc7b46",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MSF.ASSEMBLY",
"OA-FLOW.MSF.TRANSPORT",
"OA-FLOW.MC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 396,
"definition_context": [
"OA-FLOW.MSF.CONVENTIONS"
],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 253,
"through_line": 397,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
},
{
"source": "src/OA-FLOW-MSF.md",
"line": 85,
"through_line": 105,
"description": "Earlier same-lesson definitions and specified measurable data; not a proved premise."
}
]
},
{
"id": "OA-FLOW.MSF.SCOPE",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.boundary",
"start_line": 398,
"end_line": 404,
"range_sha256": "1a64b60441f229d23f19cbce8d69fd0e37f9158eb9378135ea2bc188e2acb75a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 398,
"through_line": 404,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.COUNTEREXAMPLE",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.counterexample",
"start_line": 405,
"end_line": 415,
"range_sha256": "77118817bca46ba4932e963835d4b79577f8766be63cbdd3e712aa94beb0109d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MC.1",
"OA-FLOW.NC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 414,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 405,
"through_line": 415,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.MSF.LIMITS",
"unit": "OA-FLOW-MSF",
"source_file": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a",
"reader": "OA-FLOW-MSF.html#oa-flow.msf.limits",
"start_line": 416,
"end_line": 418,
"range_sha256": "2235b45e1298be68e3fa975b523be91800038c5901f52197cfe462fb092affbd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"definition_context": [],
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MSF.md",
"line": 416,
"through_line": 418,
"description": "The complete stated hypotheses, definitions and proof scope of this source item."
}
]
},
{
"id": "OA-FLOW.ISTATE.SETTING",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.setting",
"start_line": 1,
"end_line": 65,
"range_sha256": "4d94b15a67c713d8e9602d79df8fe8a4f5c71ec874249bc55d55563b441c2286",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
}
]
},
{
"id": "OA-FLOW.ISTATE.APPROXIMATION",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.approximation",
"start_line": 66,
"end_line": 107,
"range_sha256": "a65de6d3c5fa4722dd27202436bd3d3f4a05e3d5c0f6e50cd44f5c83f77f81e9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 107,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 66,
"through_line": 107,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.SETUP",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.setup",
"start_line": 108,
"end_line": 166,
"range_sha256": "ceb87094a5c9896c1f7ac6767e7cce395c3c5d6267991f53f8000812f89410b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.AT.3",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 166,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 108,
"through_line": 166,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.EXPECTATION",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.expectation",
"start_line": 167,
"end_line": 252,
"range_sha256": "431ec5637cc815c906699958422f0cef8375d7ba33fd15ece63dae5603fd14be",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.4",
"OA-FLOW.NF.6",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 252,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 167,
"through_line": 252,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.CLOSURE",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.closure",
"start_line": 253,
"end_line": 302,
"range_sha256": "a19d1bf9a1665b61c9ec2f1d1149683f1a38e4454153e92508e0383413af694f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.CP.6",
"OA-FLOW.ST.1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.NR.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 302,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 253,
"through_line": 302,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.BASE",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.base",
"start_line": 303,
"end_line": 370,
"range_sha256": "8727d9e9f4ab6c571fa4abce5d2ef42ef13bb5390862b4b0e4c45829f89fa63f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.6",
"OA-FLOW.HR.2",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.ST.2",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 370,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 303,
"through_line": 370,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.STATES",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.states",
"start_line": 371,
"end_line": 414,
"range_sha256": "32a174d82e21e0b7c2e126c1de11f3a148921e6310681b68df07395f88f74bb3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 414,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 371,
"through_line": 414,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.GNS",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.gns",
"start_line": 415,
"end_line": 470,
"range_sha256": "644e791711fde357610a6bdb7110ee7e355dc143ad61cf13095014197b615554",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 470,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 415,
"through_line": 470,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.NONUNITAL",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.nonunital",
"start_line": 471,
"end_line": 545,
"range_sha256": "2d7097dbfb788ce6e6e7187645e21fc9d219a67a89c30a43c33de1985b2db8b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.4",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES",
"OA-FLOW.ISTATE.GNS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 545,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 471,
"through_line": 545,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.ONTO",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.onto",
"start_line": 546,
"end_line": 615,
"range_sha256": "4dd0274efb44a8f99a48a3cc5d43133c54519486d88c8b6f02e39b94a75ca993",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.2",
"OA-FLOW.BD.4",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES",
"OA-FLOW.ISTATE.GNS",
"OA-FLOW.ISTATE.NONUNITAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 615,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 546,
"through_line": 615,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.COCYCLE",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.cocycle",
"start_line": 616,
"end_line": 694,
"range_sha256": "e18c84384d34371af30a4783982c2888fdd0e81202f557af779d6345a7222136",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES",
"OA-FLOW.ISTATE.GNS",
"OA-FLOW.ISTATE.NONUNITAL",
"OA-FLOW.ISTATE.ONTO"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 694,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 616,
"through_line": 694,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.EXAMPLES",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.examples",
"start_line": 695,
"end_line": 869,
"range_sha256": "9d6f83d55f7d2d0d9a398e4fc3cd9711534ae6d61abfb89d530fe59f3040b17f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ERGDEC.MULTIPLICATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES",
"OA-FLOW.ISTATE.GNS",
"OA-FLOW.ISTATE.NONUNITAL",
"OA-FLOW.ISTATE.ONTO",
"OA-FLOW.ISTATE.COCYCLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 869,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 695,
"through_line": 869,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.ISTATE.EXERCISES",
"unit": "OA-FLOW-L40",
"source_file": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd",
"reader": "OA-FLOW-L40.html#oa-flow.istate.exercises",
"start_line": 870,
"end_line": 918,
"range_sha256": "2a9e8b540ab8975e25968c55efa4b28aed1b0fbe77c23c08eda153ed12e7ff1b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.9",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ISTATE.SETUP",
"OA-FLOW.ISTATE.EXPECTATION",
"OA-FLOW.ISTATE.CLOSURE",
"OA-FLOW.ISTATE.BASE",
"OA-FLOW.ISTATE.STATES",
"OA-FLOW.ISTATE.GNS",
"OA-FLOW.ISTATE.NONUNITAL",
"OA-FLOW.ISTATE.ONTO",
"OA-FLOW.ISTATE.COCYCLE",
"OA-FLOW.ISTATE.EXAMPLES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 918,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L40.md",
"line": 1,
"through_line": 65,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, invariant state with cyclic nondegenerate GNS representation, specified invariant unital abelian commutant algebra; no ambient centrality assumption."
},
{
"source": "src/OA-FLOW-L40.md",
"line": 870,
"through_line": 918,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.SETTING",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.setting",
"start_line": 1,
"end_line": 80,
"range_sha256": "c4b96c9d3050b216b4649db21dba642198db527a498047b49864b440ddeb4ded",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
}
]
},
{
"id": "OA-FLOW.COVREP.BASE",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.base",
"start_line": 81,
"end_line": 118,
"range_sha256": "dba591ab5468501341eaeb83e9ad9bc48b7ac2afd1db94dd3adaea7129beb574",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.CSTD.SETUP",
"OA-FLOW.NR.2",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 118,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 81,
"through_line": 118,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.HILBERT",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.hilbert",
"start_line": 119,
"end_line": 188,
"range_sha256": "c2a4cd49c4e4ad9c0ba3db10fc643b34286a405f85c9723ba4630f7e3955bab4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.2",
"OA-FLOW.ISTATE.ONTO",
"OA-FLOW.IS.2",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 188,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 119,
"through_line": 188,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.REPRESENTATIONS",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.representations",
"start_line": 189,
"end_line": 265,
"range_sha256": "d9a17c70568c8d90ac784c279a7a72db8e17aaa0d9bbac882e0a006226715a30",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.BD.4",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 265,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 189,
"through_line": 265,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.NORMALIZER",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.normalizer",
"start_line": 266,
"end_line": 346,
"range_sha256": "21991940cff9965e20258684eb30b6555c9faa54cd44b8ef91948f4a73efd23c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 346,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 266,
"through_line": 346,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.COVARIANCE",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.covariance",
"start_line": 347,
"end_line": 370,
"range_sha256": "a4dfb900f38422b8fbc4f46cb94820cf839d734a6dddf2405304c47f2f9aa519",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 370,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 347,
"through_line": 370,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.STATES",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.states",
"start_line": 371,
"end_line": 462,
"range_sha256": "7a47925a61e9faa6da8791b1f0c25eb6e304e927875354afb151976ea004bc3a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.ST.1",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.5.1",
"OA-FLOW.ISTATE.GNS",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 462,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 371,
"through_line": 462,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.CANONICAL",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.canonical",
"start_line": 463,
"end_line": 572,
"range_sha256": "cf2785417dd24b53f20215368fc9a8b790023a9a9186d9eba56810cbbfede422",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.5",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 572,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 463,
"through_line": 572,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.AVERAGING",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.averaging",
"start_line": 573,
"end_line": 738,
"range_sha256": "75a0ff8de3fd360b4befc6715a1a944673cd8e7841a0d50b6b1d75ac853b9d01",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.4",
"OA-FLOW.HR.6",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.HR.3",
"OA-FLOW.SC.7",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES",
"OA-FLOW.COVREP.CANONICAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 738,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 573,
"through_line": 738,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.ORTHOGONAL",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.orthogonal",
"start_line": 739,
"end_line": 896,
"range_sha256": "cd4cce5873f8aa631b8f55e5a0297332bcf8f8a4a99363cf92f2838f3ebbb411",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.SC.5",
"OA-FLOW.CCOV.EXTREMALITY",
"OA-FLOW.GNS.4.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES",
"OA-FLOW.COVREP.CANONICAL",
"OA-FLOW.COVREP.AVERAGING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 896,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 739,
"through_line": 896,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.BARYCENTRE",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.barycentre",
"start_line": 897,
"end_line": 948,
"range_sha256": "1f74b224f5b64550971483daf6b59b3b68f31d6442f098a780a920515bf74166",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES",
"OA-FLOW.COVREP.CANONICAL",
"OA-FLOW.COVREP.AVERAGING",
"OA-FLOW.COVREP.ORTHOGONAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 948,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 897,
"through_line": 948,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.EXAMPLES",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.examples",
"start_line": 949,
"end_line": 1093,
"range_sha256": "1b047659a3d83b27e0cfd27fb6342c058e753ed9450be2a46771e2a872181e86",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.HR.5",
"OA-FLOW.HR.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.CCOV.TRANSFORMATIONS",
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES",
"OA-FLOW.COVREP.CANONICAL",
"OA-FLOW.COVREP.AVERAGING",
"OA-FLOW.COVREP.ORTHOGONAL",
"OA-FLOW.COVREP.BARYCENTRE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1093,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 949,
"through_line": 1093,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.COVREP.EXERCISES",
"unit": "OA-FLOW-L39",
"source_file": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e",
"reader": "OA-FLOW-L39.html#oa-flow.covrep.exercises",
"start_line": 1094,
"end_line": 1121,
"range_sha256": "05319614197cf95ef7be853478ea7cee99e95a47b65c2113830cb8944623471d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ISTATE.APPROXIMATION",
"OA-FLOW.GNS.5.1",
"OA-FLOW.CF.6",
"OA-FLOW.SC.4",
"OA-FLOW.COVREP.BASE",
"OA-FLOW.COVREP.HILBERT",
"OA-FLOW.COVREP.REPRESENTATIONS",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.COVREP.COVARIANCE",
"OA-FLOW.COVREP.STATES",
"OA-FLOW.COVREP.CANONICAL",
"OA-FLOW.COVREP.AVERAGING",
"OA-FLOW.COVREP.ORTHOGONAL",
"OA-FLOW.COVREP.BARYCENTRE",
"OA-FLOW.COVREP.EXAMPLES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1121,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L39.md",
"line": 1,
"through_line": 80,
"description": "Separable possibly nonunital C*-algebra, point-norm continuous action of a separable locally compact Hausdorff group, nondegenerate covariant representation on a separable Hilbert space, specified invariant unital abelian commutant diagonal; separate canonical state-space route with quasi-invariant probability and explicitly defined orthogonality criterion."
},
{
"source": "src/OA-FLOW-L39.md",
"line": 1094,
"through_line": 1121,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.SUSPENSION.SETTING",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.setting",
"start_line": 1,
"end_line": 32,
"range_sha256": "7c554218fa305b2e099c85e9ed523f6bc0c7a248ee8ddf6dac4b3cddbce5972f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 32,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.PRODUCT",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.product",
"start_line": 33,
"end_line": 122,
"range_sha256": "011c5711211d5afef1548798bed3cf0b793e6f7260f10fd451a25c30653892d0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 122,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 33,
"through_line": 122,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.PATHS",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.paths",
"start_line": 123,
"end_line": 156,
"range_sha256": "59ccb97736222f51f29458b0ac4169a77f134fefdafe0d4a06203a1bdb02c519",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.3",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 156,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 123,
"through_line": 156,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.CROSSINGS",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.crossings",
"start_line": 157,
"end_line": 203,
"range_sha256": "16ba4d8b54afefd2094e2e80b02b343c60494e210fa42ff2aa39d69594a0de71",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 203,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 157,
"through_line": 203,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.RETURNS",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.returns",
"start_line": 204,
"end_line": 233,
"range_sha256": "810b631ea1f5cbaa93ea0166ea67c46e646734dee4494c0f617ff65f72f8ab6a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 233,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 204,
"through_line": 233,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.COVER",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.cover",
"start_line": 326,
"end_line": 376,
"range_sha256": "feb80132ad5e6c36aa529a62e61c0177d32f5155b33d10ddb0ff6353791424b5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 376,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 326,
"through_line": 376,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.MEASURE",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.measure",
"start_line": 377,
"end_line": 404,
"range_sha256": "772bd7ea2454fe6c93162d7c4ad3bec1dc6cc5a5e9fe749d7764b9d2d20a2801",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.CHART"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 404,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 377,
"through_line": 404,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.CHART",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.chart",
"start_line": 234,
"end_line": 306,
"range_sha256": "ea8d19f343f95f337e1704c96bbca5bb8a663dfc6db8373aca23b6a1e7ba9a18",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 306,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 234,
"through_line": 306,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.THEOREM",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.theorem",
"start_line": 307,
"end_line": 325,
"range_sha256": "41186d6c686d95ad54f01abe9ffd0217ca764d54f7e81762455b349c17908a8d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS",
"OA-FLOW.SUSPENSION.CHART"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 325,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 307,
"through_line": 325,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../OA-ERGODIC/reader/dense-roof-groups-periods-and-eigenfunctions.html#3a4-uniform-gaps-and-the-full-suspension-chart",
"../../OA-ERGODIC/reader/dense-roof-groups-periods-and-eigenfunctions.html#suspension-coordinates-and-the-base-measure",
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.COVER-COROLLARY",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.cover-corollary",
"start_line": 405,
"end_line": 457,
"range_sha256": "de594a398f599cf29d7942f0632ab329a4a0d3009e85a15c6539080fcc3436a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.SUSPENSION.CHART",
"OA-FLOW.SUSPENSION.THEOREM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 457,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 405,
"through_line": 457,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.SUSPENSION.NONUNIQUE-BASE",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.nonunique-base",
"start_line": 458,
"end_line": 587,
"range_sha256": "afa7df5febfcb7b7f008122f7e917be59f201a11772eeb60078bcdbdc820f878",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CC.0",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SUSPENSION.CHART",
"OA-FLOW.SUSPENSION.COVER-COROLLARY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 587,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 458,
"through_line": 587,
"description": "Any irrational alpha in (0,1), normalized circle Lebesgue measure, and every positive integer q. All suspension formulas are strict Borel and hold for every real time. Base nonconjugacy holds even for invertible measure-class-preserving conjugacies modulo null sets."
}
]
},
{
"id": "OA-FLOW.SUSPENSION.EXERCISES",
"unit": "OA-FLOW-L38",
"source_file": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099",
"reader": "OA-FLOW-L38.html#oa-flow.suspension.exercises",
"start_line": 588,
"end_line": 615,
"range_sha256": "997b8931ec26d361088c9ee0a7ca12981a63197711d0237a1179f4bf0a1e5257",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SUSPENSION.PRODUCT",
"OA-FLOW.SUSPENSION.PATHS",
"OA-FLOW.SUSPENSION.CROSSINGS",
"OA-FLOW.SUSPENSION.RETURNS",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.SUSPENSION.CHART",
"OA-FLOW.SUSPENSION.THEOREM",
"OA-FLOW.SUSPENSION.COVER-COROLLARY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 615,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L38.md",
"line": 1,
"through_line": 32,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson."
},
{
"source": "src/OA-FLOW-L38.md",
"line": 588,
"through_line": 615,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
],
"external_proof_links": [
"../../NCG-FOLIATIONS/companions/polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-06",
"../../NCG-FOLIATIONS/transverse-measures-of-foliations.html#a-proof-of-the-countable-section-theorem"
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.SETTING",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.setting",
"start_line": 1,
"end_line": 46,
"range_sha256": "096366a51af77405aa948185a455bf31beb70d0113e15757fefcc341e60fca99",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.suspension",
"start_line": 47,
"end_line": 80,
"range_sha256": "c88c49637ca52aaec0f4e72b2a0ecbab4157b704a787d1f0a44566d3b42d8ec8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.THEOREM",
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 80,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 47,
"through_line": 80,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.STRICT",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.strict",
"start_line": 81,
"end_line": 103,
"range_sha256": "dda36f4235f39b51269f700a291f98b24cd33bcbcba318047ded7d6d8c2c93cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 103,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 81,
"through_line": 103,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.MARKERS",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.markers",
"start_line": 104,
"end_line": 138,
"range_sha256": "9b31df50485ea6a6f96e807c899a51e06014ad387415bcec3a3d28e8c8564881",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 138,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 104,
"through_line": 138,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.RETURNS",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.returns",
"start_line": 139,
"end_line": 316,
"range_sha256": "3111a8ab00f3e19d4c962d8151d1c7e714aca450155c9d2cafac384d26ef9f28",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 316,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 139,
"through_line": 316,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.PATHS",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.paths",
"start_line": 317,
"end_line": 372,
"range_sha256": "53db8fe2d392d561213bb76dbf012821f37ec4f57ff4888a1e11a9a64e71a7aa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 372,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 317,
"through_line": 372,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.REPAIR",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.repair",
"start_line": 373,
"end_line": 394,
"range_sha256": "8ee645d9410bf158ab2ccdb0019de6e8e4c89f4416cdd8915086e1b6a0d2588e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS",
"OA-FLOW.FLOWCOCYCLE.PATHS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 394,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 373,
"through_line": 394,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.COHOMOLOGOUS",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.cohomologous",
"start_line": 395,
"end_line": 439,
"range_sha256": "4d8f7dd7c6e6cb91b50d0c0deb86204528aceb1adb2bfbfbc527714fd06ef2a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS",
"OA-FLOW.FLOWCOCYCLE.PATHS",
"OA-FLOW.FLOWCOCYCLE.REPAIR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 439,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 395,
"through_line": 439,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.GAUGES",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.gauges",
"start_line": 440,
"end_line": 463,
"range_sha256": "5ed097a88d4185c7e7d6231511e494f3f8cc6577e8f6fbaa991f23412f71d3a5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS",
"OA-FLOW.FLOWCOCYCLE.PATHS",
"OA-FLOW.FLOWCOCYCLE.REPAIR",
"OA-FLOW.FLOWCOCYCLE.COHOMOLOGOUS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 463,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 440,
"through_line": 463,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.EXAMPLES",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.examples",
"start_line": 464,
"end_line": 522,
"range_sha256": "f6bb087eb74d9b73a1edfec1c47a749430547817c280755dea8c2c2c93879168",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS",
"OA-FLOW.FLOWCOCYCLE.PATHS",
"OA-FLOW.FLOWCOCYCLE.REPAIR",
"OA-FLOW.FLOWCOCYCLE.COHOMOLOGOUS",
"OA-FLOW.FLOWCOCYCLE.GAUGES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 522,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 464,
"through_line": 522,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.FLOWCOCYCLE.EXERCISES",
"unit": "OA-FLOW-L37",
"source_file": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b",
"reader": "OA-FLOW-L37.html#oa-flow.flowcocycle.exercises",
"start_line": 523,
"end_line": 600,
"range_sha256": "e76a969c1bcd0d6a50da3c42164076db18a951b34cfaa69546d279fccc9ae6c4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.KERNEL.CODING",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FLOWCOCYCLE.SUSPENSION",
"OA-FLOW.FLOWCOCYCLE.STRICT",
"OA-FLOW.FLOWCOCYCLE.MARKERS",
"OA-FLOW.FLOWCOCYCLE.RETURNS",
"OA-FLOW.FLOWCOCYCLE.PATHS",
"OA-FLOW.FLOWCOCYCLE.REPAIR",
"OA-FLOW.FLOWCOCYCLE.COHOMOLOGOUS",
"OA-FLOW.FLOWCOCYCLE.GAUGES",
"OA-FLOW.FLOWCOCYCLE.EXAMPLES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 600,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L37.md",
"line": 1,
"through_line": 46,
"description": "Strict jointly Borel nonsingular properly ergodic real action on a nonzero standard sigma-finite measured space; exact hypotheses and quantifiers of this lesson. For cocycle reduction: arbitrary Polish target P, Borel normal subgroup H and its closure K, jointly Borel fixed-pair almost-everywhere cocycles with the fixed-time closure condition."
},
{
"source": "src/OA-FLOW-L37.md",
"line": 523,
"through_line": 600,
"description": "Explicit hypotheses and complete argument of this section, including its stated exceptional-set quantifiers."
}
]
},
{
"id": "OA-FLOW.GI.BIDUAL",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.bidual",
"start_line": 499,
"end_line": 550,
"range_sha256": "4dd7ee92df1824290500ef9fbfdda937a30344bbdf1db4fd3824d142b042184a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 550,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L35.md",
"line": 499,
"through_line": 550,
"description": "Complete explicit section hypotheses and proof; arbitrary C*-algebra/representation scope and separate von Neumann normality scope as stated. No separability, faithfulness or nondegeneracy inferred."
}
]
},
{
"id": "OA-FLOW.GI.REPRESENTATIONS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.representations",
"start_line": 551,
"end_line": 655,
"range_sha256": "675d0404a3f69d557dc42e419f45d6a330a123f9a47447882634dc9ba7a1328b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.BIDUAL",
"OA-FLOW.GI.INNERNESS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 655,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L35.md",
"line": 551,
"through_line": 655,
"description": "Complete explicit section hypotheses and proof; arbitrary C*-algebra/representation scope and separate von Neumann normality scope as stated. No separability, faithfulness or nondegeneracy inferred."
}
]
},
{
"id": "OA-FLOW.GI.DUALBIMODULES",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.dualbimodules",
"start_line": 656,
"end_line": 722,
"range_sha256": "c7824a18326fa1f4e7cd4ee27e371a99948cea36a626be671cec06ed613191d7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.BIDUAL",
"OA-FLOW.L34.6",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 722,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L35.md",
"line": 656,
"through_line": 722,
"description": "Complete explicit section hypotheses and proof; arbitrary C*-algebra/representation scope and separate von Neumann normality scope as stated. No separability, faithfulness or nondegeneracy inferred."
}
]
},
{
"id": "OA-FLOW.GI.NORMALITY-BEFORE-INNERNESS",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.normality-before-innerness",
"start_line": 723,
"end_line": 863,
"range_sha256": "0eb85858c21c241dc4376f9a902f1bbc52a8292f08dd00b3fc6416acd2db67e3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L34.4",
"OA-FLOW.CP.6",
"OA-FLOW.NF.6",
"OA-FLOW.NF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 863,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L35.md",
"line": 723,
"through_line": 863,
"description": "Complete explicit section hypotheses and proof; arbitrary C*-algebra/representation scope and separate von Neumann normality scope as stated. No separability, faithfulness or nondegeneracy inferred."
}
]
},
{
"id": "OA-FLOW.GI.REPRESENTATION-EXAMPLE",
"unit": "OA-FLOW-L35",
"source_file": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4",
"reader": "OA-FLOW-L35.html#oa-flow.gi.representation-example",
"start_line": 864,
"end_line": 1021,
"range_sha256": "5388a76dcd8e75a0a5e661964183da49cc53261dfd96b6381889c4f89f9f0a90",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GI.REPRESENTATIONS",
"OA-FLOW.GI.DUALBIMODULES"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1021,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L35.md",
"line": 864,
"through_line": 1021,
"description": "Complete explicit section hypotheses and proof; arbitrary C*-algebra/representation scope and separate von Neumann normality scope as stated. No separability, faithfulness or nondegeneracy inferred."
}
]
},
{
"id": "OA-FLOW.BP.SETTING",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.setting",
"start_line": 1,
"end_line": 13,
"range_sha256": "e562ba8597569a144eb4dfbb7132f41cad2cc3002317efb7860e1e123b8ead2d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.DOMAINS",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.domains",
"start_line": 14,
"end_line": 207,
"range_sha256": "92bf5f0e3c533f94a453ca83dda5c5ff86093a47b3dc75e5be3524cf7f8538e7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.ST.2",
"OA-FLOW.L34.3",
"OA-FLOW.CF.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.UC.0",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 207,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 14,
"through_line": 207,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.EVOLUTION",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.evolution",
"start_line": 208,
"end_line": 331,
"range_sha256": "a3cb6c4be7e644c5ac1d692a6adf006118f5763fee9d575d29f0e74f937dca25",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 331,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 208,
"through_line": 331,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.PERTURBATION",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.perturbation",
"start_line": 332,
"end_line": 431,
"range_sha256": "2c7291658980dba8c0aeb57d008c82d5871684c92dd097b1e9420fc02f550a0f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.1",
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.ST.2",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 431,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 332,
"through_line": 431,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.SMOOTHING",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.smoothing",
"start_line": 432,
"end_line": 551,
"range_sha256": "76e2875c8fcf0a3bf58b1f86637eee2e2316fde328731fc5625eab7b4cee910e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 551,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 432,
"through_line": 551,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.CALCULUS",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.calculus",
"start_line": 552,
"end_line": 670,
"range_sha256": "34fa9651c2389b455fc86d55e1f7029e1883b9d56a25a9c0fdba874a9f15051e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.3",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION",
"OA-FLOW.BP.SMOOTHING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 670,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 552,
"through_line": 670,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.SELFADJOINT",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.selfadjoint",
"start_line": 671,
"end_line": 756,
"range_sha256": "5b6841b7094efbe8b6180121706e0c213046731c8e0e31ea37052406c5f9183a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF1",
"OA-FLOW.UC.0",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.SF.SF3",
"OA-FLOW.SF.SF2",
"OA-FLOW.L157.3",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION",
"OA-FLOW.BP.SMOOTHING",
"OA-FLOW.BP.CALCULUS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 756,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 671,
"through_line": 756,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.COUNTEREXAMPLE",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.counterexample",
"start_line": 757,
"end_line": 875,
"range_sha256": "eecf914cb8137aae888f6a0f7cb96f62b74738c8a35d9129d325acb331fac96a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.8",
"OA-FLOW.L157.2",
"OA-FLOW.SC.0",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION",
"OA-FLOW.BP.SMOOTHING",
"OA-FLOW.BP.CALCULUS",
"OA-FLOW.BP.SELFADJOINT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 875,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 757,
"through_line": 875,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.EXERCISES",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.exercises",
"start_line": 876,
"end_line": 961,
"range_sha256": "11902606bfc3f9e5b7544a9c6bf4467a361c519eeb77150effd8fab89d32d84c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION",
"OA-FLOW.BP.SMOOTHING",
"OA-FLOW.BP.CALCULUS",
"OA-FLOW.BP.SELFADJOINT",
"OA-FLOW.BP.COUNTEREXAMPLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 961,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 876,
"through_line": 961,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.BP.READING",
"unit": "OA-FLOW-BP",
"source_file": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484",
"reader": "OA-FLOW-BP.html#oa-flow.bp.reading",
"start_line": 962,
"end_line": 965,
"range_sha256": "348ae46018548f0d9a133920b7d02134fdef6505cd280b5c4c55825dca9145a5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.1",
"OA-FLOW.L157.3",
"OA-FLOW.AL.1",
"OA-FLOW.ST.2",
"OA-FLOW.AT.1",
"OA-FLOW.CF.1",
"OA-FLOW.L34.3",
"OA-FLOW.AT.5",
"OA-FLOW.RF.1",
"OA-FLOW.RF.5",
"OA-FLOW.L157.2",
"OA-FLOW.BP.DOMAINS",
"OA-FLOW.BP.EVOLUTION",
"OA-FLOW.BP.PERTURBATION",
"OA-FLOW.BP.SMOOTHING",
"OA-FLOW.BP.CALCULUS",
"OA-FLOW.BP.SELFADJOINT",
"OA-FLOW.BP.COUNTEREXAMPLE",
"OA-FLOW.BP.EXERCISES"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-BP.md",
"line": 1,
"through_line": 13,
"description": "Arbitrary von Neumann algebra, point-ultraweakly continuous normal real action; intrinsic sigma-strong-star derivatives. Extra coefficient and smoothness hypotheses are specified in each result."
},
{
"source": "src/OA-FLOW-BP.md",
"line": 962,
"through_line": 965,
"description": "Exact hypotheses and argument of this section."
}
]
},
{
"id": "OA-FLOW.L29.SETTING",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-setting",
"start_line": 1,
"end_line": 33,
"range_sha256": "ec496595e281f6b90eb152e3dfdf8307cad147dac4a622f8e3b58adab36d5454",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.1",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-1",
"start_line": 34,
"end_line": 106,
"range_sha256": "0635b92df665ab699c6715a385368e44ba82fdb417f61fa4ef1801dc62af76ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UC.0",
"OA-FLOW.ST.2",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 106,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 34,
"through_line": 106,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.2",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-2",
"start_line": 107,
"end_line": 180,
"range_sha256": "15991c2ade2c17e20334c8ad2d26f9b3b6950a2a2dbac3550926f23bd1f90549",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.AT.3",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 180,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 107,
"through_line": 180,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.3",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-3",
"start_line": 181,
"end_line": 266,
"range_sha256": "0769d5b34e7539aac7d2d511090131384f823f4743ac2e92b71e9f6af01832df",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.NF.6",
"OA-FLOW.ST.1",
"OA-FLOW.CP.6",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 266,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 181,
"through_line": 266,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.4",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-4",
"start_line": 267,
"end_line": 310,
"range_sha256": "85eed3ba1be2982eed9930f4091873793a2e38404feb5a74da0fd6cde580c44b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 310,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 267,
"through_line": 310,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.5",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-5",
"start_line": 311,
"end_line": 432,
"range_sha256": "95d8d143f4e16daf5c60a714c382afe71348e77e884005e94fcdd62dec6cf908",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.2",
"OA-FLOW.NR.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.NR.4",
"OA-FLOW.DA.ACTION",
"OA-FLOW.DA.FIXED",
"OA-FLOW.SP.3",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3",
"OA-FLOW.L29.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 432,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 311,
"through_line": 432,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.6",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-6",
"start_line": 433,
"end_line": 525,
"range_sha256": "01aae8c954182bfd6e9444987efadb3b9a32998a15b40c6553e719e35ec37379",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.5",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3",
"OA-FLOW.L29.4",
"OA-FLOW.L29.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 525,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 433,
"through_line": 525,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.7",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-7",
"start_line": 526,
"end_line": 636,
"range_sha256": "0693863f653b805ce064183d096fcb32881606f9fdc84719a363787a8f1c82ca",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.UC.0",
"OA-FLOW.SP.3",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3",
"OA-FLOW.L29.4",
"OA-FLOW.L29.5",
"OA-FLOW.L29.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 636,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 526,
"through_line": 636,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.8",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-8",
"start_line": 637,
"end_line": 719,
"range_sha256": "c079f6edc42e1020ca0369095fc6d4a8aa9192450d8260a9c8848958a3e91140",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.5",
"OA-FLOW.NF.4",
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3",
"OA-FLOW.L29.4",
"OA-FLOW.L29.5",
"OA-FLOW.L29.6",
"OA-FLOW.L29.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 719,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 637,
"through_line": 719,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L29.READING",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-reading",
"start_line": 720,
"end_line": 723,
"range_sha256": "879d173c5ecf17f169efc4061fe51234a1d7673a0efa50b0967655ea477c8669",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.ST.2",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.NR.3",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.L29.2",
"OA-FLOW.L29.3",
"OA-FLOW.L29.4",
"OA-FLOW.L29.5",
"OA-FLOW.L29.6",
"OA-FLOW.L29.7",
"OA-FLOW.L29.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L29.md",
"line": 1,
"through_line": 33,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. Zero endpoint specified in Section 5."
},
{
"source": "src/OA-FLOW-L29.md",
"line": 720,
"through_line": 723,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.SETTING",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-setting",
"start_line": 1,
"end_line": 36,
"range_sha256": "a242fa64e5ec634dbec2f881fd7d789231e336c7855507856f367182da6af0b7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.1",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-1",
"start_line": 37,
"end_line": 84,
"range_sha256": "59630a2b025695fec87d3403e0f6b642162e5f813fc3cd909d35c50497516fe3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.XGAPS.CSTAR.H0",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 84,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 37,
"through_line": 84,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.2",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-2",
"start_line": 85,
"end_line": 234,
"range_sha256": "a370e8851794c4f6a11804a3f34baa2b937819dd63938762f49f401d159a3405",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.NF.6",
"OA-FLOW.ST.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.L29.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.L29.1",
"OA-FLOW.WRC.SPECTRAL",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 234,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 85,
"through_line": 234,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.3",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-3",
"start_line": 235,
"end_line": 279,
"range_sha256": "c8ae273147a0b37af109464179e2797271ffc8423ce45a2e552739dc7c2bfe2b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.9",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 279,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 235,
"through_line": 279,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.4",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-4",
"start_line": 280,
"end_line": 359,
"range_sha256": "e417f8709eb187014e88b08b14046febea68340351f4718c57f1d72df556acd2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.1",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.EP.6",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 359,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 280,
"through_line": 359,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.5",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-5",
"start_line": 360,
"end_line": 439,
"range_sha256": "ae478bee34388daf5c7b2669ff7b7accd5a539d0481e4df6ad5e76bb57e1d29a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L29.2",
"OA-FLOW.AM.3",
"OA-FLOW.GDA.4",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 439,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 360,
"through_line": 439,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.6",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-6",
"start_line": 440,
"end_line": 508,
"range_sha256": "ec29ca8504f6d647e4786494970d6d58d19b37853bda1dda77383d42fb6504b6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L29.2",
"OA-FLOW.NF.6",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4",
"OA-FLOW.L31.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 508,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 440,
"through_line": 508,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.7",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-7",
"start_line": 509,
"end_line": 614,
"range_sha256": "700ad406d37928932ab63eb22eafa65ee26d89b28957b53782a67ad991f624a7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.HAAR",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4",
"OA-FLOW.L31.5",
"OA-FLOW.L31.6",
"OA-FLOW.SC.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 614,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 509,
"through_line": 586,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.8",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-8",
"start_line": 615,
"end_line": 720,
"range_sha256": "d27b598ab510a6e0da25b5611e01d5f2a8188a73a3f2464374ab83965cbdd6d5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.ND.WEYL",
"OA-FLOW.SC.8",
"OA-FLOW.L29.7",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4",
"OA-FLOW.L31.5",
"OA-FLOW.L31.6",
"OA-FLOW.L31.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 720,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 587,
"through_line": 692,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.9",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-9",
"start_line": 721,
"end_line": 796,
"range_sha256": "9ee38c66082dd2e361769a4dea3e2c154ad85b20955bb20b33c5f6310883e2cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.WEYL",
"OA-FLOW.NF.4",
"OA-FLOW.EP.4",
"OA-FLOW.SC.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.EP.3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4",
"OA-FLOW.L31.5",
"OA-FLOW.L31.6",
"OA-FLOW.L31.7",
"OA-FLOW.L31.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 796,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 693,
"through_line": 768,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L31.READING",
"unit": "OA-FLOW-L31",
"source_file": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da",
"reader": "OA-FLOW-L31.html#l31-reading",
"start_line": 797,
"end_line": 800,
"range_sha256": "f41486dead26900046c6c071ff44de6655303135d10bc5e3d21243ddceb76152",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.L29.5",
"OA-FLOW.GDA.4",
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.AM.3",
"OA-FLOW.L31.1",
"OA-FLOW.L31.2",
"OA-FLOW.L31.3",
"OA-FLOW.L31.4",
"OA-FLOW.L31.5",
"OA-FLOW.L31.6",
"OA-FLOW.L31.7",
"OA-FLOW.L31.8",
"OA-FLOW.L31.9"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L31.md",
"line": 1,
"through_line": 36,
"description": "Arbitrary LCA group, paired locally determined Haar measures, arbitrary nonzero von Neumann algebra and continuous normal dual-group action with a strong eigenunitary representation. The zero-algebra endpoint is separated from the faithful Haar-coordinate assertion."
},
{
"source": "src/OA-FLOW-L31.md",
"line": 769,
"through_line": 772,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.SETTING",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-setting",
"start_line": 1,
"end_line": 35,
"range_sha256": "4be33dac809e740dc548fb37f12992e34a40e5d2c4ac4089e4883878373f5ee7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.1",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-1",
"start_line": 36,
"end_line": 139,
"range_sha256": "82fc641fe4d127a452a597d6b8d24dd800cbd499478f71e27f2049e051f60447",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.7.1",
"OA-FLOW.CF.8",
"OA-FLOW.VD.2",
"OA-FLOW.GNS.3.2",
"OA-FLOW.CP.6",
"OA-FLOW.GW.4",
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 139,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 36,
"through_line": 139,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.2",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-2",
"start_line": 140,
"end_line": 232,
"range_sha256": "3233e1c72ebdedd64fdc3e1a24ab06964b1c4fbd434e6a478b04ca780573ef81",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.2",
"OA-FLOW.CA.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 232,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 140,
"through_line": 232,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.3",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-3",
"start_line": 233,
"end_line": 321,
"range_sha256": "a1c9e55ec3d5ec81cc5ba473494e21b1743295f72b279b26531bba3d1ed06dcf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.4",
"OA-FLOW.EP.1",
"OA-FLOW.EP.5",
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 321,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 233,
"through_line": 321,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.4",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-4",
"start_line": 322,
"end_line": 448,
"range_sha256": "b415127b4f7dfeafa2dfb3b0baac7961abf2b5560bebcce57d485647a41904c0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NF.5",
"OA-FLOW.ST.2",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.NF.6",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.GW.1",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 448,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 322,
"through_line": 448,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.5",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-5",
"start_line": 449,
"end_line": 609,
"range_sha256": "85649009a21daf3eb1e66607b2df81ffd8fc122c62abc1947ef537c2b2dd28a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.3",
"OA-FLOW.CI.3",
"OA-FLOW.SF.SF1",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.MW.4",
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3",
"OA-FLOW.TW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 609,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 449,
"through_line": 609,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.6",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-6",
"start_line": 610,
"end_line": 783,
"range_sha256": "2732c6880e04463416d1ed8c20b27f9cde287ec72bc07f0f1c435e3896402d56",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.4",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3",
"OA-FLOW.TW.4",
"OA-FLOW.TW.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 783,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 610,
"through_line": 783,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.7",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-7",
"start_line": 784,
"end_line": 853,
"range_sha256": "57660e24979a52f370ba5925284727775e5f20b7ae21a03b6aaf3b10dcc5eedf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3",
"OA-FLOW.TW.4",
"OA-FLOW.TW.5",
"OA-FLOW.TW.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 853,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 784,
"through_line": 853,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.8",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-8",
"start_line": 854,
"end_line": 997,
"range_sha256": "fe694a1a4e7449b148927933362f308572eb8f7d766abc6e55c0579261b05793",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.BC.5",
"OA-FLOW.BC.4",
"OA-FLOW.HR.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GW.4",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3",
"OA-FLOW.TW.4",
"OA-FLOW.TW.5",
"OA-FLOW.TW.6",
"OA-FLOW.TW.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 997,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 854,
"through_line": 997,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TW.READING",
"unit": "OA-FLOW-TW",
"source_file": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397",
"reader": "OA-FLOW-TW.html#tw-reading",
"start_line": 998,
"end_line": 1001,
"range_sha256": "db6c9ae7d813f44b16da0368c085382b6dab5519d777af2c6e74c9c89a113939",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.1",
"OA-FLOW.NF.5",
"OA-FLOW.WR.3",
"OA-FLOW.MW.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.2",
"OA-FLOW.TW.3",
"OA-FLOW.TW.4",
"OA-FLOW.TW.5",
"OA-FLOW.TW.6",
"OA-FLOW.TW.7",
"OA-FLOW.TW.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TW.md",
"line": 1,
"through_line": 35,
"description": "Arbitrary nonzero von Neumann algebra M, faithful normal semifinite phi, arbitrary nonzero Hilbert E and arbitrary orthonormal index set I. Inner products linear in first variable; nonnegative sums are suprema over finite subsets. Zero algebra/Hilbert endpoints are separated."
},
{
"source": "src/OA-FLOW-TW.md",
"line": 998,
"through_line": 1001,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.SETTING",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-setting",
"start_line": 1,
"end_line": 15,
"range_sha256": "4bd6491be718858600a77f2ea349955bf259fbbeb37b3e54c88fe81e5527845b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 15,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.1",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-1",
"start_line": 16,
"end_line": 106,
"range_sha256": "05018999be15ab45c44f599f7a3c06904ce87905a1c7026b1456555f28b76604",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.5",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.2",
"OA-FLOW.TW.4",
"OA-FLOW.TW.5",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.HR.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 106,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 16,
"through_line": 106,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.2",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-2",
"start_line": 107,
"end_line": 197,
"range_sha256": "6bef78c8e7fa5132b03432ef2cfa29af8c53f31f6787f2bd9f3d7d5b82d5752a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.7",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.ST.2",
"OA-FLOW.BC.4",
"OA-FLOW.UR.3",
"OA-FLOW.UR.5",
"OA-FLOW.CT.1",
"OA-FLOW.BC.5",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.GDA.7",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 197,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 107,
"through_line": 197,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.3",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-3",
"start_line": 198,
"end_line": 367,
"range_sha256": "f1ee41259f8e470edeb89fa5f317d3193f40a6fc0177405fcdc56ad55e6ca716",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.VECTOR",
"OA-FLOW.HR.5",
"OA-FLOW.NR.1",
"OA-FLOW.TW.1",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.CT.1",
"OA-FLOW.NC.2",
"OA-FLOW.NC.3",
"OA-FLOW.MC.1",
"OA-FLOW.GRP.ALGEBRA",
"OA-FLOW.AT.4",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 367,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 198,
"through_line": 367,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.4",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-4",
"start_line": 368,
"end_line": 488,
"range_sha256": "8f82b4abc8176a340a7640b7256afb54c932bf7972428525f3fbbb53ad6e38f1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TW.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.UR.5",
"OA-FLOW.SP.4",
"OA-FLOW.SF.SF3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 488,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 368,
"through_line": 488,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.5",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-5",
"start_line": 489,
"end_line": 671,
"range_sha256": "66a5aa23f2375324ec46c80ccad2a07a118eba937a01f8ed7f7c4c754f01c1f6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.TW.4",
"OA-FLOW.MC.5",
"OA-FLOW.NR.1",
"OA-FLOW.EW.2",
"OA-FLOW.EW.3",
"OA-FLOW.NF.5",
"OA-FLOW.GDA.7",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3",
"OA-FLOW.L33.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 671,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 489,
"through_line": 671,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.6",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-6",
"start_line": 673,
"end_line": 805,
"range_sha256": "aece503d1902f45f64311e8ca35d5f13b0db3cb965503cd63f585bdc9a00c729",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.UR.5",
"OA-FLOW.GW.2",
"OA-FLOW.BC.5",
"OA-FLOW.BC.4",
"OA-FLOW.TW.1",
"OA-FLOW.TW.6",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3",
"OA-FLOW.L33.4",
"OA-FLOW.L33.5",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.TW.3",
"OA-FLOW.GW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 805,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 673,
"through_line": 779,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.7",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-7",
"start_line": 806,
"end_line": 971,
"range_sha256": "4463fb9b57d50e759a0e6b47162b4d78d82c35c9163deff7c2f9c0d7a31c0d1f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TW.1",
"OA-FLOW.CZ.6",
"OA-FLOW.BC.4",
"OA-FLOW.TD.6",
"OA-FLOW.TW.2",
"OA-FLOW.DA.VECTOR",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3",
"OA-FLOW.L33.4",
"OA-FLOW.L33.5",
"OA-FLOW.L33.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 971,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 780,
"through_line": 945,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.8",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-8",
"start_line": 972,
"end_line": 1107,
"range_sha256": "a0ea5849eac8c7d50b54d40b51be8be4879abc0c8c0e603705c8e923781dc7bf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EW.2",
"OA-FLOW.BC.5",
"OA-FLOW.DA.HAAR",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.DA.INVARIANCE",
"OA-FLOW.TW.6",
"OA-FLOW.BC.4",
"OA-FLOW.GDA.9",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3",
"OA-FLOW.L33.4",
"OA-FLOW.L33.5",
"OA-FLOW.L33.6",
"OA-FLOW.L33.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1107,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 946,
"through_line": 1081,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L33.READING",
"unit": "OA-FLOW-L33",
"source_file": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb",
"reader": "OA-FLOW-L33.html#l33-reading",
"start_line": 1108,
"end_line": 1111,
"range_sha256": "371ee6b9344487029ea25253162efbf148dcdc461b30b7726a8fbd131a777bf5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TW.2",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.GDW.6",
"OA-FLOW.TW.4",
"OA-FLOW.HR.9",
"OA-FLOW.L33.1",
"OA-FLOW.L33.2",
"OA-FLOW.L33.3",
"OA-FLOW.L33.4",
"OA-FLOW.L33.5",
"OA-FLOW.L33.6",
"OA-FLOW.L33.7",
"OA-FLOW.L33.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 1111,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L33.md",
"line": 1,
"through_line": 15,
"description": "Arbitrary LCA group G with Plancherel-compatible Haar measures, arbitrary nonzero von Neumann M, faithful normal semifinite phi, and point-ultraweakly continuous action alpha. No countability, state, or action-invariance hypothesis. Negative dual characters and usual rank-one-normalized trace; zero algebra endpoint separated."
},
{
"source": "src/OA-FLOW-L33.md",
"line": 1082,
"through_line": 1085,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.SETTING",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-setting",
"start_line": 1,
"end_line": 26,
"range_sha256": "f370c91021dd31aa900f1900aef983587d532de896c5d22745d55abb06508a9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.1",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-1",
"start_line": 27,
"end_line": 61,
"range_sha256": "c343fd06fb6f7afd968c80a852e042250327ab1ef77f2d8a249ab5145fd59bfd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.3",
"OA-FLOW.NWR.4",
"OA-FLOW.NWR.6",
"OA-FLOW.FR.1",
"OA-FLOW.NWR.2",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.DA.EQUALITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 61,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 27,
"through_line": 61,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.2",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-2",
"start_line": 62,
"end_line": 113,
"range_sha256": "b58fafe75404468a9691f65b28783c9d1d7cd5e28cafbe09c600effd8c4bb80f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.6",
"OA-FLOW.FR.2",
"OA-FLOW.NWR.3",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 113,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 62,
"through_line": 113,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.3",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-3",
"start_line": 114,
"end_line": 141,
"range_sha256": "dd92064a944e0a76b1c98fe0a6e1b18e048c00070c44a98621191ef1e2d54544",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.HAAR",
"OA-FLOW.GDA.8",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 141,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 114,
"through_line": 141,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.4",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-4",
"start_line": 142,
"end_line": 185,
"range_sha256": "81b9b4c97b7dcd82fe1058abeabb0abba041568f578b3815b00fded9abe28b9f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.HAAR",
"OA-FLOW.GW.2",
"OA-FLOW.DA.EXERCISES",
"OA-FLOW.SF.SF1",
"OA-FLOW.GW.4",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 185,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 142,
"through_line": 185,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.5",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-5",
"start_line": 186,
"end_line": 245,
"range_sha256": "a168d59747d61109f014ae0b462b1960e0fc0ca24c1cadb3b5cd47c9a770638b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3",
"OA-FLOW.L32.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 245,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 186,
"through_line": 245,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.6",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-6",
"start_line": 246,
"end_line": 302,
"range_sha256": "d971ea3b7c1c0b6f7e3031d48ef259fdb852417276b7cdf05a9042e112a1bb77",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.3",
"OA-FLOW.FF.4",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.EP.5",
"OA-FLOW.FF.1",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3",
"OA-FLOW.L32.4",
"OA-FLOW.L32.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 302,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 246,
"through_line": 302,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.7",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-7",
"start_line": 303,
"end_line": 363,
"range_sha256": "ce0e5592cbbea931f5e70519e52e1db35a2d7a62041f5d8170fa8f65ba21f297",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WS.1",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3",
"OA-FLOW.L32.4",
"OA-FLOW.L32.5",
"OA-FLOW.L32.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 363,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 303,
"through_line": 363,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.8",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-8",
"start_line": 364,
"end_line": 420,
"range_sha256": "c8886d73fef9426600fe1ac0d0fbdc22c6ddc7e18fab5e1ed78674198e4d210d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NWR.1",
"OA-FLOW.NWR.2",
"OA-FLOW.GDA.8",
"OA-FLOW.FR.2",
"OA-FLOW.EP.5",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3",
"OA-FLOW.L32.4",
"OA-FLOW.L32.5",
"OA-FLOW.L32.6",
"OA-FLOW.L32.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 420,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 364,
"through_line": 420,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L32.READING",
"unit": "OA-FLOW-L32",
"source_file": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92",
"reader": "OA-FLOW-L32.html#l32-reading",
"start_line": 421,
"end_line": 424,
"range_sha256": "468cbe5977bb4679687125071ca172248117ddd37260b11a85505f259abb5279",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.2",
"OA-FLOW.NWR.6",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.5",
"OA-FLOW.L32.1",
"OA-FLOW.L32.2",
"OA-FLOW.L32.3",
"OA-FLOW.L32.4",
"OA-FLOW.L32.5",
"OA-FLOW.L32.6",
"OA-FLOW.L32.7",
"OA-FLOW.L32.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L32.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary LCA G and von Neumann M, point-ultraweakly continuous action, paired Haar measures and negative dual characters. Normal semifinite weights may be nonfaithful or zero; explicitly broader normal-weight identities marked locally. No countability or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L32.md",
"line": 421,
"through_line": 424,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.SETTING",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-setting",
"start_line": 1,
"end_line": 23,
"range_sha256": "04c37a4befa01ce55251a9d361d8686bb0651a042c58f92f11e23f20c77b8964",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.1",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-1",
"start_line": 24,
"end_line": 107,
"range_sha256": "e2ae0780ca1b2ba285de447703ce8df9cc9f13914283a3bb2ddbd364284f7774",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.EP.4",
"OA-FLOW.EP.2",
"OA-FLOW.EP.1",
"OA-FLOW.SC.4",
"OA-FLOW.ST.2",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 107,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 24,
"through_line": 107,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.2",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-2",
"start_line": 108,
"end_line": 175,
"range_sha256": "da3598a45dfd7dbf474846f416b663c44852095a10a2a2a226eb98ce750b4a35",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.PC.3",
"OA-FLOW.CF.1",
"OA-FLOW.PC.1",
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 175,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 108,
"through_line": 175,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.3",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-3",
"start_line": 176,
"end_line": 220,
"range_sha256": "0bf1bb7c3d1952eabd832204cfaa2ce36f4221dcb3b18eead23ffeb01e06cc61",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.EP.6",
"OA-FLOW.TD.4",
"OA-FLOW.TD.6",
"OA-FLOW.EP.4",
"OA-FLOW.TD.2",
"OA-FLOW.TD.5",
"OA-FLOW.AT.5",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 220,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 176,
"through_line": 220,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.4",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-4",
"start_line": 221,
"end_line": 254,
"range_sha256": "64924181a4c8dc135314b2feb971f6aee1c9378d367efb5ece92913f724ea732",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SF.SF1",
"OA-FLOW.SC.5",
"OA-FLOW.NR.3",
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.L19.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 254,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 221,
"through_line": 254,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.5",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-5",
"start_line": 255,
"end_line": 323,
"range_sha256": "e2ac43a1e643142cc55d71f8cf14a579f633d2aebcaba5191b65421a0d02be0e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.FF.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.FF.1",
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.L19.3",
"OA-FLOW.L19.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 323,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 255,
"through_line": 323,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.6",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-6",
"start_line": 324,
"end_line": 412,
"range_sha256": "74c86e764a6f3ccf2339321ce850d51fbc93769a979e8ead8232579b475be44d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.L19.3",
"OA-FLOW.L19.4",
"OA-FLOW.L19.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 412,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 324,
"through_line": 412,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.7",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-7",
"start_line": 413,
"end_line": 491,
"range_sha256": "056c75e7da14b3f9efcc2e487fce5d80649b305310fdb81a0e241474a7c363b8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.5",
"OA-FLOW.EP.6",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.L19.3",
"OA-FLOW.L19.4",
"OA-FLOW.L19.5",
"OA-FLOW.L19.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 491,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 413,
"through_line": 491,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L19.READING",
"unit": "OA-FLOW-L19",
"source_file": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317",
"reader": "OA-FLOW-L19.html#l19-reading",
"start_line": 492,
"end_line": 497,
"range_sha256": "ac2e916441729a5282b5aff128e90ad5b259ff4faea89365cb49cc75d77cb242",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.4",
"OA-FLOW.EP.6",
"OA-FLOW.AT.5",
"OA-FLOW.FR.1",
"OA-FLOW.TD.6",
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.L19.3",
"OA-FLOW.L19.4",
"OA-FLOW.L19.5",
"OA-FLOW.L19.6",
"OA-FLOW.L19.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L19.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann Q, point-ultraweakly continuous real action theta, given faithful normal semifinite trace tau with tau composed theta_s = exp(-s) tau. Section1 first proves a criterion for any real action. Lebesgue normalization; no countability, factor, finite trace or faithful-state hypothesis."
},
{
"source": "src/OA-FLOW-L19.md",
"line": 492,
"through_line": 497,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.SETTING",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-setting",
"start_line": 1,
"end_line": 31,
"range_sha256": "2daf8bdc58b249ec2c0b5639f8cbf7f3d4848a2d17594cb19299800cedf5245b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.1",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-1",
"start_line": 32,
"end_line": 69,
"range_sha256": "c9d39feb8fe38aa367a245664b23941792dc84c385f39acafa959faa582f6130",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.RF.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.GDA.8",
"OA-FLOW.GDW.7",
"OA-FLOW.BC.4",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.CZ.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 69,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 32,
"through_line": 69,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.2",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-2",
"start_line": 70,
"end_line": 97,
"range_sha256": "5a48f22ce9ce1144fa01730f570aaa02164676d2a3172faa2f01894673031a86",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.5",
"OA-FLOW.KT.5",
"OA-FLOW.CZ.6",
"OA-FLOW.BC.4",
"OA-FLOW.GDA.7",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CORE.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 97,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 70,
"through_line": 97,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.3",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-3",
"start_line": 98,
"end_line": 203,
"range_sha256": "12a401e5804fbac58b7642a1a4e1b7ae97d5318fa7d1f5fe3f337e4d2cba3b68",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.2",
"OA-FLOW.CZ.1",
"OA-FLOW.CZ.3",
"OA-FLOW.GW.2",
"OA-FLOW.SF.SF1",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 203,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 98,
"through_line": 203,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.4",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-4",
"start_line": 204,
"end_line": 232,
"range_sha256": "7da3b0ea30ceeeac68c3eca6a478c35279122067de717bfd8659851e2ead5d04",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.RF.5",
"OA-FLOW.CZ.7",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 232,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 204,
"through_line": 232,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.5",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-5",
"start_line": 233,
"end_line": 306,
"range_sha256": "0fb7c5f89ba16d5564e1a5fe508209bcc567f9bd48b3d879cea9fd789bc4426e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.4",
"OA-FLOW.NR.5",
"OA-FLOW.NR.4",
"OA-FLOW.DA.ACTION",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 306,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 233,
"through_line": 306,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.6",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-6",
"start_line": 307,
"end_line": 394,
"range_sha256": "90f6038fa7cbf05a50dc8988f8799b7fff3fbdbe54720ab8f550b113356f3421",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.4",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.1",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.GDA.9",
"OA-FLOW.CZ.6",
"OA-FLOW.BC.4",
"OA-FLOW.BC.5",
"OA-FLOW.GDA.7",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 394,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 307,
"through_line": 394,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.7",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-7",
"start_line": 395,
"end_line": 440,
"range_sha256": "8b33505c9125695ddaac6cb6b32bb2773ac9b0e7cca1866bce683d14b66b05d7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.NR.4",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 440,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 395,
"through_line": 440,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.8",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-8",
"start_line": 441,
"end_line": 564,
"range_sha256": "9c92c14d724e4bdca646c553cf3414a6e665a9862e5c50e2c87d6f059c188d68",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.5",
"OA-FLOW.NR.4",
"OA-FLOW.CZ.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NC.4",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 564,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 441,
"through_line": 564,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.9",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-9",
"start_line": 565,
"end_line": 699,
"range_sha256": "d7673f95dedb4e6d5ac8686b9225c4f68f7b92f0551bb2655e0f473fac80b92e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.3",
"OA-FLOW.ND.WEYL",
"OA-FLOW.EP.1",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.FF.1",
"OA-FLOW.EP.5",
"OA-FLOW.FF.2",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 699,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 565,
"through_line": 699,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.10",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-10",
"start_line": 700,
"end_line": 778,
"range_sha256": "cdbb493cf0b2a1d8d6792ce4fb0cae26f101cbb10c53a19dfe5f0df6e0e547ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.6",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 778,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 700,
"through_line": 778,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.11",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-11",
"start_line": 779,
"end_line": 867,
"range_sha256": "28e89084ca1560edd0c742409706f3b756c0317dc78f10c3fb9e86b0e445b5d1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.WEYL",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.9",
"OA-FLOW.CORE.10"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 867,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 779,
"through_line": 867,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CORE.READING",
"unit": "OA-FLOW-CORE",
"source_file": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372",
"reader": "OA-FLOW-CORE.html#core-reading",
"start_line": 868,
"end_line": 873,
"range_sha256": "02ea3e886b570a0fde31d644fcabe72bae77ced855323afb937bf2d1f35a06e2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.8",
"OA-FLOW.CZ.2",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.9",
"OA-FLOW.CORE.10",
"OA-FLOW.CORE.11"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CORE.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary von Neumann M with faithful normal semifinite weight phi, modular crossed product by the real line with dt and dual ds/(2pi). General core trace and faithful-weight chart changes. No countability, factor, separability or faithful-state hypothesis; additional tracial or finite-matrix model conditions are stated locally."
},
{
"source": "src/OA-FLOW-CORE.md",
"line": 868,
"through_line": 873,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.SETTING",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-setting",
"start_line": 1,
"end_line": 26,
"range_sha256": "bd06a59911e51f37274e54cba74427cb0f6acd805fa520d2efa7dda44816924b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.1",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-1",
"start_line": 27,
"end_line": 58,
"range_sha256": "cdd3491faea223e27b0310589fd877a55c509269be7f4e2bf2c0aa5519e0e5cb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDW.7",
"OA-FLOW.BC.5",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NC.4",
"OA-FLOW.CZ.5",
"OA-FLOW.GDW.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 58,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 27,
"through_line": 58,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.2",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-2",
"start_line": 59,
"end_line": 96,
"range_sha256": "d86833ad4c534d4e03367de687062812ee06fde74a002bffef445cf68eb3a528",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.4",
"OA-FLOW.CZ.5",
"OA-FLOW.KT.5",
"OA-FLOW.RF.5",
"OA-FLOW.CZ.7",
"OA-FLOW.TD.5",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 96,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 59,
"through_line": 96,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.3",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-3",
"start_line": 97,
"end_line": 133,
"range_sha256": "06a16fdb7810446e3615537ef4d46595aacbc1d62e83be3558be6fdfc7977bc2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L19.3",
"OA-FLOW.WRC.UNITARY",
"OA-FLOW.FCT.8",
"OA-FLOW.PC.5",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 133,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 97,
"through_line": 133,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.4",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-4",
"start_line": 134,
"end_line": 179,
"range_sha256": "48e6da6b1a8ab5cabaf73d570f8bd909835f2f72dbe64beb692f11ec5afd139f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L31.2",
"OA-FLOW.RF.5",
"OA-FLOW.FF.1",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 179,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 134,
"through_line": 179,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.5",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-5",
"start_line": 180,
"end_line": 225,
"range_sha256": "17d5e21c2671887ced51bda7c4f3fb8cf16b98e50983076d93620675a603c3f4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CZ.2",
"OA-FLOW.KT.5",
"OA-FLOW.L31.2",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 225,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 180,
"through_line": 225,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.6",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-6",
"start_line": 226,
"end_line": 337,
"range_sha256": "4bd066b29229cc26d7471f140d9c38409cc6bdf47e96aff8ab58f44316e2ba20",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.2",
"OA-FLOW.FCT.6",
"OA-FLOW.PC.4",
"OA-FLOW.PC.8",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 337,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 226,
"through_line": 337,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.7",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-7",
"start_line": 338,
"end_line": 570,
"range_sha256": "101894bffbd21a72169c8d422c75e0f05dbe90a8cd1919e83baba7034fb4d0b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FCT.6",
"OA-FLOW.FR.1",
"OA-FLOW.EP.6",
"OA-FLOW.TD.4",
"OA-FLOW.TD.6",
"OA-FLOW.GW.4",
"OA-FLOW.PC.4",
"OA-FLOW.PC.2",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5",
"OA-FLOW.L18.6",
"OA-FLOW.PC.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 570,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 338,
"through_line": 570,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.8",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-8",
"start_line": 571,
"end_line": 616,
"range_sha256": "cc78411631a6b9071b5c2dae9dc4622ba4b982044da097aa141cb2a90544e890",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.5",
"OA-FLOW.EP.1",
"OA-FLOW.L19.4",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5",
"OA-FLOW.L18.6",
"OA-FLOW.L18.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 616,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 571,
"through_line": 616,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.9",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-9",
"start_line": 617,
"end_line": 721,
"range_sha256": "03abd26a760a658f5a93699dbd3f2867850193bab51cfdc5018606f9a54f858b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.FF.2",
"OA-FLOW.NR.3",
"OA-FLOW.FF.1",
"OA-FLOW.ND.WEYL",
"OA-FLOW.WRC.UNITARY",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5",
"OA-FLOW.L18.6",
"OA-FLOW.L18.7",
"OA-FLOW.L18.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 721,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 617,
"through_line": 721,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.10",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-10",
"start_line": 722,
"end_line": 879,
"range_sha256": "be5f21069cacd668ceb51b0d898899edb88021d1f0f8edce34caf8edc79d9672",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L19.3",
"OA-FLOW.TD.5",
"OA-FLOW.L19.4",
"OA-FLOW.WRC.UNITARY",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5",
"OA-FLOW.L18.6",
"OA-FLOW.L18.7",
"OA-FLOW.L18.8",
"OA-FLOW.L18.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 879,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 722,
"through_line": 879,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L18.READING",
"unit": "OA-FLOW-L18",
"source_file": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a",
"reader": "OA-FLOW-L18.html#l18-reading",
"start_line": 880,
"end_line": 885,
"range_sha256": "1f49dae59896bc1c514f5282289b9eebb4412202140ea47908c77115c3570b33",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L19.1",
"OA-FLOW.L31.2",
"OA-FLOW.FCT.6",
"OA-FLOW.GDW.6",
"OA-FLOW.L18.1",
"OA-FLOW.L18.2",
"OA-FLOW.L18.3",
"OA-FLOW.L18.4",
"OA-FLOW.L18.5",
"OA-FLOW.L18.6",
"OA-FLOW.L18.7",
"OA-FLOW.L18.8",
"OA-FLOW.L18.9",
"OA-FLOW.L18.10"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L18.md",
"line": 1,
"through_line": 26,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau; point-ultraweak continuous real action theta with tau theta_s=exp(-s)tau. Normal regular M=N crossed_theta R; negative dual action. Full central-corner and typeIII criterion, with no countability, factor, separability or faithful-state hypothesis. Local models have their additional explicit hypotheses."
},
{
"source": "src/OA-FLOW-L18.md",
"line": 880,
"through_line": 885,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.SETTING",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-setting",
"start_line": 1,
"end_line": 38,
"range_sha256": "690c5b17e98969d016c894ee8224081b8c24320710b4e5a30caa93833829c006",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.1",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-1",
"start_line": 39,
"end_line": 80,
"range_sha256": "32c4018cc4de6445fa0551b38a63b93a800b6d1c163ab8b7d3d15e733e99b9cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L19.1",
"OA-FLOW.L19.2",
"OA-FLOW.FR.1",
"OA-FLOW.EP.5",
"OA-FLOW.EP.6",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 80,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 39,
"through_line": 80,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.2",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-2",
"start_line": 81,
"end_line": 146,
"range_sha256": "33113da9127f3786614566ca11a98107fe7e664b65bdf7bbd737bacfddb464aa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.4",
"OA-FLOW.TD.5",
"OA-FLOW.TD.6",
"OA-FLOW.CZ.7",
"OA-FLOW.SF.SF1",
"OA-FLOW.KT.5",
"OA-FLOW.CZ.5",
"OA-FLOW.OT.5",
"OA-FLOW.BC.5",
"OA-FLOW.GDA.8",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 146,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 81,
"through_line": 146,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.3",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-3",
"start_line": 147,
"end_line": 167,
"range_sha256": "257a10ae3c28f5c49fc14f8f77e97940a9670b60848ecb442e31b14249d03598",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L29.4",
"OA-FLOW.L29.5",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 167,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 147,
"through_line": 167,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.4",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-4",
"start_line": 168,
"end_line": 231,
"range_sha256": "89389d20bd836111cbe2e077ade87c8a9a3e3ad1eac37749ddd460bb6103e0d4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EP.4",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.1",
"OA-FLOW.RF.5",
"OA-FLOW.CORE.8",
"OA-FLOW.CZ.7",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.1",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 231,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 168,
"through_line": 231,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.5",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-5",
"start_line": 232,
"end_line": 310,
"range_sha256": "fec2c5ca1586e5d7a3c06fc6b4d8e6012a2bfbe78d7f42f23846d1e0188d1d3e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.5",
"OA-FLOW.CZ.6",
"OA-FLOW.BC.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.BC.5",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 310,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 232,
"through_line": 310,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.6",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-6",
"start_line": 311,
"end_line": 407,
"range_sha256": "6cc970ed494aa23b44719d443fe026a9d08bba515adf1ce4e2c8e1f2ee097d48",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.2",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.EP.1",
"OA-FLOW.CORE.8",
"OA-FLOW.CZ.7",
"OA-FLOW.TD.5",
"OA-FLOW.NR.4",
"OA-FLOW.BC.5",
"OA-FLOW.DA.FIXED",
"OA-FLOW.CORE.1",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 407,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 311,
"through_line": 407,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.7",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-7",
"start_line": 408,
"end_line": 484,
"range_sha256": "3807996cfd6b6e8b81e80b8abbe05795b3373bd875cad4ff7dacb6555e0dcf9f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5",
"OA-FLOW.L30.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 484,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 408,
"through_line": 484,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.8",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-8",
"start_line": 485,
"end_line": 538,
"range_sha256": "afebae6353d82f2845d9074fd57871bc0f4075a7e4c6827b2d7d9b3840a7905d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.FF.1",
"OA-FLOW.CORE.9",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5",
"OA-FLOW.L30.6",
"OA-FLOW.L30.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 538,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 485,
"through_line": 538,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.9",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-9",
"start_line": 539,
"end_line": 707,
"range_sha256": "9afce3ec891e45331dc427d8c376f07275b8ca272ccdb88b870dfad630615334",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.6",
"OA-FLOW.CORE.10",
"OA-FLOW.CORE.9",
"OA-FLOW.EP.5",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5",
"OA-FLOW.L30.6",
"OA-FLOW.L30.7",
"OA-FLOW.L30.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 707,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 539,
"through_line": 707,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.10",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-10",
"start_line": 708,
"end_line": 801,
"range_sha256": "1cdab183557077223ef15406dca94051b5dfb59b3e041cc7f95f2050f36ee921",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.6",
"OA-FLOW.L18.6",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5",
"OA-FLOW.L30.6",
"OA-FLOW.L30.7",
"OA-FLOW.L30.8",
"OA-FLOW.L30.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 801,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 708,
"through_line": 801,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.L30.READING",
"unit": "OA-FLOW-L30",
"source_file": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d",
"reader": "OA-FLOW-L30.html#l30-reading",
"start_line": 802,
"end_line": 809,
"range_sha256": "0fde4211e958ecff4ca0298f26ca8f343971a4f93bce2520257e6e00153f8522",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.OT.5",
"OA-FLOW.CZ.2",
"OA-FLOW.CORE.2",
"OA-FLOW.L18.6",
"OA-FLOW.CORE.1",
"OA-FLOW.L30.1",
"OA-FLOW.L30.2",
"OA-FLOW.L30.3",
"OA-FLOW.L30.4",
"OA-FLOW.L30.5",
"OA-FLOW.L30.6",
"OA-FLOW.L30.7",
"OA-FLOW.L30.8",
"OA-FLOW.L30.9",
"OA-FLOW.L30.10"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L30.md",
"line": 1,
"through_line": 38,
"description": "Arbitrary nonzero von Neumann N with faithful normal semifinite trace tau and continuous real action theta scaling it by exp(-s); M=N^theta. Any faithful normal semifinite baseweight psi on M, core with dt and dualds/(2pi). Full normal core-converse preserving specifiedtrace, charts and functor, no factor/countability/separability/typeIII assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-L30.md",
"line": 802,
"through_line": 809,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.SETTING",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-setting",
"start_line": 1,
"end_line": 25,
"range_sha256": "6eed4eebce8addc430e611c49adf8ec073b8725d0f16a736895f9646d4ae7f6f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.1",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-1",
"start_line": 26,
"end_line": 156,
"range_sha256": "253e0af13b95971a872cf30bc34563f70a0ad4e88059b5ca92a01492cc24a962",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.2",
"OA-FLOW.PC.5",
"OA-FLOW.BD.1",
"OA-FLOW.NCF.1",
"OA-FLOW.CP.6",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 156,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 26,
"through_line": 156,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.2",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-2",
"start_line": 157,
"end_line": 210,
"range_sha256": "72ac138bb9ed8461b0a913a67608f8d52697a05e94ee33d6b6537b631bb79a99",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6",
"OA-FLOW.CDEC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 210,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 157,
"through_line": 210,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.3",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-3",
"start_line": 211,
"end_line": 271,
"range_sha256": "cb43661086edebb6b457f04fe1890253f6913c4ab0503b02d7a93cd8b8dfb95b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.CORE.8",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.L18.8",
"OA-FLOW.L18.6",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 271,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 211,
"through_line": 271,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.4",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-4",
"start_line": 272,
"end_line": 506,
"range_sha256": "c9dd975a6cd07cd01b7ca05148f08f6b881a77169201b9bb312beaedc3fde3d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L30.5",
"OA-FLOW.L30.7",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.7",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.L30.6",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.NCF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 506,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 272,
"through_line": 506,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.5",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-5",
"start_line": 507,
"end_line": 560,
"range_sha256": "51dd84b237d5b9dfef58fc016719f47585a475c16d328edb0b2e5f2a9ebe78ad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.4",
"OA-FLOW.L30.6",
"OA-FLOW.CORE.8",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.CDEC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 560,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 507,
"through_line": 560,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.6",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-6",
"start_line": 561,
"end_line": 600,
"range_sha256": "1fe45a93dc3d2d3c01ca671a4c31abeb95fc5fc3283ceffe3cc072bf7a6e2a75",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L18.1",
"OA-FLOW.CORE.8",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.CDEC.4",
"OA-FLOW.CDEC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 600,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 561,
"through_line": 600,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.7",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-7",
"start_line": 601,
"end_line": 701,
"range_sha256": "1945db56f0119c0911ff851307707c18aaec21de1e3582dc080f46a17cb6b545",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.9",
"OA-FLOW.PC.0",
"OA-FLOW.L30.6",
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.CDEC.4",
"OA-FLOW.CDEC.5",
"OA-FLOW.CDEC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 701,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 601,
"through_line": 701,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.8",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-8",
"start_line": 702,
"end_line": 779,
"range_sha256": "adcc10bc191355a3f1f4c7fb08c6d757bfed5abbf17fcecc57b3619dd8d79b43",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.8",
"OA-FLOW.L30.6",
"OA-FLOW.CORE.8",
"OA-FLOW.L30.8",
"OA-FLOW.L30.7",
"OA-FLOW.L18.6",
"OA-FLOW.PC.5",
"OA-FLOW.CORE.7",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.CDEC.4",
"OA-FLOW.CDEC.5",
"OA-FLOW.CDEC.6",
"OA-FLOW.CDEC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 779,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 702,
"through_line": 779,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.CDEC.CENTRAL-ORBIT",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-central-orbit",
"start_line": 780,
"end_line": 839,
"range_sha256": "58eb44978c8056aa2130e27d76aec8fa344d0ef3aa7317931f57e226f5c9abf3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.1",
"OA-FLOW.NR.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 839,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 780,
"through_line": 797,
"description": "Section 9 assumes a point-ultraweakly continuous centrally ergodic real action. Section 10 assumes the crossed product is a type III factor and e is a nonzero projection of the coefficient center. No separability or sigma-finiteness is assumed."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 780,
"through_line": 839,
"description": "Complete local proof, including rational orbit coverage, central disjointization, countable orthogonal partial-isometry sums, exact corner isomorphism and the noncentral counterexample."
}
]
},
{
"id": "OA-FLOW.CDEC.FULL-CENTRAL-CORNER",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-full-central-corner",
"start_line": 840,
"end_line": 913,
"range_sha256": "559ac69e5ceb34e22e37168d292b807c2f2f618fbc741f67f2665417f5ab5e64",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CDEC.CENTRAL-ORBIT",
"OA-FLOW.PC.1",
"OA-FLOW.PC.5",
"OA-FLOW.CDEC.2",
"OA-FLOW.NF.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 913,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 780,
"through_line": 797,
"description": "Section 9 assumes a point-ultraweakly continuous centrally ergodic real action. Section 10 assumes the crossed product is a type III factor and e is a nonzero projection of the coefficient center. No separability or sigma-finiteness is assumed."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 840,
"through_line": 913,
"description": "Complete local proof, including rational orbit coverage, central disjointization, countable orthogonal partial-isometry sums, exact corner isomorphism and the noncentral counterexample."
}
]
},
{
"id": "OA-FLOW.CDEC.READING",
"unit": "OA-FLOW-CDEC",
"source_file": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e",
"reader": "OA-FLOW-CDEC.html#cdec-reading",
"start_line": 914,
"end_line": 921,
"range_sha256": "d43b9c01a4eba0bc88250aa342b4d926d0c7d473911a88c5e4509f9a46ca5bcf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.5",
"OA-FLOW.PC.8",
"OA-FLOW.CORE.7",
"OA-FLOW.L30.7",
"OA-FLOW.L30.6",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.CDEC.3",
"OA-FLOW.CDEC.4",
"OA-FLOW.CDEC.5",
"OA-FLOW.CDEC.6",
"OA-FLOW.CDEC.7",
"OA-FLOW.CDEC.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CDEC.md",
"line": 1,
"through_line": 25,
"description": "Arbitrary nonzero von Neumann algebras, no factor/state/countability/separability assumption. Properly infinite absorption and type III corner equivalence; faithful normal semifinite trace-scaling systems with exp(-s) law and type III output, full continuous-decomposition existence and specified-trace system conjugacy. K=L2(real,dr), core dt and dual ds/(2pi). Model hypotheses local."
},
{
"source": "src/OA-FLOW-CDEC.md",
"line": 914,
"through_line": 921,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.SETTING",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-setting",
"start_line": 1,
"end_line": 23,
"range_sha256": "ee25bfab7ef48a03ceda26a281949175bafca73fc03023082cf3eb4c5fa30e27",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.1",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-1",
"start_line": 24,
"end_line": 87,
"range_sha256": "369a2199f18c11c7f407ac891071cc85f60c2a2c01b4dd830035d110ffe7a62c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FR.1",
"OA-FLOW.NWR.4",
"OA-FLOW.CZ.5",
"OA-FLOW.RF.5",
"OA-FLOW.CORE.5",
"OA-FLOW.BC.1",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 87,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 24,
"through_line": 87,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.2",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-2",
"start_line": 88,
"end_line": 127,
"range_sha256": "d2b35c80b48d6b5ded715439f6d8024e2a463cb7671d2d284d7983db0fbaa6bc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TD.2",
"OA-FLOW.EP.5",
"OA-FLOW.CORE.2",
"OA-FLOW.NWR.4",
"OA-FLOW.TD.6",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.SCW.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 127,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 88,
"through_line": 127,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.3",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-3",
"start_line": 128,
"end_line": 166,
"range_sha256": "86438d588b1a018090f80d692251c29a8a23a66b96e91cbcc0c845d08a3e2c21",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.4",
"OA-FLOW.CZ.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 166,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 128,
"through_line": 166,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.4",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-4",
"start_line": 167,
"end_line": 455,
"range_sha256": "3c6abb67ae1eec9ee5270b67d377754cf43bdaf92e859672b72d144a3cc15ebd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NC.4",
"OA-FLOW.PC.1",
"OA-FLOW.PC.4",
"OA-FLOW.CP.6",
"OA-FLOW.RF.5",
"OA-FLOW.TD.2",
"OA-FLOW.PC.2",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 455,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 167,
"through_line": 455,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.5",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-5",
"start_line": 456,
"end_line": 598,
"range_sha256": "36fc3fa36fde7d98431fb9a1570ec581fa9f3d60a47d3dd2984d108479ad17ab",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L31.3",
"OA-FLOW.L31.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.DA.FIXED",
"OA-FLOW.TD.2",
"OA-FLOW.TD.6",
"OA-FLOW.NWR.3",
"OA-FLOW.NWR.6",
"OA-FLOW.L32.1",
"OA-FLOW.TD.5",
"OA-FLOW.CORE.7",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3",
"OA-FLOW.SCW.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 598,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 456,
"through_line": 598,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.6",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-6",
"start_line": 599,
"end_line": 693,
"range_sha256": "5c7270b3a942beab6bc02e78a76460296e4d1957ab3b81695040797977c9400a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.6",
"OA-FLOW.EP.5",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3",
"OA-FLOW.SCW.4",
"OA-FLOW.SCW.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 693,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 599,
"through_line": 693,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.7",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-7",
"start_line": 694,
"end_line": 801,
"range_sha256": "fa92358a11c818f1af1764751170361c083d84d331c6032ed988e0ad5811353a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.9",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3",
"OA-FLOW.SCW.4",
"OA-FLOW.SCW.5",
"OA-FLOW.SCW.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 801,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 694,
"through_line": 801,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.8",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-8",
"start_line": 802,
"end_line": 920,
"range_sha256": "febd68dd4159a87f395f9c33e28dd82aac2e955286a224298ffdd5a0c5612652",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.6",
"OA-FLOW.CORE.10",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3",
"OA-FLOW.SCW.4",
"OA-FLOW.SCW.5",
"OA-FLOW.SCW.6",
"OA-FLOW.SCW.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 920,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 802,
"through_line": 920,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SCW.READING",
"unit": "OA-FLOW-SCW",
"source_file": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9",
"reader": "OA-FLOW-SCW.html#scw-reading",
"start_line": 921,
"end_line": 928,
"range_sha256": "0f8d534b1a10f65016eb3ff6d44974420d0149b12e2b42309ca8933fcfe1dfcf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.NWR.1",
"OA-FLOW.EP.5",
"OA-FLOW.SCW.1",
"OA-FLOW.SCW.2",
"OA-FLOW.SCW.3",
"OA-FLOW.SCW.4",
"OA-FLOW.SCW.5",
"OA-FLOW.SCW.6",
"OA-FLOW.SCW.7",
"OA-FLOW.SCW.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SCW.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero; arbitrary normal semifinite weights with support e, faithful reference weights, full intrinsic-core corners and whole-cone density/trace normalization, arbitrary-cardinal center transport, supported coordinate converse and naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-SCW.md",
"line": 921,
"through_line": 928,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.SETTING",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-setting",
"start_line": 1,
"end_line": 20,
"range_sha256": "a1c0ebaaa758b1ac07d4b9135037398689a6ab852b7c35e90519fc240ffc2ad2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.1",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-1",
"start_line": 21,
"end_line": 60,
"range_sha256": "f4eaef5cc97672db6f5a5fc02a6adc6a51917ea90bbaec2351075701cd63d192",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 60,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 21,
"through_line": 60,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.2",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-2",
"start_line": 61,
"end_line": 129,
"range_sha256": "10bb553b5ef883b85d4a1b5803a75f5ab745da9857ee001fb3650859bfd11b19",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.4",
"OA-FLOW.CORE.5",
"OA-FLOW.BC.5",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 129,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 61,
"through_line": 129,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.3",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-3",
"start_line": 130,
"end_line": 195,
"range_sha256": "3cb781023bb830811951ccde84a6c32cabaf5f4f680b1e5a8b4f60ce23bfdf33",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 195,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 130,
"through_line": 195,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.4",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-4",
"start_line": 196,
"end_line": 267,
"range_sha256": "281f7a8338694ae7e939a6fd9703cfa47bbf1d91cd9df925451cfb47704b0478",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.2",
"OA-FLOW.CP.6",
"OA-FLOW.BC.4",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 267,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 196,
"through_line": 267,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.5",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-5",
"start_line": 268,
"end_line": 384,
"range_sha256": "013d7f2d1188ed4a2f50eea0eab9e202768dbc1b954143c38c377545c995ce4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 384,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 268,
"through_line": 384,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.6",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-6",
"start_line": 385,
"end_line": 539,
"range_sha256": "3a9796e30e4985cfd1e0cd4cd3e513fd6f90d34773c264e233bab6475d0162c2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.5",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.4",
"OA-FLOW.GRD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 539,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 385,
"through_line": 539,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.7",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-7",
"start_line": 540,
"end_line": 659,
"range_sha256": "998f7d9ed5feb8e43092ba86a341e75e92926307f4b2f8279d31ec9346f3b4cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CZ.6",
"OA-FLOW.CZ.5",
"OA-FLOW.BC.5",
"OA-FLOW.CORE.11",
"OA-FLOW.BC.6",
"OA-FLOW.CORE.10",
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.4",
"OA-FLOW.GRD.5",
"OA-FLOW.GRD.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 659,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 540,
"through_line": 659,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.8",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-8",
"start_line": 660,
"end_line": 739,
"range_sha256": "a3a5ca4523b20c8ba2d2ee1ff63dc0dac42c5993c077d40500510ff182e236db",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.7",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.4",
"OA-FLOW.GRD.5",
"OA-FLOW.GRD.6",
"OA-FLOW.GRD.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 739,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 660,
"through_line": 739,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.GRD.READING",
"unit": "OA-FLOW-GRD",
"source_file": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31",
"reader": "OA-FLOW-GRD.html#grd-reading",
"start_line": 740,
"end_line": 747,
"range_sha256": "b70092c52b3b0a8f3d011805695a5f91170f0c5087f1ce45fd8b040d90a4b3db",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.DA.FIXED",
"OA-FLOW.FR.1",
"OA-FLOW.SCW.1",
"OA-FLOW.CF.3",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.1",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.4",
"OA-FLOW.GRD.5",
"OA-FLOW.GRD.6",
"OA-FLOW.GRD.7",
"OA-FLOW.GRD.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GRD.md",
"line": 1,
"through_line": 20,
"description": "Arbitrary von Neumann algebra M including zero; faithful normal semifinite weight charts, canonical dual Banach graded fibers, ordered product and star, bounded operator topologies and full Mackey topology, bounded spectral realization and normal-isomorphism naturality. No factor, countability, separability or faithful-state assumption; model hypotheses local."
},
{
"source": "src/OA-FLOW-GRD.md",
"line": 740,
"through_line": 747,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.SETTING",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-setting",
"start_line": 1,
"end_line": 23,
"range_sha256": "9a80ea47228cdab44b057599e1b338c915e1a5f9012a2ed082253c7c2985716a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.5",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.1",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-1",
"start_line": 24,
"end_line": 97,
"range_sha256": "f07bd3b81e11dd0bdf32337daa087b579dd12a61814530bf67c6b51e1df3aa02",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.5",
"OA-FLOW.CP.6",
"OA-FLOW.L34.3",
"OA-FLOW.GRD.4",
"OA-FLOW.SC.4",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.SC.2",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 97,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 24,
"through_line": 97,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.2",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-2",
"start_line": 98,
"end_line": 150,
"range_sha256": "0ba37d7d9323452bf83db2286106c94b1b89eb1c561e6941a642a65bf97c2fdf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.5",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1",
"OA-FLOW.SEC.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 150,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 98,
"through_line": 150,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.3",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-3",
"start_line": 151,
"end_line": 302,
"range_sha256": "7778f6a75e921d150d138d9eeef1a8b1b5134495a890e38423e9c70f50c7928d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.CP.6",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.MW.4",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1",
"OA-FLOW.FR.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 302,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 151,
"through_line": 302,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.4",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-4",
"start_line": 303,
"end_line": 455,
"range_sha256": "cc22d272b903f93b5229cf74bdfa1b1bb3dd6d4b428f440a904f6170531e2694",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.3",
"OA-FLOW.MW.4",
"OA-FLOW.ST.2",
"OA-FLOW.GRD.5",
"OA-FLOW.AT.4",
"OA-FLOW.FF.1",
"OA-FLOW.SC.2",
"OA-FLOW.FF.2",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.SC.4",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.SC.5",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 455,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 303,
"through_line": 455,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.5",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-5",
"start_line": 456,
"end_line": 598,
"range_sha256": "2cf5e2013b9e693fbbb04e1e7fd1d453dd4918f41b1afd0d3feb1d99e6955f13",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.3",
"OA-FLOW.FF.1",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.5",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3",
"OA-FLOW.SEC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 598,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 456,
"through_line": 598,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.6",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-6",
"start_line": 599,
"end_line": 715,
"range_sha256": "a3fe3daf18657add784e9aef5094fc990e6a0ace5f28dc7a8963e85889ac9e29",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.5",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.2",
"OA-FLOW.GRD.6",
"OA-FLOW.FR.1",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3",
"OA-FLOW.SEC.4",
"OA-FLOW.SEC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 715,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 599,
"through_line": 715,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.7",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-7",
"start_line": 716,
"end_line": 828,
"range_sha256": "6230d3efd9f0a0e2561ffc43250847ba1b6c792afb2bdac0f557729ff4de58dd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.KT.5",
"OA-FLOW.GRD.5",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.FF.1",
"OA-FLOW.FF.2",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3",
"OA-FLOW.SEC.4",
"OA-FLOW.SEC.5",
"OA-FLOW.SEC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 828,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 716,
"through_line": 828,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.8",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-8",
"start_line": 829,
"end_line": 921,
"range_sha256": "e4ed517b9b5213a65776b283ef5cf72c7301acc36b11db1a05fbd4eda73fc3ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.6",
"OA-FLOW.GRD.7",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.5",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3",
"OA-FLOW.SEC.4",
"OA-FLOW.SEC.5",
"OA-FLOW.SEC.6",
"OA-FLOW.SEC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 921,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 829,
"through_line": 921,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SEC.READING",
"unit": "OA-FLOW-SEC",
"source_file": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529",
"reader": "OA-FLOW-SEC.html#sec-reading",
"start_line": 922,
"end_line": 925,
"range_sha256": "714dc8e8060db695532685915197dcb3b54d7c5d6339ccdd269da2c16f5c3c6f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.5",
"OA-FLOW.SC.2",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.FF.2",
"OA-FLOW.FF.1",
"OA-FLOW.SEC.1",
"OA-FLOW.SEC.2",
"OA-FLOW.SEC.3",
"OA-FLOW.SEC.4",
"OA-FLOW.SEC.5",
"OA-FLOW.SEC.6",
"OA-FLOW.SEC.7",
"OA-FLOW.SEC.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SEC.md",
"line": 1,
"through_line": 23,
"description": "Arbitrary von Neumann algebra M including zero, faithful normal semifinite weight chart, locally Mackey-Lusin graded sections on R with completed Lebesgue dt and integrable scalar norm, modulo common almost-everywhere equality. Complete weak-star convolution algebra and normal-isomorphism naturality; arbitrary faithful Hilbert representation, no separability or norm-Bochner assumption. Model hypotheses local."
},
{
"source": "src/OA-FLOW-SEC.md",
"line": 922,
"through_line": 925,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.SETTING",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-setting",
"start_line": 1,
"end_line": 31,
"range_sha256": "b4d493d8d31b6297175903174f55e04ca4cf59a2c390660eb7580bb0cf0aadbc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.1",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-1",
"start_line": 32,
"end_line": 112,
"range_sha256": "07ffc37b8dee46959a846b368bb0ed9fc5b0384009aac1a07613f5b4b4ae7434",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 112,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 32,
"through_line": 112,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.2",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-2",
"start_line": 113,
"end_line": 170,
"range_sha256": "34a84ad5e974621242c914994efc6c806e1c1ce119879b237a5a5f1109432f03",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 170,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 113,
"through_line": 170,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.3",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-3",
"start_line": 171,
"end_line": 314,
"range_sha256": "088130e26548d8436db326de82a99ea933d551f2bdd8c4fd07cc157800b4a1b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.3",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 314,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 171,
"through_line": 314,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.4",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-4",
"start_line": 315,
"end_line": 476,
"range_sha256": "c28f7a3b0e727eb89cfb78fa8d22716d6eaaec78152f54c64c8cdaeb272ddf7b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.FF.2",
"OA-FLOW.SEC.4",
"OA-FLOW.SEC.1",
"OA-FLOW.AT.4",
"OA-FLOW.FF.1",
"OA-FLOW.SEC.3",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 476,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 315,
"through_line": 476,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.5",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-5",
"start_line": 477,
"end_line": 580,
"range_sha256": "136ae9fe3531a192bae21b2107534c65a3a3cfd6b84ae376e9b9a17b8b5a5539",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRD.3",
"OA-FLOW.AT.4",
"OA-FLOW.GRD.5",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 580,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 477,
"through_line": 580,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.6",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-6",
"start_line": 581,
"end_line": 738,
"range_sha256": "738522debafec9420e93f49539aa3cbad6751a680991148020a6230b962ee48b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.ST.2",
"OA-FLOW.SEC.3",
"OA-FLOW.CP.5",
"OA-FLOW.CCM.7",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 738,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 581,
"through_line": 738,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.7",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-7",
"start_line": 739,
"end_line": 920,
"range_sha256": "73a20c174a3372bd6d221988baa261066fc3acb18b27112ad9843e0855417a34",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.SQUARES",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.DA.DOMAINS",
"OA-FLOW.GDA.8",
"OA-FLOW.GDW.7",
"OA-FLOW.CORE.7",
"OA-FLOW.NR.4",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.2",
"OA-FLOW.CORE.3",
"OA-FLOW.CORE.4",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5",
"OA-FLOW.HLB.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 920,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 739,
"through_line": 920,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.8",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-8",
"start_line": 921,
"end_line": 1195,
"range_sha256": "9a68e61719a055c7fd64adef39f41536a32aefe128950c41f83cbf48413909fd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.5",
"OA-FLOW.GRD.6",
"OA-FLOW.SEC.6",
"OA-FLOW.CORE.8",
"OA-FLOW.CORE.6",
"OA-FLOW.CORE.1",
"OA-FLOW.CORE.2",
"OA-FLOW.TD.4",
"OA-FLOW.TD.2",
"OA-FLOW.TD.6",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5",
"OA-FLOW.HLB.6",
"OA-FLOW.HLB.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1195,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 921,
"through_line": 1195,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.9",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-9",
"start_line": 1196,
"end_line": 1359,
"range_sha256": "37794799268873a2f0fede8a84bafa50da4d126b773cc727d20e5b3f947ff99b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.3",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.CORE.9",
"OA-FLOW.BC.6",
"OA-FLOW.CORE.10",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5",
"OA-FLOW.HLB.6",
"OA-FLOW.HLB.7",
"OA-FLOW.HLB.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1359,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 1196,
"through_line": 1359,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.10",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-10",
"start_line": 1360,
"end_line": 1424,
"range_sha256": "0fee5bade1dbd06a23a0fe5c310e458a2c3f5bbbb5299c529515f2d642c03710",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SEC.3",
"OA-FLOW.GRD.2",
"OA-FLOW.SEC.5",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5",
"OA-FLOW.HLB.6",
"OA-FLOW.HLB.7",
"OA-FLOW.HLB.8",
"OA-FLOW.HLB.9"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1424,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 1360,
"through_line": 1424,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.HLB.READING",
"unit": "OA-FLOW-HLB",
"source_file": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a",
"reader": "OA-FLOW-HLB.html#hlb-reading",
"start_line": 1425,
"end_line": 1430,
"range_sha256": "1ca1848d2a04fa5377045ed3e9b6dc3e2aa92f31858dd3e401a67b8e02ecc732",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SEC.5",
"OA-FLOW.CCM.7",
"OA-FLOW.CORE.8",
"OA-FLOW.GRD.2",
"OA-FLOW.FR.1",
"OA-FLOW.BC.4",
"OA-FLOW.HLB.1",
"OA-FLOW.HLB.2",
"OA-FLOW.HLB.3",
"OA-FLOW.HLB.4",
"OA-FLOW.HLB.5",
"OA-FLOW.HLB.6",
"OA-FLOW.HLB.7",
"OA-FLOW.HLB.8",
"OA-FLOW.HLB.9",
"OA-FLOW.HLB.10"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-HLB.md",
"line": 1,
"through_line": 31,
"description": "Arbitrary faithful normal unital representation M in B(H), N=(M commutant)op with complex-linear ordinary opposite; faithful normal semifinite weight charts and their spatial derivatives, full Hilbert fibers and L2 sections using completed Lebesgue dt. Both typed actions, full regular commutants, normalized dual weights/core, and specified-unitary represented functor. No factor, countability, separability or faithful-state assumption; zero case immediate and model hypotheses local."
},
{
"source": "src/OA-FLOW-HLB.md",
"line": 1425,
"through_line": 1430,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.SETTING",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-setting",
"start_line": 1,
"end_line": 20,
"range_sha256": "91d8788fc10f2ac19d696971fa207393e0c6923a5c6bb322bc9f5b4af5cabbd4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.1",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-1",
"start_line": 21,
"end_line": 297,
"range_sha256": "09601603f794d4d29b99f11f078c13a43ab16c22a81a76909660fb657e524f5e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7",
"OA-FLOW.CF.8",
"OA-FLOW.CV.1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 297,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 21,
"through_line": 297,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.2",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-2",
"start_line": 298,
"end_line": 371,
"range_sha256": "77ba85c549e5c71866c1c7c9f6c79ca45df895c1eabff6ee4a97d4ab8a3eb820",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.SC.7",
"OA-FLOW.REL.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 371,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 298,
"through_line": 371,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.3",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-3",
"start_line": 372,
"end_line": 530,
"range_sha256": "c7cb483f38768da059f7b98eb2e9d3edca20f13f7a80f4100a58f332843326cc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 530,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 372,
"through_line": 530,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.4",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-4",
"start_line": 531,
"end_line": 620,
"range_sha256": "272c82efb73ece4c042fea010807f11fdbfddb501c1950caf02aa684e02df665",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 620,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 531,
"through_line": 620,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.5",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-5",
"start_line": 621,
"end_line": 886,
"range_sha256": "0601155347aae2bf7ca381385927fa09d193fe3afe4a38958f6b625ffa12fecd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GW.4",
"OA-FLOW.GW.1",
"OA-FLOW.GW.3",
"OA-FLOW.SF.SF1",
"OA-FLOW.CI.1",
"OA-FLOW.CI.3",
"OA-FLOW.MW.4",
"OA-FLOW.WR.3",
"OA-FLOW.WR.5",
"OA-FLOW.MF06.8",
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 886,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 621,
"through_line": 886,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.6",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-6",
"start_line": 887,
"end_line": 1076,
"range_sha256": "7cf426f79f3b38ce1e93ce0c15b59eb8f8a240618b40d5e7183e28e5208fcc8d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.6",
"OA-FLOW.BC.6",
"OA-FLOW.GW.4",
"OA-FLOW.TD.6",
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.4",
"OA-FLOW.REL.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1076,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 887,
"through_line": 1076,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.7",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-7",
"start_line": 1077,
"end_line": 1175,
"range_sha256": "9c8699389378ecb720943fc13a98b2e5c2cd091e00fde3d11c1211e94675fc6f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.4",
"OA-FLOW.REL.5",
"OA-FLOW.REL.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1175,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 1077,
"through_line": 1175,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.REL.READING",
"unit": "OA-FLOW-REL",
"source_file": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f",
"reader": "OA-FLOW-REL.html#rel-reading",
"start_line": 1176,
"end_line": 1181,
"range_sha256": "4097e8d3ac8d0d8bfcef8edbdff86fa591dc9364040724d2f8c3c713567c16c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.7",
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.4",
"OA-FLOW.REL.5",
"OA-FLOW.REL.6",
"OA-FLOW.REL.7"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-REL.md",
"line": 1,
"through_line": 20,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Distinct orbit points index fibers; right counting integrates the second coordinate, left counting the first; delta=d nu_l/d nu_r on a common invariant conull set. Whole orbit algebra, diagonal expectation, GNS/Tomita graph domains and modular group. Zero measure has the zero representation; model hypotheses are local."
},
{
"source": "src/OA-FLOW-REL.md",
"line": 1176,
"through_line": 1181,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.SETTING",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-setting",
"start_line": 1,
"end_line": 23,
"range_sha256": "e56ef28727deca083d6c6f7912781e0bfb83413b96f056c7be9590dd4a48adc7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.1",
"OA-FLOW.REL.5"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.1",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-1",
"start_line": 24,
"end_line": 120,
"range_sha256": "0cf2cfe1dd5d229d364acdf4e0fb025a1decb2a1fb210ea23778495a7472c74a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.1",
"OA-FLOW.FF.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 120,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 24,
"through_line": 120,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.2",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-2",
"start_line": 121,
"end_line": 215,
"range_sha256": "0e7d1ede35b6ae9daa0232fa3e675f9e763c275feea919c6b3333a670b02c6d9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.4",
"OA-FLOW.REL.1",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 215,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 121,
"through_line": 215,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.3",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-3",
"start_line": 216,
"end_line": 353,
"range_sha256": "215905edb695a7dc4efb954f3e97591b98b860036b50f1aae0e3efdcc5eb1b25",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.FF.2",
"OA-FLOW.CP.6",
"OA-FLOW.REL.1",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 353,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 216,
"through_line": 353,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.4",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-4",
"start_line": 354,
"end_line": 529,
"range_sha256": "978f9c14d264d83e6b7b05bdbbeb7b1624443884b9600696f9d7a6667922acc3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.DA.ACTION",
"OA-FLOW.FF.3",
"OA-FLOW.REL.5",
"OA-FLOW.SF.SF1",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.CORE.1",
"OA-FLOW.REL.1",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.SF.SB0",
"OA-FLOW.SC.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 529,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 354,
"through_line": 529,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.5",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-5",
"start_line": 530,
"end_line": 593,
"range_sha256": "b5f0ff29540fa1f556104b4ecf9e17756b54471401b41fa0f7efc7b7c8eec5b9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.8",
"OA-FLOW.DA.AVERAGE",
"OA-FLOW.DA.EQUALITY",
"OA-FLOW.CORE.2",
"OA-FLOW.CZ.2",
"OA-FLOW.L18.2",
"OA-FLOW.KERNEL.INTEGRATION",
"OA-FLOW.REL.1",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 593,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 530,
"through_line": 593,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.6",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-6",
"start_line": 594,
"end_line": 835,
"range_sha256": "81be69c5cf966d4ce713e051f3f24dc80f72ef8c996e725f2481bde3ad7a70ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.2",
"OA-FLOW.REL.1",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.FF.3",
"OA-FLOW.REL.5",
"OA-FLOW.NR.4",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4",
"OA-FLOW.LFT.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 835,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 594,
"through_line": 835,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.7",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-7",
"start_line": 836,
"end_line": 1052,
"range_sha256": "d20aa7962d3f5902a4822c9084d89d616f0654a37532cfd5e185b8d106e8bd10",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.6",
"OA-FLOW.FF.2",
"OA-FLOW.REL.4",
"OA-FLOW.REL.5",
"OA-FLOW.TD.2",
"OA-FLOW.REL.1",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4",
"OA-FLOW.LFT.5",
"OA-FLOW.LFT.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1052,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 836,
"through_line": 1052,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.8",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-8",
"start_line": 1053,
"end_line": 1142,
"range_sha256": "f986e2c182c60a24734477670fb59dc59a0347e4fd25d8faf50985163e9588dd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.3",
"OA-FLOW.REL.1",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4",
"OA-FLOW.LFT.5",
"OA-FLOW.LFT.6",
"OA-FLOW.LFT.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1142,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 1053,
"through_line": 1142,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.LFT.MEASURE-CHANGE",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-measure-change",
"start_line": 1144,
"end_line": 1354,
"range_sha256": "2a14c4607d34d524e42610d76204508522e6979ec4e211a9e3705b2201c42bb4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.1",
"OA-FLOW.REL.2",
"OA-FLOW.REL.3",
"OA-FLOW.REL.4",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4",
"OA-FLOW.LFT.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1354,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1144,
"through_line": 1354,
"description": "Finite or countably infinite groups act nonsingularly on a standard sigma-finite measure space. The replacement measure is equivalent and sigma-finite, with a strictly positive finite Borel density after a common invariant conull restriction. Neither density nor reciprocal need be bounded. Nonergodic actions and stabilizers are allowed. All trace identities include infinite values, and the transported positive generator has its full multiplication domain."
}
]
},
{
"id": "OA-FLOW.LFT.READING",
"unit": "OA-FLOW-LFT",
"source_file": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7",
"reader": "OA-FLOW-LFT.html#lft-reading",
"start_line": 1355,
"end_line": 1362,
"range_sha256": "57c1e3e2c1eb8791342d4aaf60520efb93c1776ede72ea9a71b33e531b381ec2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.2",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.REL.1",
"OA-FLOW.REL.5",
"OA-FLOW.LFT.1",
"OA-FLOW.LFT.2",
"OA-FLOW.LFT.3",
"OA-FLOW.LFT.4",
"OA-FLOW.LFT.5",
"OA-FLOW.LFT.6",
"OA-FLOW.LFT.7",
"OA-FLOW.LFT.8"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-LFT.md",
"line": 1,
"through_line": 23,
"description": "Finite or countably infinite group acting by nonsingular Borel bijections on a standard sigma-finite measure space; stabilizers and nonergodic actions allowed. Lift on X times positive reals has base measure mu d lambda and cocycle delta=d nu_l/d nu_r. Full lifted algebra, basepoint trace, dilation fixed algebra, onto Fourier core, entire normalized trace, and separately defined continuous product representation. Group measure dt and dual measure ds/(2pi). Zero measure gives zero algebras; example hypotheses are local."
},
{
"source": "src/OA-FLOW-LFT.md",
"line": 1355,
"through_line": 1362,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.SETTING",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-setting",
"start_line": 1,
"end_line": 51,
"range_sha256": "e60a331f7c53701f590e55c9fd5315592b8fa00c8ef40580bd1e98365722f04a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.1",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-1",
"start_line": 52,
"end_line": 146,
"range_sha256": "a9351ae655ccb666b646104b5b8950295825cfc9ca614b2bf95274967f1e484e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L18.1",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.ND.ORBITS",
"OA-FLOW.CONNES.CENTERKERNEL",
"OA-FLOW.L115.CONVENTIONS",
"OA-FLOW.CDEC.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 146,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 52,
"through_line": 146,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.2",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-2",
"start_line": 147,
"end_line": 240,
"range_sha256": "2833a02823db921cd896dbb104faa57e26e877dfee0962332d4c149458ed7e2e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.BC.4",
"OA-FLOW.UR.0",
"OA-FLOW.UR.5",
"OA-FLOW.COCINT.REDUCTION",
"OA-FLOW.L120.TRANSPORT",
"OA-FLOW.FR.1",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 240,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 147,
"through_line": 240,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.3",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-3",
"start_line": 241,
"end_line": 322,
"range_sha256": "76c3790c0373faeb2f787ad95be1d4fbfa0ec494650479990c73b2dd4f4369db",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MT.1",
"OA-FLOW.MT.2",
"OA-FLOW.MT.3",
"OA-FLOW.MT.4",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1",
"OA-FLOW.MIV.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 322,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 241,
"through_line": 322,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.4",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-4",
"start_line": 323,
"end_line": 371,
"range_sha256": "9cbdb3cd6703be34040b56c43b7c7e14b6ae849cdeaf83691c4ad20dd900e8bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1",
"OA-FLOW.MIV.2",
"OA-FLOW.MIV.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 371,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 323,
"through_line": 371,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.5",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-5",
"start_line": 372,
"end_line": 443,
"range_sha256": "d10a19db6c5b205d39e647d1e99861aa6e3e089fc7d4c17519dd68622799ea0c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.4",
"OA-FLOW.CZ.0",
"OA-FLOW.PW.1",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1",
"OA-FLOW.MIV.2",
"OA-FLOW.MIV.3",
"OA-FLOW.MIV.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 443,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 372,
"through_line": 443,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.6",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-6",
"start_line": 444,
"end_line": 661,
"range_sha256": "7c38723a578d814e937544237f17de90ec66fa71c8bd756621212deadddd2a44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.LFT.7",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.FF.2",
"OA-FLOW.ND.WEYL",
"OA-FLOW.TD.1",
"OA-FLOW.MG.1",
"OA-FLOW.BC.4",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1",
"OA-FLOW.MIV.2",
"OA-FLOW.MIV.3",
"OA-FLOW.MIV.4",
"OA-FLOW.MIV.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 661,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 444,
"through_line": 661,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.7",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-7",
"start_line": 662,
"end_line": 734,
"range_sha256": "c9c54f25d3fea640c7f41b8c0fcb74b50bee9d5aa4c1e71f94236ef41d65dead",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.1",
"OA-FLOW.MIV.1",
"OA-FLOW.MIV.2",
"OA-FLOW.MIV.3",
"OA-FLOW.MIV.4",
"OA-FLOW.MIV.5",
"OA-FLOW.MIV.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 734,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1,
"through_line": 51,
"description": "N is an arbitrary nonzero von Neumann algebra with a point-ultraweak continuous real action theta and faithful normal semifinite trace tau satisfying tau theta_s=exp(-s) tau. P=N crossed_theta R is a type III factor, with no separability or sigma-finiteness assumption; C=Z(N) is centrally ergodic. Full S invariant intersects all faithful n.s.f. weights; T is the inner modular-time group. The general factor identity and individual type III zero-spectrum assertions state their broader hypotheses locally. Commutative-flow models give conditional type III consequences; the separately specified translation example is semifinite."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 662,
"through_line": 734,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.PRODUCT",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-product-times",
"start_line": 735,
"end_line": 858,
"range_sha256": "fd8f7dd0a2bd94f600917bbd4a9237c6218884f90b94e0d5e282233e774bf8ad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.MIV.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 858,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 735,
"through_line": 858,
"description": "Countable product of finite matrix factors with strictly positive normalized diagonal densities, represented in the faithful product-state GNS space. No assumption about factor type."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 735,
"through_line": 858,
"description": "Complete section statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.DISPERSION",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-product-dispersion",
"start_line": 859,
"end_line": 916,
"range_sha256": "c16fd7e808d7355f653dbd3763e8cb999852d3d1340255040c99b834cc6e90c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.MIV.PRODUCT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 916,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 735,
"through_line": 858,
"description": "Countable product of finite matrix factors with strictly positive normalized diagonal densities, represented in the faithful product-state GNS space. No assumption about factor type."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 859,
"through_line": 916,
"description": "Complete section statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.PRODUCTEXAMPLES",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-product-examples",
"start_line": 917,
"end_line": 1006,
"range_sha256": "9fefa387117b797dbaa65c25dc44aa3a25bb7829234d591f75af9092ade7ea82",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.MIV.6",
"OA-FLOW.MIV.PRODUCT",
"OA-FLOW.MIV.DISPERSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1006,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 735,
"through_line": 858,
"description": "Countable product of finite matrix factors with strictly positive normalized diagonal densities, represented in the faithful product-state GNS space. No assumption about factor type."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 917,
"through_line": 1006,
"description": "Complete section statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.MIV.COUNTABLESUPPORT",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-countable-support",
"start_line": 1007,
"end_line": 1042,
"range_sha256": "4dd5d19a99066665958519cf3571bff7d0918366cb9d9befbb7d1cb11b435650",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1042,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1007,
"through_line": 1014,
"description": "Arbitrary index set of finite matrix factors with faithful normalized diagonal states; full product representation and coordinate subproducts, including empty index set."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 1007,
"through_line": 1042,
"description": "Complete section hypotheses, statement, proof and examples."
}
]
},
{
"id": "OA-FLOW.MIV.ARBITRARYTIMES",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-arbitrary-times",
"start_line": 1043,
"end_line": 1101,
"range_sha256": "3b111cae945861e6f0aa5cfc6e8a99dcf5b705d49b409da7e98fc084b6d55948",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.COUNTABLESUPPORT",
"OA-FLOW.MIV.PRODUCT",
"OA-FLOW.MIV.DISPERSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1101,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1007,
"through_line": 1014,
"description": "Arbitrary index set of finite matrix factors with faithful normalized diagonal states; full product representation and coordinate subproducts, including empty index set."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 1043,
"through_line": 1101,
"description": "Complete section hypotheses, statement, proof and examples."
}
]
},
{
"id": "OA-FLOW.MIV.COUNTABLEDETECTION",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-countable-detection",
"start_line": 1102,
"end_line": 1176,
"range_sha256": "32d638bb6c4eb8ec5514417959ed5049054be9aaef4306fc03e024ed89cb8f6c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1",
"OA-FLOW.MIV.ARBITRARYTIMES",
"OA-FLOW.MIV.DISPERSION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1176,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1007,
"through_line": 1014,
"description": "Arbitrary index set of finite matrix factors with faithful normalized diagonal states; full product representation and coordinate subproducts, including empty index set."
},
{
"source": "src/OA-FLOW-MIV.md",
"line": 1102,
"through_line": 1176,
"description": "Complete section hypotheses, statement, proof and examples."
}
]
},
{
"id": "OA-FLOW.MIV.AP.ACTION",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-almost-periodic-action",
"start_line": 1177,
"end_line": 1233,
"range_sha256": "4b3be84aec0f2ae57ade12c535e5281cc272b5464b15e5cde6e23a466cef96a2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.WR.3",
"OA-FLOW.WR.5",
"OA-FLOW.MF06.POLAR",
"OA-FLOW.MW.4",
"OA-FLOW.SF.SF1",
"OA-FLOW.CF.4",
"OA-FLOW.CF.5",
"OA-FLOW.XGAPS.CSTAR.H3",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GNS.7.1",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1233,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1177,
"through_line": 1233,
"description": "A specified faithful normal almost-periodic state on an arbitrary von Neumann algebra; pure point modular decomposition. No factoriality or separability, except in the separately stated separable-predual type III amplification corollary."
}
]
},
{
"id": "OA-FLOW.MIV.AP.TRACE",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-almost-periodic-trace",
"start_line": 1234,
"end_line": 1312,
"range_sha256": "7097f959ac829e0c4275b7e8063785c9f4b0d3b5001642a19cc2bfb57c9aafca",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.AP.ACTION",
"OA-FLOW.NR.3",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1312,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1177,
"through_line": 1233,
"description": "A specified faithful normal almost-periodic state on an arbitrary von Neumann algebra; pure point modular decomposition. No factoriality or separability, except in the separately stated separable-predual type III amplification corollary."
}
]
},
{
"id": "OA-FLOW.MIV.AP.RECOVERY",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-almost-periodic-recovery",
"start_line": 1313,
"end_line": 1366,
"range_sha256": "9aabef19c3f8ca067b36bc910c9b70fcb56f693d06163cf94dc856aeb8702c73",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.AP.ACTION",
"OA-FLOW.MIV.AP.TRACE",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.ST.1",
"OA-FLOW.CP.6",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1366,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1177,
"through_line": 1233,
"description": "A specified faithful normal almost-periodic state on an arbitrary von Neumann algebra; pure point modular decomposition. No factoriality or separability, except in the separately stated separable-predual type III amplification corollary."
}
]
},
{
"id": "OA-FLOW.MIV.AP.EXAMPLES",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-almost-periodic-examples",
"start_line": 1367,
"end_line": 1408,
"range_sha256": "8734cd766f5fb427373b0bff04931e35bf62af3f2d5ae443d6063c04213404e6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.AP.ACTION",
"OA-FLOW.MIV.AP.TRACE",
"OA-FLOW.MIV.AP.RECOVERY",
"OA-FLOW.MIV.PRODUCT",
"OA-FLOW.MIV.ARBITRARYTIMES",
"OA-FLOW.MIV.COUNTABLEDETECTION",
"OA-FLOW.NR.4",
"OA-FLOW.MW.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1408,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1177,
"through_line": 1233,
"description": "A specified faithful normal almost-periodic state on an arbitrary von Neumann algebra; pure point modular decomposition. No factoriality or separability, except in the separately stated separable-predual type III amplification corollary."
}
]
},
{
"id": "OA-FLOW.MIV.AP.FACTOR",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-extremal-core",
"start_line": 1409,
"end_line": 1494,
"range_sha256": "5b3d3dd5f7f690e811814765c63927736e2094005326af1ded79fcbdd9035058",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.AP.ACTION",
"OA-FLOW.MIV.AP.TRACE",
"OA-FLOW.MIV.AP.RECOVERY",
"OA-FLOW.CZ.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1494,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1409,
"through_line": 1494,
"description": "A specified faithful normal almost-periodic state with factorial centralizer, on an arbitrary von Neumann algebra. Type II infinity and the type III dichotomy additionally assume that the algebra is type III; the integer crossed-product and generating-isometry conclusions additionally assume a nonzero discrete logarithmic eigenvalue group. No separable-predual hypothesis."
}
]
},
{
"id": "OA-FLOW.MIV.AP.TYPE",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-extremal-spectrum",
"start_line": 1495,
"end_line": 1588,
"range_sha256": "9441c41c9bad619fb5c6b8a2660ea6bc9cb5e97ff7b69568ed85736faac71430",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.AP.FACTOR",
"OA-FLOW.MIV.AP.ACTION",
"OA-FLOW.MIV.AP.TRACE",
"OA-FLOW.MIV.AP.RECOVERY",
"OA-FLOW.MIV.2",
"OA-FLOW.MIV.3",
"OA-FLOW.MIV.4",
"OA-FLOW.MG.1",
"OA-FLOW.CDEC.1",
"OA-FLOW.CDEC.2",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.CF.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1588,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1409,
"through_line": 1494,
"description": "A specified faithful normal almost-periodic state with factorial centralizer, on an arbitrary von Neumann algebra. Type II infinity and the type III dichotomy additionally assume that the algebra is type III; the integer crossed-product and generating-isometry conclusions additionally assume a nonzero discrete logarithmic eigenvalue group. No separable-predual hypothesis."
}
]
},
{
"id": "OA-FLOW.MIV.TENSOR.TIMES",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-tensor-times",
"start_line": 1589,
"end_line": 1617,
"range_sha256": "7f43fe286593c135d2de26e31d6ff49974f833c9f36d479cc5e5edd0d8b79041",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.5",
"OA-FLOW.FR.1"
],
"external_proof_links": [
"../../OA-MOD/OA-MOD-TG.html#oa-mod-tg-04",
"../../OA-APPROX/infinite-tensor-products.html#lemma-7-1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1617,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1589,
"through_line": 1617,
"description": "Nonzero arbitrary factors with faithful normal semifinite reference weights. No separability or sigma-finiteness requirement."
}
]
},
{
"id": "OA-FLOW.MIV.PERIODIC.STATE",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-periodic-realization",
"start_line": 1618,
"end_line": 1747,
"range_sha256": "44b3aaf2f7627eec620bddaf9636f462747273be0f9fdb870335d0702b2e7efd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.5",
"OA-FLOW.MIV.ARBITRARYTIMES",
"OA-FLOW.MIV.DISPERSION",
"OA-FLOW.MIV.COUNTABLESUPPORT",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8"
],
"external_proof_links": [
"../../OA-APPROX/infinite-tensor-products.html#proposition-5-2",
"../../OA-APPROX/infinite-tensor-products.html#theorem-6-1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1747,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1618,
"through_line": 1747,
"description": "An arbitrary index set of finite matrix algebras with strictly positive normalized diagonal densities and their faithful pointed product representation; P>0. The equivalent conditions and all summability and lattice qualifications are in the complete section."
}
]
},
{
"id": "OA-FLOW.MIV.ABSORPTION.TIMES",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-absorption-periods",
"start_line": 1748,
"end_line": 1824,
"range_sha256": "ce17d0568006d0c8040ea3b9f6e930608e209fa1d1ba131151b7316a58057afd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.TENSOR.TIMES",
"OA-FLOW.MIV.PERIODIC.STATE",
"OA-FLOW.MIV.PRODUCT",
"OA-FLOW.MIV.PRODUCTEXAMPLES",
"OA-FLOW.MIV.5"
],
"external_proof_links": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1824,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1748,
"through_line": 1824,
"description": "A nonzero arbitrary factor for the absorption obstruction, 00, and the all-faithful-normal-semifinite-weight definition of the positive intrinsic spectrum."
}
]
},
{
"id": "OA-FLOW.MIV.BLOCK.RATIOS",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-block-ratios",
"start_line": 1985,
"end_line": 2079,
"range_sha256": "63f71d837c366fdfbcfc9e908523bbfe29115c4e2d56c1c2fff8b4fceb6e1774",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.CORNER.RATIOS",
"OA-FLOW.MIV.PRODUCT.RELATION",
"OA-FLOW.MIV.3"
],
"external_proof_links": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 2079,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 1985,
"through_line": 2079,
"description": "Exactly the countable faithful finite-matrix product of Section23. The block equivalence is for 01. The endpoint discussion and constant two-level example retain their stated hypotheses."
}
]
},
{
"id": "OA-FLOW.MIV.LATTICE.CORNER",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-lattice-corner",
"start_line": 2080,
"end_line": 2162,
"range_sha256": "bcdbf9aed32af21dadf20abd9f2dbc86629d5202272f15a26d59fad54ab11808",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.PRODUCT"
],
"external_proof_links": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 2162,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 2080,
"through_line": 2162,
"description": "Fixed 00, r not equal to 1."
}
]
},
{
"id": "OA-FLOW.MIV.TYPEI.ABSORPTION",
"unit": "OA-FLOW-MIV",
"source_file": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80",
"reader": "OA-FLOW-MIV.html#miv-typei-absorption",
"start_line": 2554,
"end_line": 2632,
"range_sha256": "6e7b40ba9cd5118b642491f7442ae49d5f5b7530ce0745dde2cc20c7db4b0e4f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MIV.TYPEI.CORNER",
"OA-FLOW.MIV.CORNER.INVARIANTS",
"OA-FLOW.MIV.ABSORPTION.CHARACTERIZATION",
"OA-FLOW.PW.1",
"OA-FLOW.PW.2",
"OA-FLOW.PW.3",
"OA-FLOW.PC.7"
],
"external_proof_links": [
"../../OA-APPROX/infinite-tensor-products.html#theorem-5-1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 2632,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-MIV.md",
"line": 2554,
"through_line": 2632,
"description": "The absorption equivalence uses the countable separably represented type-I product of Section 29 and 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 16,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.1",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-1",
"start_line": 17,
"end_line": 40,
"range_sha256": "783ebefe437bb2251adecacbe9024c36eac4c2a6164ba22684618b61aed37aea",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DDP.EXISTENCE",
"OA-FLOW.DDP.RECOGNITION",
"OA-FLOW.CIM.1",
"OA-FLOW.CIM.2",
"OA-FLOW.CIM.3"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 17,
"through_line": 40,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.2",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-2",
"start_line": 41,
"end_line": 84,
"range_sha256": "b267ebc88dd4aaacf477c0d8522dde860527cadaf34da9a9da0b0e2b5e4bfdf6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.8",
"OA-FLOW.CIM.2",
"OA-FLOW.PC.7",
"OA-FLOW.PC.3",
"OA-FLOW.NWR.1",
"OA-FLOW.CT.1",
"OA-FLOW.CZ.0",
"OA-FLOW.DDP.RECOGNITION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 84,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 41,
"through_line": 84,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.3",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-3",
"start_line": 85,
"end_line": 153,
"range_sha256": "51a47d4bd824e6f5f349892fa08a142b6e5a21d6ef702173c4d1471f6b8655b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DDP.RECOGNITION",
"OA-FLOW.FF.3",
"OA-FLOW.CORE.1",
"OA-FLOW.PF.1",
"OA-FLOW.PDM.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 153,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 85,
"through_line": 153,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.4",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-4",
"start_line": 154,
"end_line": 184,
"range_sha256": "fcd1785f90205af55834d05c1da4a79331906b84b25f8aa25f9e163da6ef0dfa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.5",
"OA-FLOW.CIM.1",
"OA-FLOW.CIM.2",
"OA-FLOW.CIM.3",
"OA-FLOW.CORE.8",
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 154,
"through_line": 184,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.5",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-5",
"start_line": 185,
"end_line": 218,
"range_sha256": "462560594e65dbbeb8dfcde0d67403e20364febe6a3dd696e48b3e30aaa8fc96",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DDP.RECOGNITION",
"OA-FLOW.CT.1",
"OA-FLOW.CORE.8",
"OA-FLOW.CIM.3",
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 218,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 185,
"through_line": 218,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.6",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-6",
"start_line": 219,
"end_line": 265,
"range_sha256": "0b57bdf987809701c419bcb031de036bfe10b9f7178d0771ecccf819d0e15e66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CT.1",
"OA-FLOW.DDP.EXISTENCE",
"OA-FLOW.CORE.5",
"OA-FLOW.DDP.CORNERS",
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4",
"OA-FLOW.PDM.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 265,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 219,
"through_line": 265,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.7",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-7",
"start_line": 266,
"end_line": 359,
"range_sha256": "105753097cfbb8e3f82893407184977112d6426f0d63d262f6fa27df55a26108",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DDP.EXISTENCE",
"OA-FLOW.DDP.RECOGNITION",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.5",
"OA-FLOW.DD.6",
"OA-FLOW.CT.3",
"OA-FLOW.PC.7",
"OA-FLOW.CT.1",
"OA-FLOW.CZ.6",
"OA-FLOW.CORE.5",
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4",
"OA-FLOW.PDM.5",
"OA-FLOW.PDM.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 359,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 266,
"through_line": 359,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.8",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-8",
"start_line": 360,
"end_line": 376,
"range_sha256": "21a2199ddadb22915c21f425ce367197eb0c1e235d6667cb9f35524d9998446d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4",
"OA-FLOW.PDM.5",
"OA-FLOW.PDM.6",
"OA-FLOW.PDM.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 376,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 360,
"through_line": 376,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.BOUNDARIES",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-boundaries",
"start_line": 377,
"end_line": 390,
"range_sha256": "2b9b4fa9d714bd5ccd7ee948bd21d69005fafa2bb5023cabfb3cde610c0fef06",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4",
"OA-FLOW.PDM.5",
"OA-FLOW.PDM.6",
"OA-FLOW.PDM.7",
"OA-FLOW.PDM.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 390,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 377,
"through_line": 390,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.PDM.READING",
"unit": "OA-FLOW-PDM",
"source_file": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9",
"reader": "OA-FLOW-PDM.html#pdm-reading",
"start_line": 391,
"end_line": 396,
"range_sha256": "f4992cf39211ef3755a1b9714c611860cd8e357ed729e278f20428916d57eafa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DDP.RECOGNITION",
"OA-FLOW.DDP.CENTER",
"OA-FLOW.CIM.2",
"OA-FLOW.PDM.2",
"OA-FLOW.PDM.3",
"OA-FLOW.PDM.4",
"OA-FLOW.PDM.5",
"OA-FLOW.PDM.6",
"OA-FLOW.PDM.7",
"OA-FLOW.PDM.8",
"OA-FLOW.PDM.BOUNDARIES"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-PDM.md",
"line": 1,
"through_line": 40,
"description": "Nonzero type III-lambda factor, 00. Full normal core-center equality, normalized canonical phase, negative dual convention and chart/support-choice independence. Higher-period finite root alternatives and literal T=L cases; matrix-weight example on any supplied separable type III-lambda factor with exact least period kP, lacunarity and properly infinite nonfactor centralizer. No assertion of scalar-scaling automorphism existence for every factor; zero/nonpositive-scalar boundaries proved separately."
},
{
"source": "src/OA-FLOW-PDM.md",
"line": 391,
"through_line": 396,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.STATEMENT",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-statement",
"start_line": 1,
"end_line": 50,
"range_sha256": "ccd0cb22fea42b6c1983675ea3c8784121fe251088e1c6ea8dc3b77c252c6823",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.TENSORS",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-tensors",
"start_line": 51,
"end_line": 137,
"range_sha256": "c78a6c78e0c0b0ed75db3726bef66830f5eebc6220ad89f3e091fd57ac9fe4d2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TW.2",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.CZ.5",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.BC.4",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 137,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 51,
"through_line": 137,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.SHEAR",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-shear",
"start_line": 138,
"end_line": 208,
"range_sha256": "62b9d4afe29eba06178b5d0728e6cfbb3c248ff702edfe7f80673543d84eebc8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 208,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 138,
"through_line": 208,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.FIXED",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-fixed",
"start_line": 209,
"end_line": 244,
"range_sha256": "7ca5d04fcc4deb422697a7ded76cfc06d29081d08d98370aac0d424395f9f3e0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.ND.WEYL",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 244,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 209,
"through_line": 244,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.CENTER",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-center",
"start_line": 245,
"end_line": 285,
"range_sha256": "4d63a0572acd2f65c1d37089abc0df8c241da6d5251709fa42d86765f988ecc1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.RCC.5",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 285,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 245,
"through_line": 285,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.NATURAL",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-natural",
"start_line": 286,
"end_line": 336,
"range_sha256": "fc9f9a562067edef913487f8013ea35dcdaba1e39a8eb2d40e52f75271108a8a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.7",
"OA-FLOW.NR.4",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED",
"OA-FLOW.TF.CENTER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 336,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 286,
"through_line": 336,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.DUALITY",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-duality",
"start_line": 337,
"end_line": 369,
"range_sha256": "834ec946a255e088566a0f0c5d84f383fb096da758fc1ff5409c0002a437d5ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.DUALITY",
"OA-FLOW.ND.INVERSE",
"OA-FLOW.CORE.9",
"OA-FLOW.L30.5",
"OA-FLOW.CDEC.6",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED",
"OA-FLOW.TF.CENTER",
"OA-FLOW.TF.NATURAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 369,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 337,
"through_line": 369,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.MODELS",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-models",
"start_line": 370,
"end_line": 425,
"range_sha256": "30f38e106d47549b37902ab793af067caad5010701ddd5ae2253188dcd75c93e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ND.WEYL",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED",
"OA-FLOW.TF.CENTER",
"OA-FLOW.TF.NATURAL",
"OA-FLOW.TF.DUALITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 425,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 370,
"through_line": 425,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.SCOPE",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-scope",
"start_line": 426,
"end_line": 434,
"range_sha256": "1236c04e5bc3e4694bb544f2de5cdcb4457b0b705f85a9e01619993c4e6cd1d7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CGF.5",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED",
"OA-FLOW.TF.CENTER",
"OA-FLOW.TF.NATURAL",
"OA-FLOW.TF.DUALITY",
"OA-FLOW.TF.MODELS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 434,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 426,
"through_line": 434,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.TF.READING",
"unit": "OA-FLOW-TF",
"source_file": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad",
"reader": "OA-FLOW-TF.html#tf-reading",
"start_line": 435,
"end_line": 442,
"range_sha256": "1cef021d0b08deda062309be72dd1140867f45657a3142b539907b27d527587e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DA.FIXED",
"OA-FLOW.RCC.5",
"OA-FLOW.ND.DUALITY",
"OA-FLOW.CORE.7",
"OA-FLOW.CORE.5",
"OA-FLOW.CORE.8",
"OA-FLOW.FR.1",
"OA-FLOW.TF.TENSORS",
"OA-FLOW.TF.SHEAR",
"OA-FLOW.TF.FIXED",
"OA-FLOW.TF.CENTER",
"OA-FLOW.TF.NATURAL",
"OA-FLOW.TF.DUALITY",
"OA-FLOW.TF.MODELS",
"OA-FLOW.TF.SCOPE"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TF.md",
"line": 1,
"through_line": 50,
"description": "Arbitrary nonzero von Neumann algebras M1,M2, spatial tensors and intrinsic cores, negative dual convention theta_s(lambda_t)=exp(-ist)lambda_t. Opposite action delta_t=theta1_t tensor theta2_-t. Full normal isomorphism C(M1 tensor M2) to (C(M1) tensor C(M2))^delta; entire center (A1 tensor A2)^delta; residual first-factor action, equivalently half-time diagonal. No separability, faithful-state, factorial-core or type III hypothesis. Complete product-weight cocycle, normal shear/inverse, full fixed algebra and relative commutant, intrinsic naturality/modulus/flip/finite reassociation; independent double-duality alternative, canonical-trace restriction boundary and exact models. Continuous-carrier interpretation retains CGF separable-predual properly infinite scope. Zero algebra treated by unique zero maps; no topological-modulus, classification or infinite-tensor claim."
},
{
"source": "src/OA-FLOW-TF.md",
"line": 435,
"through_line": 442,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.SETTING",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-setting",
"start_line": 1,
"end_line": 91,
"range_sha256": "896bd76e7e2721436d71158e3ae39f5013ba9a01a4a0733ce2955f54d3fc37c1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.LIFT",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-lift",
"start_line": 92,
"end_line": 360,
"range_sha256": "3513148666d8da4bfd37173fa69f3bba12105f9be91dcfb6d43c63d2a781b130",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 360,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 92,
"through_line": 360,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.COVER",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-cover",
"start_line": 361,
"end_line": 397,
"range_sha256": "16108fb1eb03b825e27d819b30f718d24d803399e053fe87115a16b74c383117",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 397,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 361,
"through_line": 397,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.IW",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-iw",
"start_line": 398,
"end_line": 421,
"range_sha256": "f365603b627776582397616bdf87fbadf4d33d97f180f0cec567920a9dfb4df3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 421,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 398,
"through_line": 421,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.CST",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-cst",
"start_line": 422,
"end_line": 497,
"range_sha256": "232498da7fa01b04827fb5c83a91bbd9465ddd973a900935aa207d6c78ae505d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 497,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 422,
"through_line": 497,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.CROSS",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-cross",
"start_line": 498,
"end_line": 596,
"range_sha256": "50cb4fabf4a24a48bd5834fc0abdf9a89126375e6b0449fb45a972b0d2d8df70",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ST.2",
"OA-FLOW.NR.1",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 596,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 498,
"through_line": 596,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.CENTER",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-center",
"start_line": 597,
"end_line": 603,
"range_sha256": "a550c260acd3d8116682cfbea8460dd1499389f77fe4b8f8f3df3619392f3385",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 603,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 597,
"through_line": 603,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.TYPE",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-type",
"start_line": 604,
"end_line": 671,
"range_sha256": "9ddb394a7397d4a2d501c2a41f4dfe593af4d31c9bc1a221569c7042e856c962",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.8",
"OA-FLOW.PC.2",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 671,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 604,
"through_line": 671,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.MODELS",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-models",
"start_line": 672,
"end_line": 747,
"range_sha256": "09d56d90eaeff9fe14dad58be5f81e89c858212f6c9d070a762fd4d6c8bdad09",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 747,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 672,
"through_line": 747,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.DIAGNOSTICS",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-diagnostics",
"start_line": 748,
"end_line": 802,
"range_sha256": "92fdc9d0fa3d32d622927e04d36eda2ecd8be4643fa4c5268133ff81d3f0f06d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SR.MODELS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 802,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 748,
"through_line": 802,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.FIGURE",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-figure",
"start_line": 803,
"end_line": 809,
"range_sha256": "4dd86812775a3bd3c4e2cc59d794f98b4242e25cf3fc105de5880ed659a90ed3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SR.MODELS",
"OA-FLOW.SR.DIAGNOSTICS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 809,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 803,
"through_line": 809,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SR.SOURCES",
"unit": "OA-FLOW-SR",
"source_file": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae",
"reader": "OA-FLOW-SR.html#sr-sources",
"start_line": 810,
"end_line": 819,
"range_sha256": "ae89aa48a329149c6d1f441b68dd6da25ac4163d92e4965d40d86161cee262ec",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.SUSPENSION.MEASURE",
"OA-FLOW.IW.3",
"OA-FLOW.CSTD.STATIC",
"OA-FLOW.CSTD.NORMALIZER",
"OA-FLOW.CST.5",
"OA-FLOW.NR.1",
"OA-FLOW.NR.5",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.ND.MULTIPLICATION",
"OA-FLOW.ND.WEYL",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.PC.2",
"OA-FLOW.SR.LIFT",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SR.MODELS",
"OA-FLOW.SR.DIAGNOSTICS",
"OA-FLOW.SR.FIGURE"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SR.md",
"line": 1,
"through_line": 91,
"description": "P nonzero with separable predual; separable LCH acting G on specified standard sigma-finite nonsingular jointly Borel base; arbitrary LCH target H and Borel cocycle, or explicit regular-measurable version. HS1-HS6 prove automatic vector measurability at full target scope. Lifted real system has properly ergodic center, specified roof r bounded below by positive delta and return transformation T. Two full normal product constructions, exact center/deck action, normal regular crossed-product inverses, all homogeneous coarse types I/II/III. No preservation of finite/infinite subtype, no trace required on P, no III0 classification claimed here."
},
{
"source": "src/OA-FLOW-SR.md",
"line": 810,
"through_line": 819,
"description": "Full section hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.ZDC.SETTING",
"unit": "OA-FLOW-ZDC",
"source_file": "src/OA-FLOW-ZDC.md",
"source_sha256": "7136d488d023a539642e99c2ec177f6d42a103f1affdd50212b5ffc2e0b1eff2",
"reader": "OA-FLOW-ZDC.html#zdc-setting",
"start_line": 1,
"end_line": 32,
"range_sha256": "7f721f1ba927a48f1aab603b149bbd85302050d07f74ed0899e5066c6895913c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.5",
"OA-FLOW.MIV.1",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SUSPENSION.COVER"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ZDC.md",
"line": 1,
"through_line": 32,
"description": "All algebras have separable predual. Every nonzero type III0 factor has a full integer crossed-product presentation with nonfactor-allowed type II-infinity coefficient, nonatomic center, centrally ergodic automorphism and specified faithful normal semifinite trace contracting by a central density 00, or a finite strictly positive measurable central width h. All formulas hold on the full positive cone, including infinite trace values."
}
]
},
{
"id": "OA-FLOW.ZDC.5",
"unit": "OA-FLOW-ZDC",
"source_file": "src/OA-FLOW-ZDC.md",
"source_sha256": "7136d488d023a539642e99c2ec177f6d42a103f1affdd50212b5ffc2e0b1eff2",
"reader": "OA-FLOW-ZDC.html#zdc-5",
"start_line": 345,
"end_line": 426,
"range_sha256": "0a882a92819c73c5267f2d2de81e316ea1e8b9ca566b3c9f361223fec8fb26e2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CSTD.TRANSPORT",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.5",
"OA-FLOW.MIV.1",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.ZDC.1",
"OA-FLOW.ZDC.2",
"OA-FLOW.ZDC.3",
"OA-FLOW.ZDC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 426,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ZDC.md",
"line": 1,
"through_line": 32,
"description": "All algebras have separable predual. Every nonzero type III0 factor has a full integer crossed-product presentation with nonfactor-allowed type II-infinity coefficient, nonatomic center, centrally ergodic automorphism and specified faithful normal semifinite trace contracting by a central density 0=delta>0 and a countably strict integer action. All fiber weight equalities hold on whole positive cones."
}
]
},
{
"id": "OA-FLOW.ZDC.7",
"unit": "OA-FLOW-ZDC",
"source_file": "src/OA-FLOW-ZDC.md",
"source_sha256": "7136d488d023a539642e99c2ec177f6d42a103f1affdd50212b5ffc2e0b1eff2",
"reader": "OA-FLOW-ZDC.html#zdc-7",
"start_line": 714,
"end_line": 806,
"range_sha256": "1efab427dd92a3814a4e7e03039e58a9a2217fe758257c8a51129573dbfb4215",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.4",
"OA-FLOW.NR.5",
"OA-FLOW.PC.7",
"OA-FLOW.CORE.8",
"OA-FLOW.CDEC.3",
"OA-FLOW.L18.5",
"OA-FLOW.MIV.1",
"OA-FLOW.SR.COVER",
"OA-FLOW.SR.IW",
"OA-FLOW.SR.CST",
"OA-FLOW.SR.CROSS",
"OA-FLOW.SR.CENTER",
"OA-FLOW.SR.TYPE",
"OA-FLOW.SUSPENSION.COVER",
"OA-FLOW.ZDC.1",
"OA-FLOW.ZDC.2",
"OA-FLOW.ZDC.3",
"OA-FLOW.ZDC.4",
"OA-FLOW.ZDC.5",
"OA-FLOW.ZDC.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 806,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ZDC.md",
"line": 1,
"through_line": 32,
"description": "All algebras have separable predual. Every nonzero type III0 factor has a full integer crossed-product presentation with nonfactor-allowed type II-infinity coefficient, nonatomic center, centrally ergodic automorphism and specified faithful normal semifinite trace contracting by a central density 0=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Arbitrary group G and normal N; or standard Borel groups with a Borel normal subgroup and jointly Borel maps. Central circle kernel and action restricting to conjugation by lifts."
}
]
},
{
"id": "OA-FLOW.CPP.GAUGE",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-gauge",
"start_line": 114,
"end_line": 184,
"range_sha256": "a306286a57b66217f42f97cd907b7133e45fec48642940be5916346773f7fc01",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 184,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 114,
"through_line": 184,
"range_sha256": "a306286a57b66217f42f97cd907b7133e45fec48642940be5916346773f7fc01",
"description": "Normalized phase changes, algebraic or Borel, and compatible-extension isomorphisms fixing central kernel, quotient and G-action. Pullback along the specified homomorphism."
}
]
},
{
"id": "OA-FLOW.CPP.COMPACT",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-compact",
"start_line": 185,
"end_line": 221,
"range_sha256": "37f9f9d08818ec1f3c2664d90111890c7a519ad0e2e65e29d34d7c9e01f0c667",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 221,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 185,
"through_line": 221,
"range_sha256": "37f9f9d08818ec1f3c2664d90111890c7a519ad0e2e65e29d34d7c9e01f0c667",
"description": "Countable discrete G, any normal N, pointwise topology on circle-valued functions."
}
]
},
{
"id": "OA-FLOW.CPP.INTRINSIC",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-intrinsic",
"start_line": 222,
"end_line": 319,
"range_sha256": "664970ee5fb495f91c0342de9cfb1d154692392233b9a812d89193020369f9f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.BRL.INTRINSIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 319,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 222,
"through_line": 319,
"range_sha256": "664970ee5fb495f91c0342de9cfb1d154692392233b9a812d89193020369f9f0",
"description": "Nonzero factor M with separable predual; Aut(M) u-topology, U(M) strong topology, Inn(M) inherited Borel structure; normal unital *-isomorphisms."
}
]
},
{
"id": "OA-FLOW.CPP.ACTIONS",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-actions",
"start_line": 320,
"end_line": 388,
"range_sha256": "53007ccf3bc2bb3482963c1cadbc7ed8dc2285831f31d6317a384cd1a4ed1035",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.CPP.INTRINSIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 388,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 320,
"through_line": 388,
"range_sha256": "53007ccf3bc2bb3482963c1cadbc7ed8dc2285831f31d6317a384cd1a4ed1035",
"description": "Countable discrete action on a nonzero separable-predual factor; normalized genuine unitary 1-cocycle, normal *-isomorphism."
}
]
},
{
"id": "OA-FLOW.CPP.CYCLIC",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-cyclic",
"start_line": 389,
"end_line": 445,
"range_sha256": "ffe561dc8e5fa5c2aeabddc60dae8bff0897689b64e49c7ac9c49ad2e8abbf2f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.CPP.COMPACT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 445,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 389,
"through_line": 445,
"range_sha256": "ffe561dc8e5fa5c2aeabddc60dae8bff0897689b64e49c7ac9c49ad2e8abbf2f",
"description": "G=Z, N=pZ, integer p>=0, compatible inner-action identity imposed. Positive and zero cases distinguished."
}
]
},
{
"id": "OA-FLOW.CPP.PERIOD",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-period",
"start_line": 446,
"end_line": 520,
"range_sha256": "0bad267ac02986b24b874ea6a4867ce428a924d906ea9bc683d3b9998043ed9c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.INTRINSIC",
"OA-FLOW.CPP.ACTIONS",
"OA-FLOW.CPP.CYCLIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 520,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 446,
"through_line": 520,
"range_sha256": "0bad267ac02986b24b874ea6a4867ce428a924d906ea9bc683d3b9998043ed9c",
"description": "Automorphism of a nonzero separable-predual factor; finite outer period p>=1 or p=0 separately."
}
]
},
{
"id": "OA-FLOW.CPP.DIAGNOSTICS",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-diagnostics",
"start_line": 521,
"end_line": 544,
"range_sha256": "303961ea49753cd473820bacf24fe290dd5795908ea7bd2b2167fb579315e770",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.GAUGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 544,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 521,
"through_line": 544,
"range_sha256": "303961ea49753cd473820bacf24fe290dd5795908ea7bd2b2167fb579315e770",
"description": "S3 with right-to-left permutation composition; exact normalized fourth-root-valued phase change."
}
]
},
{
"id": "OA-FLOW.CPP.READING",
"unit": "OA-FLOW-CPP",
"source_file": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317",
"reader": "OA-FLOW-CPP.html#cp-reading",
"start_line": 545,
"end_line": 549,
"range_sha256": "e843b212f3766dbe783dc4a28ffe17f011fcbb947f1aba6c7ddaa9cb2cf6e03d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-CPP.md",
"line": 1,
"through_line": 113,
"range_sha256": "eabe2a8786266b04951cc3dd1a74cbf69ec1c97247956561f19c1c743a99ff77",
"description": "Compatible central circle extensions and their actions for arbitrary G and normal N; Borel version for standard Borel groups and Borel normal N. Compact metrizable Hausdorff quotient only for countable discrete G. Intrinsic implementer class and action/automorphism applications assume a nonzero factor with separable predual, with inherited Borel structure on Inn(M), not inherited Polishness or a continuous section. Genuine normalized unitary cocycles. Cyclic p>=1 has T[p]; p=0 has the trivial group. The inner-action identity is imposed and the weaker definition is explicitly refuted. No Exercises4-5 realization or classification is asserted."
},
{
"source": "src/OA-FLOW-CPP.md",
"line": 545,
"through_line": 549,
"range_sha256": "e843b212f3766dbe783dc4a28ffe17f011fcbb947f1aba6c7ddaa9cb2cf6e03d",
"description": "Bibliography only; not an earlier proof premise."
}
]
},
{
"id": "OA-FLOW.TRP.SETTING",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-setting",
"start_line": 1,
"end_line": 55,
"range_sha256": "c8d0529438dfea880ab9b3c2dd2aec5b80e65bd61c886a2d261ff387d9e22015",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 55,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.TWO-ACTIONS",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-two-actions",
"start_line": 56,
"end_line": 129,
"range_sha256": "4c61bc574db695984c223c70abeefb3d7ffcaa9880bca0524e16bfb683900e64",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 129,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.TRACE",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-trace",
"start_line": 130,
"end_line": 151,
"range_sha256": "876c880014f3d52c9285fadd968a807d6c226d32ea2ff3b2b72e3178e1307706",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.TWO-ACTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 151,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.FOURIER",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-fourier",
"start_line": 152,
"end_line": 195,
"range_sha256": "1f11efd3c4f5419f519f788ac2e214f3576fe58212a1079f6d1639c285e80143",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.TWO-ACTIONS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 195,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.SYMMETRY",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-symmetry",
"start_line": 196,
"end_line": 231,
"range_sha256": "f193401581697c09c5e9724abfac4d606dc58918eb5ac89489fae6b20cd0ee52",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.TWO-ACTIONS",
"OA-FLOW.TRP.FOURIER"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 231,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.KERNEL",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-kernel",
"start_line": 232,
"end_line": 265,
"range_sha256": "29c5f4c13113bf3747f7112ddeb218cf996cb967bf09f1c68efb9a0aae16b6ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.TRP.FOURIER",
"OA-FLOW.TRP.SYMMETRY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 265,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.GAUGE",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-gauge",
"start_line": 266,
"end_line": 284,
"range_sha256": "3f521c11ec5804088f12b57b02ef5209ce39bb464b346911276278c504a00408",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.TRP.SYMMETRY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 284,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.BERNOULLI",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-bernoulli",
"start_line": 285,
"end_line": 312,
"range_sha256": "ecd4db427475ae770009b679989cc0a23573acfae4b329b928275824ef509a7d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.KERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 312,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.FINITE-AFD",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-finite-afd",
"start_line": 313,
"end_line": 325,
"range_sha256": "e10ae19cf64d823d93a309a5cc921e35829ad4bfa9c5653b7fa67b49d051516a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.BERNOULLI",
"OA-FLOW.TRP.KERNEL",
"OA-FLOW.TRP.SYMMETRY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 325,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.OBSTRUCTION",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-obstruction",
"start_line": 326,
"end_line": 374,
"range_sha256": "e3b6ecfefc8c8e2127ad58689820dd153ed8353dae304247a228b7b16cc2bc44",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.KERNEL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 374,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.MODELS",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-models",
"start_line": 375,
"end_line": 426,
"range_sha256": "5479c03ac412b3aef2c79ba59632410a6e40d0ac500a053ad410623e7d473415",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.FINITE-AFD",
"OA-FLOW.TRP.OBSTRUCTION",
"OA-FLOW.TRP.GAUGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 426,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.TRP.INPUT-BOUNDARY",
"unit": "OA-FLOW-TRP",
"source_file": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588",
"reader": "OA-FLOW-TRP.html#trp-input-boundary",
"start_line": 427,
"end_line": 449,
"range_sha256": "e9fa8c32d91069a59fe2af37d6f24095e3c74079d726558071179bee6a4f5903",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-TRP.md",
"line": 1,
"through_line": 55,
"description": "Countable discrete G, normal N, normalized compatible scalar pair; supplied faithful normal tracial state on a nonzero finite factor and trace-preserving normal action. Outerness and Bernoulli/finite-N hypotheses enter only where explicitly stated."
}
]
},
{
"id": "OA-FLOW.NB.THEOREM",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-theorem",
"start_line": 1,
"end_line": 38,
"range_sha256": "5f9a0550ead8e6aa6c71146ba738f53f454e529cff8eea29b81900182b7eb9d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.PRODUCT",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-product",
"start_line": 39,
"end_line": 72,
"range_sha256": "689f87af3b4ff5d8c73786dc43a6b156ccab1732e9767d6bb3fb12028d06264c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 72,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.MODULAR",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-modular",
"start_line": 73,
"end_line": 110,
"range_sha256": "b483dc09695ecf14c67002936a6bbca752e48cf4015c7d915583effd1845368b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 110,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.FIXED-FACTOR",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-fixed-factor",
"start_line": 111,
"end_line": 128,
"range_sha256": "128089fca680a3dce41b9870fa294d402f84bab8cc040536451f3e56788b1ecd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.MODULAR"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 128,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.TAILS",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-tails",
"start_line": 129,
"end_line": 162,
"range_sha256": "e28061895b5a37ed184fc78ad5753feeb7e7a98d63abfda60a9ae6a488b77d74",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.MODULAR",
"OA-FLOW.NB.FIXED-FACTOR",
"OA-FLOW.CP.1",
"OA-FLOW.CP.2",
"OA-FLOW.CP.3",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 162,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.CROSSED",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-crossed",
"start_line": 163,
"end_line": 190,
"range_sha256": "cce0793fd935bc560b8470a3ffb8c7c0f21ac8ef74cf6e5f1faf8d4eb2d5869a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.TAILS",
"OA-FLOW.TRP.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 190,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.FOURIER",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-fourier",
"start_line": 191,
"end_line": 214,
"range_sha256": "10d53fb6a0f07168abcab83bb025f8195990ad329c1e9da118bde2082e918cbc",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.FIXED-FACTOR",
"OA-FLOW.NB.TAILS",
"OA-FLOW.NB.CROSSED"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 214,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.CROSSED-MODULAR",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-crossed-modular",
"start_line": 215,
"end_line": 247,
"range_sha256": "402f249dd1a24e31854103e916de0c13cfb3642854f8016df3aa603efa7b14d5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.MODULAR",
"OA-FLOW.NB.CROSSED",
"OA-FLOW.NB.FOURIER",
"OA-FLOW.TRP.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 247,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.CORNER-SPECTRUM",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-corner-spectrum",
"start_line": 248,
"end_line": 283,
"range_sha256": "d27837fbfe62405527e86d806d7ec15824efc70d932bd8f405f31db5d9bac6b4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.FIXED-FACTOR",
"OA-FLOW.NB.CROSSED-MODULAR",
"OA-FLOW.AL.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 283,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
},
{
"line": 251,
"through_line": 251,
"scope": "0 < lambda < 1 for the lattice and all-weight subtype claims."
}
]
},
{
"id": "OA-FLOW.NB.ALL-WEIGHTS",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-all-weights",
"start_line": 284,
"end_line": 314,
"range_sha256": "481932252523787ec1e1b1a5f7586e4ca9e88d7062ab2c2001083ea15a3cd1fa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.MODULAR",
"OA-FLOW.NB.FOURIER",
"OA-FLOW.NB.CROSSED-MODULAR",
"OA-FLOW.NB.CORNER-SPECTRUM",
"OA-FLOW.MG.0",
"OA-FLOW.MG.1",
"OA-FLOW.MG.3",
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.BC.3",
"OA-FLOW.KU.1",
"OA-FLOW.KU.2",
"OA-FLOW.KU.3",
"OA-FLOW.CS.0",
"OA-FLOW.CS.1",
"OA-FLOW.CS.2",
"OA-FLOW.CS.3",
"OA-FLOW.CS.4",
"OA-FLOW.MW.4",
"OA-FLOW.KT.2",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8",
"OA-FLOW.L18.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 314,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
},
{
"line": 251,
"through_line": 251,
"scope": "0 < lambda < 1 for the lattice and all-weight subtype claims."
}
]
},
{
"id": "OA-FLOW.NB.TRACE-ENDPOINT",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-trace-endpoint",
"start_line": 315,
"end_line": 321,
"range_sha256": "b4616f5bc717f355b5a862875c1d8dafa67f881f1d5e887f31d0809b3f8b21a6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.MODULAR",
"OA-FLOW.NB.FIXED-FACTOR",
"OA-FLOW.NB.TAILS",
"OA-FLOW.NB.CROSSED",
"OA-FLOW.NB.FOURIER",
"OA-FLOW.NB.CROSSED-MODULAR",
"OA-FLOW.NB.CORNER-SPECTRUM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 321,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
},
{
"line": 316,
"through_line": 316,
"scope": "lambda = 1 on the same original sites."
}
]
},
{
"id": "OA-FLOW.NB.DIAGNOSTICS",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-diagnostics",
"start_line": 322,
"end_line": 336,
"range_sha256": "174b2b3e8160b5132e7130dc44edf36de4239a3eaad93ae689d031251a60f77f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NB.PRODUCT",
"OA-FLOW.NB.MODULAR",
"OA-FLOW.NB.FIXED-FACTOR",
"OA-FLOW.NB.TAILS",
"OA-FLOW.NB.CROSSED",
"OA-FLOW.NB.FOURIER",
"OA-FLOW.NB.CROSSED-MODULAR",
"OA-FLOW.NB.CORNER-SPECTRUM",
"OA-FLOW.NB.ALL-WEIGHTS",
"OA-FLOW.NB.TRACE-ENDPOINT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 336,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.NB.SOURCES",
"unit": "OA-FLOW-NB",
"source_file": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203",
"reader": "OA-FLOW-NB.html#nb-sources",
"start_line": 337,
"end_line": 343,
"range_sha256": "8a86a465c7f29cd4345abf8ead64c7261f70b9d563f99fa64978a6fc0abe8b56",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"line": 1,
"through_line": 38,
"scope": "Countable infinite discrete G, original sites G, and 0 < lambda <= 1; the theorem statement is a hypothesis/definition locator, never a proved-premise dependency."
}
]
},
{
"id": "OA-FLOW.TG.GOAL",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-goal",
"start_line": 1,
"end_line": 23,
"range_sha256": "07b2de66904304166c05a0a760d5120c94d4f2817df98a366294527f45028c15",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 3385
},
{
"id": "OA-FLOW.TG.BOREL-TOOLS",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-borel-tools",
"start_line": 24,
"end_line": 43,
"range_sha256": "94e141ec5a5b72d2d3c20b8ca8a34933dd075cd14167a914c07c44572405bcbe",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 43,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 3456
},
{
"id": "OA-FLOW.TG.INITIAL-SECTION",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-initial-section",
"start_line": 44,
"end_line": 88,
"range_sha256": "b280f3a1d6cc6cead50f0ac3548578140ac67f4fa29c5b9027929ae6aeb02e6d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 88,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 7457
},
{
"id": "OA-FLOW.TG.POSITIVE-ARITHMETIC",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-positive-arithmetic",
"start_line": 89,
"end_line": 115,
"range_sha256": "50da3930898a7d6f2ce6594825b032b87144c9faefb32bde1c96ff86ebdf8fa7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 115,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 2661
},
{
"id": "OA-FLOW.TG.CORRIDORS",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-corridors",
"start_line": 116,
"end_line": 134,
"range_sha256": "8cba1eec4abefbab3eab17f1cdc66b0b037737cf0b44ae9c123ea2098a25aa36",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.POSITIVE-ARITHMETIC"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 134,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 3481
},
{
"id": "OA-FLOW.TG.FREQUENCY",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-frequency-control",
"start_line": 135,
"end_line": 172,
"range_sha256": "62a11f3b29d226677cf5fe44c1d3c3bc70f5587e91683c21e6dc099eb4e71e9a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.POSITIVE-ARITHMETIC",
"OA-FLOW.TG.CORRIDORS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 172,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 4615
},
{
"id": "OA-FLOW.TG.REFINEMENT",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-propagation",
"start_line": 173,
"end_line": 217,
"range_sha256": "68712b2ce97da62520271b1725f53303fd188db88876157663808d3a9017a4f2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.POSITIVE-ARITHMETIC",
"OA-FLOW.TG.CORRIDORS",
"OA-FLOW.TG.FREQUENCY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 217,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 4442
},
{
"id": "OA-FLOW.TG.RANKED-BLOCKS",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-ranked-blocks",
"start_line": 218,
"end_line": 311,
"range_sha256": "125341a8d95ccbeb26b304d7e0723d9f6503fa5747dd485442a1e8bb5b0c6fa2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS",
"OA-FLOW.TG.INITIAL-SECTION",
"OA-FLOW.TG.POSITIVE-ARITHMETIC",
"OA-FLOW.TG.REFINEMENT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 311,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 10932
},
{
"id": "OA-FLOW.TG.RANK-LIMIT",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-rank-limit",
"start_line": 312,
"end_line": 341,
"range_sha256": "0654f67bfd7acf7dd0dc3f297a38e84139db7be76c433018d19ccd4d939e9ad3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS",
"OA-FLOW.TG.RANKED-BLOCKS"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 341,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 3387
},
{
"id": "OA-FLOW.TG.ORBIT-PIECES",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-orbit-pieces",
"start_line": 342,
"end_line": 389,
"range_sha256": "9875324aacfc399755686bd32b01f2120051a5f5fb317efc73b52af8cbdb9a0d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS",
"OA-FLOW.TG.RANK-LIMIT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 389,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 4134
},
{
"id": "OA-FLOW.TG.SPARSE",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-sparse",
"start_line": 390,
"end_line": 416,
"range_sha256": "d08c074f85ad811a049c4129bae10f015be597854c68cb3ad1febf0e316cb4d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS",
"OA-FLOW.TG.POSITIVE-ARITHMETIC",
"OA-FLOW.TG.RANK-LIMIT"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 416,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 5393
},
{
"id": "OA-FLOW.TG.ASSEMBLY",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-assembly",
"start_line": 417,
"end_line": 430,
"range_sha256": "30941d78aad77de4b3cfcf55db735a121b34d1fd2dd59c72f4b8af40e78141a4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.BOREL-TOOLS",
"OA-FLOW.TG.ORBIT-PIECES",
"OA-FLOW.TG.SPARSE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 430,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 1540
},
{
"id": "OA-FLOW.TG.DIAGNOSTICS",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-diagnostics",
"start_line": 431,
"end_line": 460,
"range_sha256": "508a5747f52a2b107b9b71c676050c28c09410ced7be183d12499170cbfa8c53",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TG.POSITIVE-ARITHMETIC",
"OA-FLOW.TG.CORRIDORS",
"OA-FLOW.TG.FREQUENCY",
"OA-FLOW.TG.REFINEMENT",
"OA-FLOW.TG.RANKED-BLOCKS",
"OA-FLOW.TG.RANK-LIMIT",
"OA-FLOW.TG.ORBIT-PIECES",
"OA-FLOW.TG.SPARSE",
"OA-FLOW.TG.ASSEMBLY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 460,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 5499
},
{
"id": "OA-FLOW.TG.SOURCES",
"unit": "OA-FLOW-TG",
"source_file": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77",
"reader": "OA-FLOW-TG.html#two-gap-prerequisites",
"start_line": 461,
"end_line": 472,
"range_sha256": "18164c0e2ba298aff0eb79100dcd31828cdb3c04ee43d5b61b7e032a2fbd57d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"line": 8,
"through_line": 22,
"scope": "Free jointly Borel real action on a standard Borel space; prescribed a,b>0 with irrational ratio; the forward statement is not an earlier proved premise."
}
],
"range_bytes": 2968
},
{
"id": "OA-FLOW.CCOV.STATECOMPACTNESS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.statecompactness",
"start_line": 486,
"end_line": 512,
"range_sha256": "12dae802e157c6aa1c1287ea00526c359060518ebd54a19a64cf301c5fcae582",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.INVARIANTSTATE",
"OA-FLOW.CCOV.EXTREMALITY",
"OA-FLOW.GNS.5.1",
"OA-FND-WT-02",
"OA-FND-WT-03",
"OA-FND-WT-06"
]
},
{
"id": "OA-FLOW.CCOV.REALKERNELHS",
"unit": "OA-FLOW-L25",
"source_file": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456",
"reader": "OA-FLOW-L25.html#oa-flow.ccov.realkernelhs",
"start_line": 596,
"end_line": 639,
"range_sha256": "99e93ba2a8cf2a0484c281c26a5498cb28892793e1c11b42a794387f865009cf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CCOV.TRANSFORMATIONS",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.SC.4",
"OA-FND-HS-04"
]
},
{
"id": "OA-FLOW.L128.FC2A",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc2a",
"start_line": 82,
"end_line": 110,
"range_sha256": "fbba418095667f9b05ed004e679b6db07c8ec4960e5a37d5681bf55c89283aad",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.SF.SF2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 110
},
{
"id": "OA-FLOW.L128.GC0",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-gc0",
"start_line": 282,
"end_line": 315,
"range_sha256": "1d9557d8f9546dbe53458885ebb06c2f5a5b728c99ec54b772e784da200776a1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2A",
"OA-FLOW.L128.TC3"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 315
},
{
"id": "OA-FLOW.L128.FC2B",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc2b",
"start_line": 382,
"end_line": 427,
"range_sha256": "bdeabb6592137f1660ba9c321215dd7a5e562ded3e44a97884ddf3796603382c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.OR0",
"OA-FLOW.CC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 427
},
{
"id": "OA-FLOW.L128.FC2C",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc2c",
"start_line": 428,
"end_line": 467,
"range_sha256": "8c2e6c18d46a125530c6cc413b1bff8a32ef542dd7776a8fcbef22d5dc487ba0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.FC2A",
"OA-FLOW.L128.FC2B",
"OA-FLOW.CCM.7",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 467
},
{
"id": "OA-FLOW.ARP.SETTING",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-setting",
"start_line": 1,
"end_line": 55,
"range_sha256": "f046c38e4152b15645cdd26d19c1c6fa2094b8c30cb51e0b699d4c23a5543fe6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.REL.1"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.DIAGONAL-FLIPS",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-diagonal-flips",
"start_line": 56,
"end_line": 175,
"range_sha256": "09d965155dedd98dba0d72bb1f214d57c80055f84a459dbf8344640b134730cd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.BERNOULLI",
"OA-FLOW.MIV.PRODUCT.RELATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 175,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.AMENABLE-FREE-ACTION",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-amenable-free-action",
"start_line": 176,
"end_line": 219,
"range_sha256": "ecdc2bd22ff6de512c3acb0f6e8b1daf6574502b21c062acc5cb83ef5145034f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ARP.DIAGONAL-FLIPS",
"OA-FLOW.TRP.SETTING"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 219,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.REGULAR-IDENTIFICATION",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-regular-identification",
"start_line": 220,
"end_line": 343,
"range_sha256": "a237eb08632bae47c3679dd713900a4b956f898b0d313b116f79ed25f4a8cf79",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ARP.DIAGONAL-FLIPS",
"OA-FLOW.ARP.AMENABLE-FREE-ACTION",
"OA-FLOW.TRP.TWO-ACTIONS",
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.FOURIER",
"OA-FLOW.TRP.SYMMETRY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 343,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.COMPATIBLE-PRIMITIVE",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-compatible-primitive",
"start_line": 344,
"end_line": 464,
"range_sha256": "59d221adb01dbae088707b219d71110a84a8c3ddaab92c57fb9e9711920314ba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ARP.REGULAR-IDENTIFICATION",
"OA-FLOW.ARP.AMENABLE-FREE-ACTION",
"OA-FLOW.CV.1",
"OA-FLOW.CV.3",
"OA-FLOW.REL.1",
"OA-FLOW.SC.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 464,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.UNTWISTING",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-untwisting",
"start_line": 465,
"end_line": 509,
"range_sha256": "de5ecedbb7fb11905e8f9283245476069bb8808acf8ca1fdc2b8c25e5d3e5ad2",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ARP.COMPATIBLE-PRIMITIVE",
"OA-FLOW.ARP.REGULAR-IDENTIFICATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 509,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.FINITE-PATTERNS",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-finite-patterns",
"start_line": 510,
"end_line": 694,
"range_sha256": "c11ce03fd29c46e2e0e8f8ca63b0474a8aa4bfbe14e0a6b196125cfbea88d8bb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.ARP.DIAGONAL-FLIPS",
"OA-FLOW.ARP.COMPATIBLE-PRIMITIVE",
"OA-FLOW.ARP.UNTWISTING",
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.KERNEL",
"OA-FLOW.TRP.BERNOULLI"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 694,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.PREREQUISITES-SOURCES",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-prerequisites-sources",
"start_line": 695,
"end_line": 721,
"range_sha256": "ee70d455a03e55cc49286624f286cbb30f9fb83ae21e41ef3eed284d29fe2de4",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.ARP.THEOREM",
"unit": "OA-FLOW-ARP",
"source_file": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d",
"reader": "OA-FLOW-ARP.html#arp-finite-patterns",
"start_line": 1,
"end_line": 694,
"range_sha256": "3cfd635e65fa9a18c8a2c63028dd4b29d711bb6a12761d978236b6573f7c5eba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.TWO-ACTIONS",
"OA-FLOW.TRP.TRACE",
"OA-FLOW.TRP.FOURIER",
"OA-FLOW.TRP.SYMMETRY",
"OA-FLOW.TRP.KERNEL",
"OA-FLOW.TRP.BERNOULLI",
"OA-FLOW.MIV.PRODUCT.RELATION",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7",
"OA-FLOW.CV.1",
"OA-FLOW.CV.3",
"OA-FLOW.REL.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 694,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-ARP.md",
"line": 9,
"through_line": 55,
"description": "Arbitrary countable discrete G; infinite amenable normal N; actual fair-tracial M2 tensor on G × positive integers; every normalized scalar pair satisfying all five CPP/TRP identities. Retained measure/operator foundations are explicitly named. The forward theorem is available only after the Section 7 proof."
}
]
},
{
"id": "OA-FLOW.NNR.CONVENTION",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-convention",
"start_line": 9,
"end_line": 46,
"range_sha256": "c365a9823cce00aec4022b58ede34ecb6247aff6a0e8ce973232f0123a1fbddb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 46
},
{
"id": "OA-FLOW.NNR.PAIR",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-pair",
"start_line": 47,
"end_line": 121,
"range_sha256": "f673677f8b7294d71426f023c07b65334532b0f754c662e729a5d4c32b5f2a93",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 121
},
{
"id": "OA-FLOW.NNR.AFD-VECTORS",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-afd-vectors",
"start_line": 122,
"end_line": 179,
"range_sha256": "04400d7958d63c0eb2063a02ca3120255f03763a425d49243d1958cb0bfb95fa",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.OBSTRUCTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 179
},
{
"id": "OA-FLOW.NNR.MEASURE",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-measure",
"start_line": 180,
"end_line": 202,
"range_sha256": "3a1427bf9507a19134f38401c8ec3c8b0e90f26d62d5f40080e5490fadf56ff7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 202
},
{
"id": "OA-FLOW.NNR.RIGIDITY",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-rigidity",
"start_line": 203,
"end_line": 310,
"range_sha256": "301812dba39a4227a4e9cd66ae4508804b413515dfedeb2cd5155ec6e86469c9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NNR.MEASURE"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 310
},
{
"id": "OA-FLOW.NNR.FINITE-DIMENSION",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-finite-dimension",
"start_line": 311,
"end_line": 331,
"range_sha256": "398e38adf44d88072972e3a6f2ce1c23e10955055dd6f83d690ac76d9d7f4531",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NNR.PAIR",
"OA-FLOW.NNR.AFD-VECTORS",
"OA-FLOW.NNR.RIGIDITY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 331
},
{
"id": "OA-FLOW.NNR.NONREALIZATION",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-nonrealization",
"start_line": 332,
"end_line": 344,
"range_sha256": "fb4ae9521f2e00bf455bfa25b271989589643c6b9034f92fc802461667e960a8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NNR.FINITE-DIMENSION",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.CPP.ACTIONS",
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 344
},
{
"id": "OA-FLOW.NNR.BOUNDARY",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-boundary",
"start_line": 345,
"end_line": 376,
"range_sha256": "4a1e584353c52d95979fcb1dc8e76006c8bec210cb5ff3978db84561f3d5b6b1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NNR.NONREALIZATION",
"OA-FLOW.NNR.PAIR",
"OA-FLOW.TRP.BERNOULLI",
"OA-FLOW.TRP.FINITE-AFD",
"OA-FLOW.TRP.OBSTRUCTION",
"OA-FLOW.ARP.THEOREM"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 376
},
{
"id": "OA-FLOW.NNR.FOUNDATIONS",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-foundations",
"start_line": 377,
"end_line": 399,
"range_sha256": "e45c8cf723979ab83e261feee0d0c4b14506c12f4007377d2f8f3a66100e454d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 399
},
{
"id": "OA-FLOW.NNR.FIGURE",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-figure",
"start_line": 400,
"end_line": 405,
"range_sha256": "5dafc911848fd2eff28a608be0f6ca76a7109659668b45d609a66cdfc5e205b3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NNR.NONREALIZATION",
"OA-FLOW.NNR.BOUNDARY"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": 405
},
{
"id": "OA-FLOW.NNR.THEOREM",
"unit": "OA-FLOW-NNR",
"source_file": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd",
"reader": "OA-FLOW-NNR.html#nnr-nonrealization",
"start_line": 1,
"end_line": 376,
"range_sha256": "744de169541ef33ae81a131b146e16ac1cbbc0bc1dfa33b04b09ead6648ab286",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CPP.COMPATIBLE",
"OA-FLOW.CPP.GAUGE",
"OA-FLOW.CPP.ACTIONS",
"OA-FLOW.TRP.SETTING",
"OA-FLOW.TRP.BERNOULLI",
"OA-FLOW.TRP.FINITE-AFD",
"OA-FLOW.TRP.OBSTRUCTION",
"OA-FLOW.ARP.THEOREM",
"OA-FLOW.FCT.0",
"OA-FLOW.FCT.1",
"OA-FLOW.FCT.2",
"OA-FLOW.FCT.3",
"OA-FLOW.FCT.4",
"OA-FLOW.FCT.5",
"OA-FLOW.FCT.6",
"OA-FLOW.FCT.7",
"OA-FLOW.FCT.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 376
},
{
"id": "OA-FLOW.MF06.POLAR",
"unit": "OA-FLOW-MF06",
"source_file": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d",
"reader": "OA-FLOW-MF06.html#oa-flow.mf06.polar-reduction",
"start_line": 83,
"end_line": 134,
"range_sha256": "85d89b0462dc56a279312e1730c73039f9b033c15fa3cf853b7e7773dee428d6",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HA-R.1",
"OA-FLOW.HA-R.2",
"OA-FLOW.HA-R.4",
"OA-FLOW.HA-R.7",
"OA-FLOW.WH04.2",
"OA-FLOW.WH04.3",
"OA-FLOW.WH04.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 134,
"definition_context": [
"OA-FLOW.MF06.1",
"OA-FLOW.MF06.2"
]
},
{
"id": "OA-FLOW.NR.6",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.6",
"start_line": 249,
"end_line": 302,
"range_sha256": "c35d47a8017f3e44d0085635880dda705583383c3e9bb8ab85f0ff86259e2733",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.5",
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.UC.2",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 302,
"range_bytes": 2962
},
{
"id": "OA-FLOW.NR.7",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#oa-flow.nr.7",
"start_line": 303,
"end_line": 358,
"range_sha256": "ee1d0023c5abcb13eec6d32445753fb26141e9afd61b63c1cf7fcf3759baac07",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.5",
"OA-FLOW.NR.6",
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.AT.1",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 358,
"range_bytes": 3466
},
{
"id": "OA-FLOW.NR.DEGENERATE-SUPPORT",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#nr-degenerate-support",
"start_line": 360,
"end_line": 396,
"range_sha256": "69497df8c3c829674fb1774ada432ca7ab2219ee67d4e685d8e75c3a56203720",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.BD.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 396,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-NR.md",
"line": 360,
"through_line": 396,
"description": "Arbitrary locally compact Hausdorff group and normal continuous action on an arbitrary von Neumann algebra. The support theorem allows any normal coefficient representation, including nonfaithful or zero ones; identification of its supported component with the original crossed product retains faithfulness. Scalar compressions use all L2 scalar vectors and full predual normality. Tensor location uses arbitrary Hilbert-space bases and finite-subset nets. The four-summand counterexample uses the trivial action of the two-element group on C."
}
],
"range_bytes": 3202
},
{
"id": "OA-FLOW.NR.COMPACT-COMPRESSIONS",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#nr-compact-compressions",
"start_line": 397,
"end_line": 483,
"range_sha256": "14f6fee16142bbe63fa448a3b2887f92daed124ab49a8c869ba4af183017205e",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.1",
"OA-FLOW.NR.3",
"OA-FLOW.CP.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 483,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-NR.md",
"line": 397,
"through_line": 483,
"description": "Arbitrary locally compact Hausdorff group and normal continuous action on an arbitrary von Neumann algebra. The support theorem allows any normal coefficient representation, including nonfaithful or zero ones; identification of its supported component with the original crossed product retains faithfulness. Scalar compressions use all L2 scalar vectors and full predual normality. Tensor location uses arbitrary Hilbert-space bases and finite-subset nets. The four-summand counterexample uses the trivial action of the two-element group on C."
}
],
"range_bytes": 4291
},
{
"id": "OA-FLOW.NR.TENSOR-LOCATION",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#nr-tensor-location",
"start_line": 484,
"end_line": 531,
"range_sha256": "3f6cfa9d24f2bcae9a3551fc7f8ca89cd6d21e90bc8c5fdec08fe4652bca1b46",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.NR.DEGENERATE-SUPPORT",
"OA-FLOW.CF.1",
"OA-FLOW.CF.8",
"OA-FLOW.BD.1",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 531,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-NR.md",
"line": 484,
"through_line": 531,
"description": "Arbitrary locally compact Hausdorff group and normal continuous action on an arbitrary von Neumann algebra. The support theorem allows any normal coefficient representation, including nonfaithful or zero ones; identification of its supported component with the original crossed product retains faithfulness. Scalar compressions use all L2 scalar vectors and full predual normality. Tensor location uses arbitrary Hilbert-space bases and finite-subset nets. The four-summand counterexample uses the trivial action of the two-element group on C."
}
],
"range_bytes": 2898
},
{
"id": "OA-FLOW.NR.DEGENERATE-EXAMPLE",
"unit": "OA-FLOW-NR",
"source_file": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc",
"reader": "OA-FLOW-NR.html#nr-degenerate-example",
"start_line": 532,
"end_line": 570,
"range_sha256": "44cebb8441d4512bebd19edc33fbac41d4a0041d9cd46f085f5467159c0d471f",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.DEGENERATE-SUPPORT",
"OA-FLOW.CF.8"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 570,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-NR.md",
"line": 532,
"through_line": 570,
"description": "Arbitrary locally compact Hausdorff group and normal continuous action on an arbitrary von Neumann algebra. The support theorem allows any normal coefficient representation, including nonfaithful or zero ones; identification of its supported component with the original crossed product retains faithfulness. Scalar compressions use all L2 scalar vectors and full predual normality. Tensor location uses arbitrary Hilbert-space bases and finite-subset nets. The four-summand counterexample uses the trivial action of the two-element group on C."
}
],
"range_bytes": 2326
},
{
"id": "OA-FLOW.GDA.13",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-13",
"start_line": 832,
"end_line": 912,
"range_sha256": "fce1198e9181f76d4ff525bc366939c6f700b25a5e397d7f3da7f25ed22be6f8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.5",
"OA-FLOW.GDA.6",
"OA-FLOW.EP.3",
"OA-FLOW.EP.6",
"OA-FLOW.NF.6",
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.1",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 895
},
{
"id": "OA-FLOW.GDA.PAIRED-COEFFICIENTS",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-paired-coefficients",
"start_line": 915,
"end_line": 1058,
"range_sha256": "e003ef930b0afa1dbb10e56488490f23e400cf5f1a44a4736cafab7a3a95fa9d",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.7",
"OA-FLOW.GDW.1",
"OA-FLOW.GDW.2",
"OA-FLOW.EP.6",
"OA-FLOW.NF.5",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1058,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"line": 915,
"through_line": 1058,
"description": "Arbitrary locally compact Hausdorff groups, point-ultraweak continuous normal actions and von Neumann algebras retain the lesson setting. Scalar comparison weights are faithful normal semifinite. Balanced matrices have finite size; Haar rescaling uses a finite positive constant. The nondiscrete boundedness characterization assumes a nonzero coefficient algebra; the final circle example uses probability Haar measure and proves its Fourier basis completeness locally."
}
]
},
{
"id": "OA-FLOW.GDA.COLUMN-DOMAINS",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-column-domains",
"start_line": 1059,
"end_line": 1147,
"range_sha256": "f0ca6f902b9a2bd6b95175b3e3838c658205bb4fd6c77352e9f88420b68cd0ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.1",
"OA-FLOW.BC.2",
"OA-FLOW.GDA.BALANCED-DUAL",
"OA-FLOW.GDW.1",
"OA-FLOW.GDA.8",
"OA-FLOW.EP.4",
"OA-FLOW.EP.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1147,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"line": 1059,
"through_line": 1147,
"description": "Arbitrary locally compact Hausdorff groups, point-ultraweak continuous normal actions and von Neumann algebras retain the lesson setting. Scalar comparison weights are faithful normal semifinite. Balanced matrices have finite size; Haar rescaling uses a finite positive constant. The nondiscrete boundedness characterization assumes a nonzero coefficient algebra; the final circle example uses probability Haar measure and proves its Fourier basis completeness locally."
}
]
},
{
"id": "OA-FLOW.GDA.HAAR-RESCALING",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-haar-rescaling",
"start_line": 1148,
"end_line": 1220,
"range_sha256": "0c7fee0f46ac7a1c9c83f5513a3d359349ea60802c7752376cc5be9a4eb853a0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.BC.5",
"OA-FLOW.GDA.7",
"OA-FLOW.GDA.8",
"OA-FLOW.GDA.UNIQUE-MAP",
"OA-FLOW.GDA.COLUMN-DOMAINS",
"OA-FLOW.GDW.5",
"OA-FLOW.GDW.6",
"OA-FLOW.SF.SB4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1220,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"line": 1148,
"through_line": 1220,
"description": "Arbitrary locally compact Hausdorff groups, point-ultraweak continuous normal actions and von Neumann algebras retain the lesson setting. Scalar comparison weights are faithful normal semifinite. Balanced matrices have finite size; Haar rescaling uses a finite positive constant. The nondiscrete boundedness characterization assumes a nonzero coefficient algebra; the final circle example uses probability Haar measure and proves its Fourier basis completeness locally."
}
]
},
{
"id": "OA-FLOW.GDA.NONDISCRETE-IDENTITY",
"unit": "OA-FLOW-GDA",
"source_file": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896",
"reader": "OA-FLOW-GDA.html#gda-nondiscrete-identity",
"start_line": 1221,
"end_line": 1306,
"range_sha256": "0d3a78c16da4f5fa97fbfe0eaf1322c4366b121e7ccca3ca4234f35aad552b41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GDA.6",
"OA-FLOW.GDA.13",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1306,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-GDA.md",
"line": 1221,
"through_line": 1306,
"description": "Arbitrary locally compact Hausdorff groups, point-ultraweak continuous normal actions and von Neumann algebras retain the lesson setting. Scalar comparison weights are faithful normal semifinite. Balanced matrices have finite size; Haar rescaling uses a finite positive constant. The nondiscrete boundedness characterization assumes a nonzero coefficient algebra; the final circle example uses probability Haar measure and proves its Fourier basis completeness locally."
}
]
},
{
"id": "OA-FLOW.L128.FC7",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc7",
"start_line": 1512,
"end_line": 1546,
"range_sha256": "97f158bfe56145daabdc55027b5194ce4713aee4c8aed1183b1f3236e48da18a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC6",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1546
},
{
"id": "OA-FLOW.L128.FC8",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc8",
"start_line": 1547,
"end_line": 1591,
"range_sha256": "021e5acfe747803e1272d8e10f18d93742c6b2447328d76531e563a31431fe82",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2A",
"OA-FLOW.L128.FC3",
"OA-FLOW.L128.FC4",
"OA-FLOW.NCF.5",
"OA-FLOW.ST.1",
"OA-FLOW.ST.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1591
},
{
"id": "OA-FLOW.L128.FC9",
"unit": "OA-FLOW-L128",
"source_file": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5",
"reader": "OA-FLOW-L128.html#l128-fc9",
"start_line": 1592,
"end_line": 1634,
"range_sha256": "1d90cd8079e54d9c4d8cb659f50e98f4606a05e7396a2d5be9aebc2c9d17a7ba",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.OR0",
"OA-FLOW.L128.TY0",
"OA-FLOW.L128.TY1",
"OA-FLOW.SC.4",
"OA-FLOW.SC.6",
"OA-FLOW.SC.7"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1634
},
{
"id": "OA-FLOW.L29.COCYCLE-CONJUGACY",
"unit": "OA-FLOW-L29",
"source_file": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759",
"reader": "OA-FLOW-L29.html#l29-cocycle-conjugacy",
"start_line": 725,
"end_line": 816,
"range_sha256": "7f95dd7e3c22c046849a876632f6b047898a07806c28093a8263f25f763ead19",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.NR.3",
"OA-FLOW.NR.4",
"OA-FLOW.NR.5",
"OA-FLOW.AT.4",
"OA-FLOW.UC.0",
"OA-FLOW.UC.1",
"OA-FLOW.UC.4",
"OA-FLOW.ST.2",
"OA-FLOW.DA.ACTION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 813
},
{
"id": "OA-FLOW.L115.GENERAL-AVERAGE",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#l115-general-average",
"start_line": 688,
"end_line": 786,
"range_sha256": "fb0f1d520aed84c2f9e8bd64a8794cfd8e0a39381c050e2324ef36ec45a81aaf",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.HR.2",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.EP.1",
"OA-FLOW.EP.2",
"OA-FLOW.EP.3",
"OA-FLOW.EP.4",
"OA-FLOW.EP.6",
"OA-FLOW.FF.6",
"OA-FLOW.WF.2",
"OA-FLOW.PC.1",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 785
},
{
"id": "OA-FLOW.L115.INHERITANCE",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#l115-integrability-inheritance",
"start_line": 787,
"end_line": 859,
"range_sha256": "ad7e3190b1a8c3d71eac0957afd91e85ea9350ad0d7f9f719d625fd6a0d7d8b1",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.L115.GENERAL-AVERAGE",
"OA-FLOW.EP.1",
"OA-FLOW.EP.4",
"OA-FLOW.EP.6",
"OA-FLOW.CP.6",
"OA-FLOW.WF.2",
"OA-FLOW.NR.1",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.NCF.1",
"OA-FLOW.CF.6",
"OA-FLOW.L29.COCYCLE-CONJUGACY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 856
},
{
"id": "OA-FLOW.CONNES.REGULAR-DOMAIN",
"unit": "OA-FLOW-L115",
"source_file": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9",
"reader": "OA-FLOW-L115.html#l115-regular-multiplier-domain",
"start_line": 861,
"end_line": 1035,
"range_sha256": "822a9b47e3498bcc7126d82b9ed1daf4f70459ad6c110b060f0873e8ea1140ac",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CONNES.REGULARINT",
"OA-FLOW.L114.AVERAGE",
"OA-FLOW.INTSPEC.DOMAIN",
"OA-FLOW.CONT.DENSITY",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.CP.6"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 1035,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-L115.md",
"line": 861,
"through_line": 878,
"description": "Arbitrary LCA group with locally determined Haar multiplication algebra on L2. The bounded linear domain is the complex span of positive bounded orbit averages; no positivity of the multiplier or countability assumption is added."
},
{
"source": "src/OA-FLOW-L115.md",
"line": 861,
"through_line": 1035,
"description": "Complete positive extended orbit value, uniform normal coefficients, recovery of compact continuous tests, radial clipping of compact phase tests and exact L1/L2 domain intersections."
}
]
},
{
"id": "OA-FLOW.TIC.SETTING",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.setting",
"start_line": 1,
"end_line": 40,
"range_sha256": "bc893853edc0f1f72168898aac8872801f856ee9bc9cdf7e9189fcdbf621ec1c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2",
"OA-FLOW.PC.3",
"OA-FLOW.PC.4",
"OA-FLOW.PC.5",
"OA-FLOW.PC.6",
"OA-FLOW.PC.7",
"OA-FLOW.AT.1",
"OA-FLOW.AT.2",
"OA-FLOW.AT.3",
"OA-FLOW.AT.4",
"OA-FLOW.AT.5",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.HR.7",
"OA-FLOW.HR.8",
"OA-FLOW.HR.9",
"OA-FLOW.GRP.HAARCONVENTIONS",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.GRP.COMPLETIONS",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.L115.GENERAL-AVERAGE",
"OA-FLOW.L115.INHERITANCE",
"OA-FLOW.DS.2",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.SP.0",
"OA-FLOW.SP.1",
"OA-FLOW.SP.2",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4",
"OA-FND-PB-09",
"OA-FND-PB-11"
]
},
{
"id": "OA-FLOW.TIC.1",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.1",
"start_line": 41,
"end_line": 67,
"range_sha256": "085ae4f43401b26c6e0f11f5504ddcf1d7fba2616d04b77bd70b51261a749e4c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.SETTING",
"OA-FLOW.L115.GENERAL-AVERAGE",
"OA-FLOW.L115.INHERITANCE",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.TIC.2",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.2",
"start_line": 68,
"end_line": 92,
"range_sha256": "fdde81cca0eb96d203febd4fb13b8399c665e8bfe0d54e82e817259acd7b70bd",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.SETTING",
"OA-FLOW.CP.4",
"OA-FLOW.CP.5",
"OA-FLOW.CP.6",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.ST.2",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.TIC.3",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.3",
"start_line": 93,
"end_line": 123,
"range_sha256": "8d15425e3accab5c669038a817f49ecdf9d75f8cbe39269afefd0aa695d28879",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.2",
"OA-FLOW.AT.1",
"OA-FLOW.AT.3",
"OA-FLOW.GRP.TRANSLATIONS"
]
},
{
"id": "OA-FLOW.TIC.4",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.4",
"start_line": 124,
"end_line": 160,
"range_sha256": "890c3265fbc4024c16392ec9d47cf36db26bc313ddc02b2dcef7b8f216d628f0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.SETTING",
"OA-FLOW.AT.1",
"OA-FLOW.ST.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.L115.INHERITANCE"
]
},
{
"id": "OA-FLOW.TIC.5",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.5",
"start_line": 161,
"end_line": 197,
"range_sha256": "ceb2a089c17a3517227c0ccef5dc1955d055988fc7a59710a2b4cab56e41c1ff",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.4",
"OA-FLOW.PC.0",
"OA-FLOW.PC.1",
"OA-FLOW.PC.2"
]
},
{
"id": "OA-FLOW.TIC.6",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.6",
"start_line": 198,
"end_line": 214,
"range_sha256": "9135e0859cf869ea3ef6b81491512aeccd1e5076cc22946243f0596943dca489",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.5",
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.PC.7"
]
},
{
"id": "OA-FLOW.TIC.7",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.7",
"start_line": 215,
"end_line": 257,
"range_sha256": "136dd84c7a04340c7cecf96dc9bcd046c26ea0a90d06ed7135c2f092c963f5a3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.4",
"OA-FLOW.TIC.5",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.2.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.PC.1",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.AT.1",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.L115.GENERAL-AVERAGE"
]
},
{
"id": "OA-FLOW.TIC.8",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.8",
"start_line": 258,
"end_line": 292,
"range_sha256": "46505beb8c89c69a7e14b83c3f1b5f69004037dc4c7bc5c373a090820fa57935",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.1",
"OA-FLOW.TIC.3",
"OA-FLOW.TIC.4",
"OA-FLOW.TIC.5",
"OA-FLOW.TIC.6",
"OA-FLOW.TIC.7",
"OA-FLOW.PC.3",
"OA-FLOW.PC.7",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.L115.INHERITANCE"
]
},
{
"id": "OA-FLOW.TIC.9",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.9",
"start_line": 293,
"end_line": 339,
"range_sha256": "fd9ae909d679c8043cde61bf2cba9ce48c6e0ef0b8115b5e543d6b6283de32f3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.2",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.HR.5",
"OA-FLOW.HR.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
]
},
{
"id": "OA-FLOW.TIC.10",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.10",
"start_line": 340,
"end_line": 357,
"range_sha256": "88f3f07524e6ff04a57d27e3f23135a52617b5364c4d4a6e1820360545937328",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.DS.2",
"OA-FLOW.ST.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.6",
"OA-FLOW.XGAPS.CSTAR.H1",
"OA-FLOW.GRP.COMPLETIONS"
]
},
{
"id": "OA-FLOW.TIC.11",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.11",
"start_line": 358,
"end_line": 393,
"range_sha256": "4018b216e76d55137da60053f80a466c2ebc64024947f9c2313516d0a0423f24",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.2",
"OA-FLOW.TIC.10",
"OA-FLOW.AT.1",
"OA-FLOW.ST.2",
"OA-FND-PB-09",
"OA-FND-PB-11",
"OA-FLOW.SP.0",
"OA-FLOW.SP.1",
"OA-FLOW.SP.2",
"OA-FLOW.SP.3",
"OA-FLOW.SP.4"
]
},
{
"id": "OA-FLOW.TIC.12",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.12",
"start_line": 394,
"end_line": 419,
"range_sha256": "9af9b0b51f48bc254e566f6ee6803f9fe7c8323223245085530acdf38fd3d0d3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.PC.3",
"OA-FLOW.PC.5",
"OA-FLOW.ND.TENSOR",
"OA-FLOW.L115.INHERITANCE"
]
},
{
"id": "OA-FLOW.TIC.13",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.13",
"start_line": 420,
"end_line": 448,
"range_sha256": "607189e6f99f58ba161c3188e4f250e1fa72af1d6f393fb0b621818fb97857c7",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.9",
"OA-FLOW.TIC.10",
"OA-FLOW.TIC.11",
"OA-FLOW.TIC.12"
]
},
{
"id": "OA-FLOW.TIC.14",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.14",
"start_line": 449,
"end_line": 461,
"range_sha256": "6e3b54a1a998482801b8932885e6333dbb808ebeab2da0a5af790a1b61d3b6d9",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.5",
"OA-FLOW.TIC.6",
"OA-FLOW.TIC.8",
"OA-FLOW.TIC.11",
"OA-FLOW.TIC.12",
"OA-FLOW.TIC.13",
"OA-FLOW.DS.2"
]
},
{
"id": "OA-FLOW.TIC.15",
"unit": "OA-FLOW-TIC",
"source_file": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767",
"reader": "OA-FLOW-TIC.html#oa-flow.tic.15",
"start_line": 462,
"end_line": 485,
"range_sha256": "74437e541eb70a9ea459713ab0ae7416301f77a11a923a98f19f2bd5c4749396",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.TIC.1",
"OA-FLOW.TIC.7",
"OA-FLOW.TIC.8",
"OA-FLOW.TIC.11",
"OA-FLOW.TIC.13"
]
},
{
"id": "OA-FLOW.SAR.SETTING",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.setting",
"start_line": 1,
"end_line": 21,
"range_sha256": "b13a2430195f8507bf75323aeae14fa848ed733c624fa63cafcb8e7f15de986a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2",
"OA-FLOW.IS.2",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.SC.2",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.NF.6",
"OA-FLOW.CC.0",
"OA-FLOW.L128.RT0",
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.TC3",
"OA-FLOW.COVREP.NORMALIZER"
],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.SPECTRAL",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.spectral",
"start_line": 22,
"end_line": 64,
"range_sha256": "73ad6d35eb3c6154a445dc97f7c34f4f5c6fca163716aa20b8358e4b1cfcdee3",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.IS.2",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 64,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 22,
"through_line": 64,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.ROTATION",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.rotation",
"start_line": 65,
"end_line": 85,
"range_sha256": "95a0a38ef532ad48ec0b9391b48dbac96fc99e2493d5f3d003c708a8bc521294",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.SPECTRAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 85,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 65,
"through_line": 85,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.TRACE",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.trace",
"start_line": 86,
"end_line": 106,
"range_sha256": "2ff871b78e928f20009e5f7943cb0ce9440c767099f0f41d077c8eb2e9d03b0b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.ROTATION",
"OA-FLOW.CC.0",
"OA-FLOW.HR.3",
"OA-FLOW.SC.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 106,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 86,
"through_line": 106,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.CROSSED",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.crossed",
"start_line": 107,
"end_line": 157,
"range_sha256": "d1e6b5f92e571a38183a9e3149ef2034ad2f8a037f1e626ef54f1aa0a1fe589c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.TRACE",
"OA-FLOW.L128.RT0",
"OA-FLOW.L128.FC0",
"OA-FLOW.L128.FC1",
"OA-FLOW.L128.FC2",
"OA-FLOW.L128.TC3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.NF.6",
"OA-FLOW.ST.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 157,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 107,
"through_line": 157,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.GENERATOR",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.generator",
"start_line": 158,
"end_line": 209,
"range_sha256": "d484fde96db39e9147382338390f76ba0fd9aac11fda13c1a1b085385a3892b5",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 209,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 158,
"through_line": 209,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.AFFINE",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.affine",
"start_line": 210,
"end_line": 239,
"range_sha256": "2203d5ace5567f7b62f52163f04b415a085c54fbae256e4d8072c8a3193a41cb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.GENERATOR",
"OA-FLOW.SAR.SPECTRAL"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 239,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 210,
"through_line": 239,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.FIELD",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.field",
"start_line": 240,
"end_line": 296,
"range_sha256": "bf915105e84e39afdb4af0e3e00fa78bfbc766cbfe45dee4d207eee3e44ee85b",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.AFFINE",
"OA-FLOW.SAR.SPECTRAL",
"OA-FLOW.COVREP.NORMALIZER",
"OA-FLOW.IS.2"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 296,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 240,
"through_line": 296,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.SAR.MODELS",
"unit": "OA-FLOW-SAR",
"source_file": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32",
"reader": "OA-FLOW-SAR.html#oa-flow.sar.models",
"start_line": 297,
"end_line": 334,
"range_sha256": "c404e36276187b03d0aa8305c3ac7c8b7ebd1f11d2aa0a2a9100517c72721e95",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SAR.ROTATION",
"OA-FLOW.SAR.CROSSED",
"OA-FLOW.SAR.FIELD",
"OA-FLOW.BD.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 334,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-SAR.md",
"line": 1,
"through_line": 21,
"description": "Global conventions and exact earlier proofs; additional assumptions are stated in the body."
},
{
"source": "src/OA-FLOW-SAR.md",
"line": 297,
"through_line": 334,
"description": "Complete current statement, hypotheses and proof."
}
]
},
{
"id": "OA-FLOW.QM.MEASURABLE",
"unit": "OA-FLOW-L43",
"source_file": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821",
"reader": "OA-FLOW-L43.html#oa-flow.qm.measurable",
"start_line": 353,
"end_line": 425,
"range_sha256": "f6c152de46d71c4df4837ef71f8f3af3c42a34bbcfa5328fe576873c6ce05751",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.QM.SETTING",
"OA-FLOW.QM.CUTOFF",
"OA-FLOW.QM.DESCENT",
"OA-FLOW.QM.CLASS",
"OA-FLOW.QM.DERIVATIVE",
"OA-FLOW.QF.6",
"OA-FLOW.QF.7"
]
},
{
"id": "OA-FLOW.EOR.0",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-0",
"start_line": 1,
"end_line": 26,
"range_sha256": "21b25c78011000a97ef0d2ce16d21ccebeb9a553362afb502d94a5b39f4e8562",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [],
"may_be_used_as_earlier_proved_premise": false,
"proof_available_after_line": null,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.1",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-1",
"start_line": 27,
"end_line": 89,
"range_sha256": "ac8bf7a4df9c6287b4acf0cea15dd88bd21f4614d8b50dc76f404c3445df8630",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.MODEL.DOMAIN",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.7",
"OA-FLOW.AT.3",
"OA-FLOW.AT.5",
"OA-FLOW.AT.6",
"OA-FLOW.GNS.4.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 89,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.2",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-2",
"start_line": 90,
"end_line": 120,
"range_sha256": "0085914135c4b8ce882350b1f256a8d87c7ed9bf05a14c6772839a5ecbffb6ee",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.8",
"OA-FLOW.CF.10",
"OA-FLOW.GNS.2.1",
"OA-FLOW.GNS.3.1",
"OA-FLOW.GNS.3.2",
"OA-FLOW.GNS.3.3",
"OA-FLOW.GNS.4.1",
"OA-FLOW.GNS.5.1",
"OA-FLOW.ST.COEFFICIENT",
"OA-FLOW.BD.1",
"OA-FLOW.BD.3",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 120,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.3",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-3",
"start_line": 121,
"end_line": 150,
"range_sha256": "764f8c593e0061e95ca16aaaa66e14c1057c97f3b42943d3e2a93842edaaea54",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EOR.1",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.HR.7",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 150,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.4",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-4",
"start_line": 151,
"end_line": 177,
"range_sha256": "88bfaae26191bd8ca9491c1c08f69a0012126a585c07e1fc5d7f67c39785869c",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EOR.1",
"OA-FLOW.EOR.2",
"OA-FLOW.EOR.3",
"OA-FLOW.CF.6",
"OA-FLOW.CF.7",
"OA-FLOW.CF.8",
"OA-FLOW.CF.9",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.BD.5",
"OA-FLOW.HR.TOPOLOGY"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 177,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.5",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-5",
"start_line": 178,
"end_line": 225,
"range_sha256": "a0631d96871d2462316bcac0384ad418a36880b6b307fdc60e6d57144c043ecb",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EOR.1",
"OA-FLOW.EOR.2",
"OA-FLOW.EOR.4",
"OA-FLOW.CP.4",
"OA-FLOW.CP.6",
"OA-FLOW.CF.8",
"OA-FLOW.HR.2",
"OA-FLOW.HR.3",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.BD.4",
"OA-FLOW.BD.5",
"OA-FLOW.ST.2",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 225,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.6",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-6",
"start_line": 226,
"end_line": 259,
"range_sha256": "4e6a0681f652ae8d2d970ac913ac9068ef4d894f5ab63c91b55a333ca7699c34",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EOR.5",
"OA-FLOW.QM.RHO",
"OA-FLOW.QM.CLASS",
"OA-FLOW.QF.1",
"OA-FLOW.QF.3",
"OA-FLOW.IW.1",
"OA-FLOW.HR.3",
"OA-FLOW.HR.5",
"OA-FLOW.HR.9",
"OA-FLOW.HR.TOPOLOGY",
"OA-FLOW.GRP.TRANSLATIONS",
"OA-FLOW.GRP.VECTORINTEGRATION"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 259,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
},
{
"id": "OA-FLOW.EOR.7",
"unit": "OA-FLOW-EOR",
"source_file": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928",
"reader": "OA-FLOW-EOR.html#eor-7",
"start_line": 260,
"end_line": 280,
"range_sha256": "d2e668d35bcd4f5cec02d9ac82cce8bbc23e3f98324dc76294e53a0b4d1ac168",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.EOR.1",
"OA-FLOW.EOR.2",
"OA-FLOW.EOR.3",
"OA-FLOW.EOR.4",
"OA-FLOW.EOR.5",
"OA-FLOW.EOR.6",
"OA-FLOW.IW.1",
"OA-FLOW.GNS.7.1"
],
"may_be_used_as_earlier_proved_premise": true,
"proof_available_after_line": 280,
"source_level_hypothesis_ranges": [
{
"source": "src/OA-FLOW-EOR.md",
"line": 1,
"through_line": 26,
"description": "The precise theorem and conventions; the bounded extension lemma explicitly states its separate general hypotheses."
}
]
}
],
"current_reader_sources": {
"OA-FLOW-CF": {
"source": "src/OA-FLOW-CF.md",
"source_sha256": "7ccbe454799f6f38015be837809a6876b8b39a7cb98c002eb7f1164de9ea8357"
},
"OA-FLOW-SC": {
"source": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637"
},
"OA-FLOW-GNS": {
"source": "src/OA-FLOW-GNS.md",
"source_sha256": "3b54bf6286dca7865eac83144405386f85c97ce81a16073ee249b6b2fe4ae4b8"
},
"OA-FLOW-TOPOLOGY": {
"source": "src/OA-FLOW-TOPOLOGY.md",
"source_sha256": "bc8a5db2cd33669d0b4eed82a56f622b185f2f5900e2fcc73ae71dfac23c6cd8"
},
"OA-FLOW-HR": {
"source": "src/OA-FLOW-HR.md",
"source_sha256": "45a21cf0bd974a9431d625517251839af3c8a3bd1fa65de4474b1ca4ae94eba5"
},
"OA-FLOW-L24": {
"source": "src/OA-FLOW-L24.md",
"source_sha256": "ce08364e03f97230551f56a2051e7fe1453a6db3a863ebbe849679606dd5d4a7"
},
"OA-FLOW-L25": {
"source": "src/OA-FLOW-L25.md",
"source_sha256": "f291cd6eb8932543ebf43d7e098421e21c077a49650ccbbf24a6d50f890e6456"
},
"OA-FLOW-HARMONIC": {
"source": "src/OA-FLOW-HARMONIC.md",
"source_sha256": "543b5cee0579fecf19b41edeb0c8d80f4299932fe4126a2f11d4aaae0a035ef2"
},
"OA-FLOW-PLANCHEREL": {
"source": "src/OA-FLOW-PLANCHEREL.md",
"source_sha256": "0824a9b7e8afc170ca61d705dbd06c06844ca597c4bfe5754be316a1e14ec5b3"
},
"OA-FLOW-HARMONIC-LATE": {
"source": "src/OA-FLOW-HARMONIC-LATE.md",
"source_sha256": "1743fe83b68c21fb2057b2e54b48b0127d2d0f946e839be483ef13ce12f36722"
},
"OA-FLOW-L138": {
"source": "src/OA-FLOW-L138.md",
"source_sha256": "fc1a411650673cbefa5117fa83bd2072a17e66e9ae962b9a2f1631771e07dda5"
},
"OA-FLOW-SF": {
"source": "src/OA-FLOW-SF.md",
"source_sha256": "b7647a8d7c2ceebc2bb67edcf1776bda86a6d7b56db1606bdef26f6105dba22f"
},
"OA-FLOW-FF": {
"source": "src/OA-FLOW-FF.md",
"source_sha256": "ddaef17a27c36b040964749e0fc26769bab5eab5eb85712ff5710893d2087f36"
},
"OA-FLOW-CP": {
"source": "src/OA-FLOW-CP.md",
"source_sha256": "502bf387088c5fc1d778569fc78ee126b76bd3b42851f3978a88ea3053175d7a"
},
"OA-FLOW-AS": {
"source": "src/OA-FLOW-AS.md",
"source_sha256": "ed52cc65c52f569ec7aa09e0633eef3058b29667995fdc64afd07f9190517449"
},
"OA-FLOW-PC": {
"source": "src/OA-FLOW-PC.md",
"source_sha256": "ddf3c5bbeffa1f46d68941d589daa9791387d95497e4661bec94cb1e76b77463"
},
"OA-FLOW-CI": {
"source": "src/OA-FLOW-CI.md",
"source_sha256": "776a46025ddc560e38207993b85be6f616ed42530a2a4754b18c84a5cdf9d401"
},
"OA-FLOW-BD": {
"source": "src/OA-FLOW-BD.md",
"source_sha256": "de8def4c42e8e7ec323efb7a923984171f8d5d1be1e9c0263b9f80f84638db6e"
},
"OA-FLOW-ST12": {
"source": "src/OA-FLOW-ST12.md",
"source_sha256": "700c9ed27a668738ba8289d3ccdede950f41ecd77847e7e1d6af4efa239d5599"
},
"OA-FLOW-HA-R": {
"source": "src/OA-FLOW-HA-R.md",
"source_sha256": "3eb5ebe45bfdcc16f4f73d0737a5717553640ab2a9f658e5830a1601f3372611"
},
"OA-FLOW-WH04": {
"source": "src/OA-FLOW-WH04.md",
"source_sha256": "b1829dd0ebc6b20e185179aeba9f750af227aa279efe5be4dab69304595ba505"
},
"OA-FLOW-WF": {
"source": "src/OA-FLOW-WF.md",
"source_sha256": "98cb83c47266fc7af561916338a6289746a57825bf4479f20fcc83b4e77fbfa8"
},
"OA-FLOW-STC": {
"source": "src/OA-FLOW-STC.md",
"source_sha256": "6905b695aad18e71172e326d2d77f4572ea4519444280bbfbe2fe9b380dd7dfe"
},
"OA-FLOW-GW": {
"source": "src/OA-FLOW-GW.md",
"source_sha256": "cf13284c8c9aace8a85a2b2b78d270d229259d964c154d15d5d16cc3cac5a9c2"
},
"OA-FLOW-NF": {
"source": "src/OA-FLOW-NF.md",
"source_sha256": "8c964f22593c8e95b128067ff537f4b836cd2e71391aad7495e29063ba10ac27"
},
"OA-FLOW-CV": {
"source": "src/OA-FLOW-CV.md",
"source_sha256": "a50eca2ce7684dbd1cae54941b0449e3428ecd4fd524edd6a6ef81a80b3d243d"
},
"OA-FLOW-EW": {
"source": "src/OA-FLOW-EW.md",
"source_sha256": "4548ac5879ea4a05d7f9c46d02bbe29ced4b476644d967a1f32378a5c8ce7ef4"
},
"OA-FLOW-EP": {
"source": "src/OA-FLOW-EP.md",
"source_sha256": "313cdf5a0426642e58d4b02e1c90d0446837255ac7e8e52ff03c2079794314c6"
},
"OA-FLOW-WR": {
"source": "src/OA-FLOW-WR.md",
"source_sha256": "2eac3d79d4939070f905836709702b694bc11bfbd995b6ada06a85db2ae6d693"
},
"OA-FLOW-WH02": {
"source": "src/OA-FLOW-WH02.md",
"source_sha256": "5ced6463ce46f108a0ce9ee23a88a83bdcd554ff2a342207870d20092cf9446f"
},
"OA-FLOW-WH04-WEIGHT": {
"source": "src/OA-FLOW-WH04-WEIGHT.md",
"source_sha256": "6411579ccc1046c13fdbf05980c8ea503cc715d6410484c8d00bc3f0fbdbede1"
},
"OA-FLOW-MF-SB": {
"source": "src/OA-FLOW-MF-SB.md",
"source_sha256": "eb3ae4f4a0a626d90200130be95ddf1b803d797138aefa7a12d5fb214f8cff46"
},
"OA-FLOW-MF06": {
"source": "src/OA-FLOW-MF06.md",
"source_sha256": "0a66baa1801ee4632764a1826df54ae5b13b7493ee824a0e54952269707fcd6d"
},
"OA-FLOW-MW": {
"source": "src/OA-FLOW-MW.md",
"source_sha256": "c700af2aea826ef944a92d7b61d8e1f6352a72902f532846d70b8d972def757c"
},
"OA-FLOW-NO": {
"source": "src/OA-FLOW-NO.md",
"source_sha256": "ebe6bdf19b0e9056f913e5255a5ee9617074fa50897073a2e27046266669be1d"
},
"OA-FLOW-KT": {
"source": "src/OA-FLOW-KT.md",
"source_sha256": "d1cc53ef17bc0c541e61d688dca2a374803189a0e3e70364eed37fec86705d72"
},
"OA-FLOW-WS": {
"source": "src/OA-FLOW-WS.md",
"source_sha256": "1e79903fa6611e4c2ea54d3a565a3b6fd539545e5c6209b584293624ce8dfc55"
},
"OA-FLOW-TD": {
"source": "src/OA-FLOW-TD.md",
"source_sha256": "7226858a35b6b7f0201ed7c69358de8f92716248c14bf23287975490cd501a59"
},
"OA-FLOW-KU": {
"source": "src/OA-FLOW-KU.md",
"source_sha256": "acabd00673100ab9d7376d2bfb0ab3c77bc08f95021c80dd3f163e2a634b13ec"
},
"OA-FLOW-BC": {
"source": "src/OA-FLOW-BC.md",
"source_sha256": "c767d3a7dc7db46291989de7725878575a43fbab637975d47df0b3c32bb89274"
},
"OA-FLOW-MA": {
"source": "src/OA-FLOW-MA.md",
"source_sha256": "e674e5058e4ee80cb1875ef61256a7dd6b4747a3f91e7462c2347f162caf836a"
},
"OA-FLOW-CZ": {
"source": "src/OA-FLOW-CZ.md",
"source_sha256": "b0330d0e49388e80057be8ec99b1d8c55478e9fded17374f0b0d299e1d9a9bb1"
},
"OA-FLOW-NC": {
"source": "src/OA-FLOW-NC.md",
"source_sha256": "7d830972c76a266eba36f773788037e9ccc2b3880734856059a6a7655d978a8d"
},
"OA-FLOW-CR": {
"source": "src/OA-FLOW-CR.md",
"source_sha256": "17ed9619d841403f0ab294bb8446ace0cfb5abe141ee8205f818bba9c399d380"
},
"OA-FLOW-MC": {
"source": "src/OA-FLOW-MC.md",
"source_sha256": "70db3017259607d548864421e3eca1382b4440dc0ef9ca369eb9ec85f57bf9dc"
},
"OA-FLOW-CC": {
"source": "src/OA-FLOW-CC.md",
"source_sha256": "55541b29ede51b6c2b4b9250c263c6b3301ce45c9d9523cac003e643f36352ec"
},
"OA-FLOW-RF": {
"source": "src/OA-FLOW-RF.md",
"source_sha256": "e86bf046ec88cca873d39f7a249aa2f85c73104ed160a7b62d1e8b67145fc61b"
},
"OA-FLOW-AL": {
"source": "src/OA-FLOW-AL.md",
"source_sha256": "af7408463871fe497b8070a5e47431f3311badb84f68bbe16d058233c67f74e6"
},
"OA-FLOW-CT": {
"source": "src/OA-FLOW-CT.md",
"source_sha256": "ffb26a3191feb5c5bdfa27475ef3e12e07134500cab4c6932a6d6072da760466"
},
"OA-FLOW-TS": {
"source": "src/OA-FLOW-TS.md",
"source_sha256": "f3ccd40e8eaedf8d0eef3959f117b6854fc1a6b1e66a0d74a8ce1643199fd2fb"
},
"OA-FLOW-OT": {
"source": "src/OA-FLOW-OT.md",
"source_sha256": "9fadb70d3003046fa7f7f8c8626081dd770ce0929038ef5bd35f7c753185d53f"
},
"OA-FLOW-CD": {
"source": "src/OA-FLOW-CD.md",
"source_sha256": "b1b9235ccc053c0720bc8d8e9fad83618c7cce604932a3d23aa63dcc7f4b7189"
},
"OA-FLOW-FD": {
"source": "src/OA-FLOW-FD.md",
"source_sha256": "e5425943d34923563b297373a0c52a43160590a8566d178173413f41d3c2997c"
},
"OA-FLOW-VD": {
"source": "src/OA-FLOW-VD.md",
"source_sha256": "b67283728f3b7389f8fc51e4515747f56c5e9970ca571d9b37a82d58a096d9c5"
},
"OA-FLOW-PW": {
"source": "src/OA-FLOW-PW.md",
"source_sha256": "60eda4f4aa857aaf60b886417592ffd9b01427e5512c3295f929973fafe3a245"
},
"OA-FLOW-PF": {
"source": "src/OA-FLOW-PF.md",
"source_sha256": "7880e3f2cc7390f2dfa421ed581f7a54b1d7ccc935ee72c056839774e04e7655"
},
"OA-FLOW-GT": {
"source": "src/OA-FLOW-GT.md",
"source_sha256": "ea6fbd271bc9d695584215eecc8e09b8f7b254c2747af3294bbe566bed9b07c7"
},
"OA-FLOW-CA": {
"source": "src/OA-FLOW-CA.md",
"source_sha256": "2a4f3145e4b1b7e89d8241ebdf240d49561c609c4e7d9ce6d46b674156a69bd9"
},
"OA-FLOW-CS": {
"source": "src/OA-FLOW-CS.md",
"source_sha256": "871cccf60847c9557677eb511a6563dcb6d4e845bb237613c649c1af0eb737d5"
},
"OA-FLOW-MG": {
"source": "src/OA-FLOW-MG.md",
"source_sha256": "96d8c542900309077367f97df94d6242096f1692298c8a89315d78d6dd79c94d"
},
"OA-FLOW-L157": {
"source": "src/OA-FLOW-L157.md",
"source_sha256": "8caf188c24eda01401af4c3566071e0391f368d777c09d093d9f0690ef8fbdd3"
},
"OA-FLOW-L158": {
"source": "src/OA-FLOW-L158.md",
"source_sha256": "ca6b893f87686dd21c6bc71c961cab8eacd7b91dd1ba1bce38333c64d524643a"
},
"OA-FLOW-L34": {
"source": "src/OA-FLOW-L34.md",
"source_sha256": "09fb67762ddd8f43d5d3c6062feb6688ab7be07874f34505a088fae9422148e6"
},
"OA-FLOW-L35": {
"source": "src/OA-FLOW-L35.md",
"source_sha256": "b68df396a7b9d0c11bd16094c2e50895be598db392d51e351e59377a03f281f4"
},
"OA-FLOW-IP": {
"source": "src/OA-FLOW-IP.md",
"source_sha256": "75911d569807faadb1745f8878f58c5c2bd6a5f0867be7f1bf3d9c68dfdcd012"
},
"OA-FLOW-MT": {
"source": "src/OA-FLOW-MT.md",
"source_sha256": "108d2a3feafc6fc58313aba2fa1fc9f0577a0a6680d86acb6d46004a612ba081"
},
"OA-FLOW-DD": {
"source": "src/OA-FLOW-DD.md",
"source_sha256": "eeeb3f15ecdaee4070c19a01b21b2eb6e44a3d582c1d149acff46c96e89c37fb"
},
"OA-FLOW-GM": {
"source": "src/OA-FLOW-GM.md",
"source_sha256": "2d60c3620b79caf47540fe1c362833373796a6c59ffb200ad66131adb0185398"
},
"OA-FLOW-Z": {
"source": "src/OA-FLOW-Z.md",
"source_sha256": "a6ef118b12a3a0f3075cf4c423f2a7ef1162555d6fa2b1cb90df225d925685f4"
},
"OA-FLOW-ST": {
"source": "src/OA-FLOW-ST.md",
"source_sha256": "f3b9b64d9615a66347e42ce1d2590e918fa6dd163260173cad30bf1c8b5898c5"
},
"OA-FLOW-L159": {
"source": "src/OA-FLOW-L159.md",
"source_sha256": "c37de2d240e12cbd1f28bbe91aad4c03981e1865200ac75c28e96232ba913886"
},
"OA-FLOW-FP": {
"source": "src/OA-FLOW-FP.md",
"source_sha256": "3b82f4b4d601c3d63c038643231ea3b08188149b513add78874bf835a57233f8"
},
"OA-FLOW-HC": {
"source": "src/OA-FLOW-HC.md",
"source_sha256": "f5f07b83bc3b414ce725a5dcd99407f4d8ae6ffc4f5355117a10ce84dc819bd5"
},
"OA-FLOW-DP": {
"source": "src/OA-FLOW-DP.md",
"source_sha256": "ae60d32ffcd2a87a9dc65a94472028562dc8e1d03c5a48febe87230b1fb4edf1"
},
"OA-FLOW-UC": {
"source": "src/OA-FLOW-UC.md",
"source_sha256": "8a7061c6bb5da2c6d3b3b6327662bdf983b018e4a9c790a94ad2c0da53b8d28c"
},
"OA-FLOW-AC": {
"source": "src/OA-FLOW-AC.md",
"source_sha256": "9e13adad00498c0ac66be1ab9477f7c55233385f2760d4b81c84f110b8f61b7a"
},
"OA-FLOW-LF": {
"source": "src/OA-FLOW-LF.md",
"source_sha256": "1c3c2333ea663be46ad74d4a1d6beed72fc7b2662f90aefc77e61bc14506da2c"
},
"OA-FLOW-GBAC": {
"source": "src/OA-FLOW-GBAC.md",
"source_sha256": "f8cca7c1733176b35ccbe415958f805f9196eca81ace36df0d8ab6de0bbeb36b"
},
"OA-FLOW-AT": {
"source": "src/OA-FLOW-AT.md",
"source_sha256": "1fb270fc44badf485ccc0cd0fb23d5b4e8bddb26fa45c66e4423ac71e02fc014"
},
"OA-FLOW-BP": {
"source": "src/OA-FLOW-BP.md",
"source_sha256": "aa35f2e8739da7bfaa43fa9ec21b8bcee8b99b40ec92c9002f350a47395a4484"
},
"OA-FLOW-NR": {
"source": "src/OA-FLOW-NR.md",
"source_sha256": "3d49bc3e847d16ee74bf94f2c6809092b4ea129f5302654979284129cac1a3cc"
},
"OA-FLOW-GL": {
"source": "src/OA-FLOW-GL.md",
"source_sha256": "bcdf710877cf1f90fca6bbcb5ebbafc661825befbee86d549819fe82b323701f"
},
"OA-FLOW-SS": {
"source": "src/OA-FLOW-SS.md",
"source_sha256": "8daf0298dc2fc04321a600b2f50baff4fe02abad4a5923951e3484fcaa518a14"
},
"OA-FLOW-FS": {
"source": "src/OA-FLOW-FS.md",
"source_sha256": "2f350714343ef25fb7b862a0076cbe3ae5eb3c15ccaf886d7276ba948822368c"
},
"OA-FLOW-CAI": {
"source": "src/OA-FLOW-CAI.md",
"source_sha256": "b3aedb35dcd9852b967d00912696375d089a810b7196a21448c6e33f21264f30"
},
"OA-FLOW-WY": {
"source": "src/OA-FLOW-WY.md",
"source_sha256": "59c84e39a6e8133eaab51c0cb57a4d469c78ae33f69dedbc208b269b51ed1c15"
},
"OA-FLOW-BS": {
"source": "src/OA-FLOW-BS.md",
"source_sha256": "9934e1ce895f0cd4efad81add696690c9ee5f16db9a660601f0eb420e1e07f2b"
},
"OA-FLOW-L94": {
"source": "src/OA-FLOW-L94.md",
"source_sha256": "ff00be5782d7c9052bd2f8e04ff239d92f702ea1cba96352b7a0c6bfe24513bd"
},
"OA-FLOW-GDW": {
"source": "src/OA-FLOW-GDW.md",
"source_sha256": "0a74d59c267dc7d0e3a4fed2fc2a77ec18e829adc9a1e40f5586efe91989f112"
},
"OA-FLOW-CCM": {
"source": "src/OA-FLOW-CCM.md",
"source_sha256": "ac6d7c55746ac39996738d31ebfae3c968af3299e91b2ea728132a0b3404429a"
},
"OA-FLOW-QF": {
"source": "src/OA-FLOW-QF.md",
"source_sha256": "d7f972074628318c876aea2166f62d433212f8d65b661c575c738a7d5d826d39"
},
"OA-FLOW-L43": {
"source": "src/OA-FLOW-L43.md",
"source_sha256": "797bb3001b8def9b6fff10cafc962f90a5018a70bc70e37a52af21e4cdeb3821"
},
"OA-FLOW-L44": {
"source": "src/OA-FLOW-L44.md",
"source_sha256": "533baf62515a90b2cfce4dab8efffd01b852840507b26ed633290c8cb47c50d6"
},
"OA-FLOW-L45": {
"source": "src/OA-FLOW-L45.md",
"source_sha256": "92fae154dc3830e554f272460de3a2bfdddf16f3171e9344c3723f4570a7fca9"
},
"OA-FLOW-L46": {
"source": "src/OA-FLOW-L46.md",
"source_sha256": "c52be1d9da5197056272f66299112e0bd1da9c4a570107fc98f0cfb98a4e426d"
},
"OA-FLOW-L95": {
"source": "src/OA-FLOW-L95.md",
"source_sha256": "93c3c3bd8192a542e8f3bb18760178c51e5b7be1a4873747393d7ab3566cd3bf"
},
"OA-FLOW-L96": {
"source": "src/OA-FLOW-L96.md",
"source_sha256": "031f2d45b1e25a2d33093986d55fd14652940806fc0afeca022a1394217e2d4b"
},
"OA-FLOW-GDA": {
"source": "src/OA-FLOW-GDA.md",
"source_sha256": "b293b4c4a2386bc8f0b7b93f5a2d8409828da9af242f45555fc317813f1bc896"
},
"OA-FLOW-NCF": {
"source": "src/OA-FLOW-NCF.md",
"source_sha256": "4f0616f30c23c1e193a6769e1a5df79b8e63171f15d116f01cdb77eccc24651b"
},
"OA-FLOW-AF": {
"source": "src/OA-FLOW-AF.md",
"source_sha256": "80fadb8f00263752b4b701ae9be1d0109b18d764dc47c6715d609606fc2deae7"
},
"OA-FLOW-L87": {
"source": "src/OA-FLOW-L87.md",
"source_sha256": "cbf1d211411f51d9d4ae882c4193e513102c099d37869b69d47783b81ad353af"
},
"OA-FLOW-L88": {
"source": "src/OA-FLOW-L88.md",
"source_sha256": "de48f95a572282f90ce153d59d1e028acfe7b4fdd008615fb3fab8632392ee37"
},
"OA-FLOW-L89": {
"source": "src/OA-FLOW-L89.md",
"source_sha256": "f48083d54c054c54f4fb193f0cc2f1156c7f12b9b413fe53b6945dce72e8846a"
},
"OA-FLOW-L90": {
"source": "src/OA-FLOW-L90.md",
"source_sha256": "9357fd6e25831a65f756198acdfbaa9ac81f29ddff5d998348a2bd006f04d003"
},
"OA-FLOW-L91": {
"source": "src/OA-FLOW-L91.md",
"source_sha256": "90ede6465bc20c581118ae53318b3ca8cbdba492b0c98e6bb537e39748489ce8"
},
"OA-FLOW-L92": {
"source": "src/OA-FLOW-L92.md",
"source_sha256": "2334a9b3a2faf564c3a972d16a576d97a067bf2b764379c4baef9e43974e4940"
},
"OA-FLOW-L93": {
"source": "src/OA-FLOW-L93.md",
"source_sha256": "6e4804e32e126460af0b6b9bd31090881b596fe3627bbf830ee77656dcd6691a"
},
"OA-FLOW-IS": {
"source": "src/OA-FLOW-IS.md",
"source_sha256": "202ff5c029a89cf12c431f0b4f7d13225dd8ed38d4bb0b069d2e159467707a37"
},
"OA-FLOW-IW": {
"source": "src/OA-FLOW-IW.md",
"source_sha256": "eae06c1eb6fedc50cef172f142c0f1c78da4af5389dbccc067173e555106e792"
},
"OA-FLOW-L47": {
"source": "src/OA-FLOW-L47.md",
"source_sha256": "7ef767fc531f5699234d4b50eb31492bdf673b29ca92d9480d3d76f0ebb13765"
},
"OA-FLOW-L51": {
"source": "src/OA-FLOW-L51.md",
"source_sha256": "65bc68bd2591e5a36f7d7cbf031eea7ee505412a646367b0966ec04866f10621"
},
"OA-FLOW-L52": {
"source": "src/OA-FLOW-L52.md",
"source_sha256": "34ee29c874201797283b5778f342ec500382637cd12df05b2f21573ecd14d2ca"
},
"OA-FLOW-L53": {
"source": "src/OA-FLOW-L53.md",
"source_sha256": "036657f25d6e5a9e96855f78d8d03926391f5892c69c7b3307871604d6a67713"
},
"OA-FLOW-L55": {
"source": "src/OA-FLOW-L55.md",
"source_sha256": "f91b3dd9689f8f802f94e157c19c1fd99636ec34118e663d3dc35f65e0bd3f06"
},
"OA-FLOW-L72": {
"source": "src/OA-FLOW-L72.md",
"source_sha256": "3a08cc7defb2715da04e639e9b139f52d286005111c8325023ecb69036fac850"
},
"OA-FLOW-EOR": {
"source": "src/OA-FLOW-EOR.md",
"source_sha256": "b8ad3d98b981271a13c5fee5e2342d9d22e28311c850520d1813e41eeb6c9928"
},
"OA-FLOW-L69": {
"source": "src/OA-FLOW-L69.md",
"source_sha256": "5365b8d1938aec1f86cabdf51471ebd0ac5a8324fe79155a9f1390ed9af909a0"
},
"OA-FLOW-L71": {
"source": "src/OA-FLOW-L71.md",
"source_sha256": "0b6201bac688d01de3f7eaee0774e0d92ce6475b7e198b6670f114df8d569c95"
},
"OA-FLOW-L73": {
"source": "src/OA-FLOW-L73.md",
"source_sha256": "12c2e155683c63465966e0a9d101ccbd134a87b71695c959e1aadf24f354a9c1"
},
"OA-FLOW-L74": {
"source": "src/OA-FLOW-L74.md",
"source_sha256": "b4cfda44a3a0a0c539fe31a478f14d0474cad0db30ffc34b559b5b5c681fec01"
},
"OA-FLOW-L75": {
"source": "src/OA-FLOW-L75.md",
"source_sha256": "797c0b1d6ab1859d902373ab420bc1e44a3b4385da99d6e485f470323544032f"
},
"OA-FLOW-L42": {
"source": "src/OA-FLOW-L42.md",
"source_sha256": "f9422e1a08dfacd381daf143e8eeab4b344439723eab4c14811fdec8f57aba5e"
},
"OA-FLOW-ED": {
"source": "src/OA-FLOW-ED.md",
"source_sha256": "9b87fae941079d02d24341be23ad459849c5a4bd1e5971a8c1bdf58048a55282"
},
"OA-FLOW-L41": {
"source": "src/OA-FLOW-L41.md",
"source_sha256": "5f2e8765af89422a070fab369bc0aec06ee1c6f7aa1ed50499ad0f20008f5fde"
},
"OA-FLOW-MSF": {
"source": "src/OA-FLOW-MSF.md",
"source_sha256": "85d2cfaeec74e106bda5fed18c1ed2ab521254623f3a43a102cb0a7096bdd38a"
},
"OA-FLOW-L40": {
"source": "src/OA-FLOW-L40.md",
"source_sha256": "ff6d19308905ac2319d0ea662b709e9e885e73e39edfa8a2716f06e19fa784cd"
},
"OA-FLOW-L39": {
"source": "src/OA-FLOW-L39.md",
"source_sha256": "c65b40ee5ddbae40d5bf643bf55b099b129957dbfaf25149f7c622cc616bc17e"
},
"OA-FLOW-L38": {
"source": "src/OA-FLOW-L38.md",
"source_sha256": "350b25105a0094f16c253ae39fcf5934ec29d2ad951303e1d5525d035ffd8099"
},
"OA-FLOW-L37": {
"source": "src/OA-FLOW-L37.md",
"source_sha256": "a6e50f46964dcbfeb09662d4678673e0b4c07533fda3fae05a346d2b9242e30b"
},
"OA-FLOW-MX": {
"source": "src/OA-FLOW-MX.md",
"source_sha256": "79f7dd491b1cbf54e9296f1444d62be43742cf9811d35b2e87d064763f52fd43"
},
"OA-FLOW-L80": {
"source": "src/OA-FLOW-L80.md",
"source_sha256": "67432774872bb8be0eee44deea134d3659c6733ef341c49100924c3103926fc4"
},
"OA-FLOW-L81": {
"source": "src/OA-FLOW-L81.md",
"source_sha256": "4eb345099de5ef62cdaa456301e44a75991ce75baa1f62c96ff9aab96e3f353b"
},
"OA-FLOW-ND": {
"source": "src/OA-FLOW-ND.md",
"source_sha256": "7602d57d2dbf9f9896d45b181e0cff85c2774ef9a5a7ac76d065379247f2c4e5"
},
"OA-FLOW-WRC": {
"source": "src/OA-FLOW-WRC.md",
"source_sha256": "3691c1b9d97e6cb05bc3446e50bf6b9d92c6b3cd4751f28be47a118a62c150e7"
},
"OA-FLOW-AM": {
"source": "src/OA-FLOW-AM.md",
"source_sha256": "90e1330c4d8bc3145a29af68d0f5d0b267adb866340aa432510553beda1219e1"
},
"OA-FLOW-DA": {
"source": "src/OA-FLOW-DA.md",
"source_sha256": "f98e46347fef265156a75925475a3d4b1df0ad33f4652532c3596a223171759c"
},
"OA-FLOW-UR": {
"source": "src/OA-FLOW-UR.md",
"source_sha256": "ea3836dce1daf9023a3cfe990180199683b7a75342d5d9040332a0dda7bf4402"
},
"OA-FLOW-FR": {
"source": "src/OA-FLOW-FR.md",
"source_sha256": "f4d43bda6a168b39efe029ce37123d8c5c083047cc54f2ee8a8afbefcfcc4de7"
},
"OA-FLOW-NWR": {
"source": "src/OA-FLOW-NWR.md",
"source_sha256": "20763aa718c05d8ff06d5432f325559b85d2fae20f6459f773a1d49d2fe8bf49"
},
"OA-FLOW-L32": {
"source": "src/OA-FLOW-L32.md",
"source_sha256": "d1b0ea330d7d7c90106eed80d164b5d77a34eea6a613f9adb656caa7613f8e92"
},
"OA-FLOW-L29": {
"source": "src/OA-FLOW-L29.md",
"source_sha256": "4452c727399c48e668d7cb4f4fc3ab48e9548cded034e3834eae4ac16ed7e759"
},
"OA-FLOW-L31": {
"source": "src/OA-FLOW-L31.md",
"source_sha256": "fe6dc775d9cc2022030aef0e5161522fadad848ce469995421a5563c361f93da"
},
"OA-FLOW-TW": {
"source": "src/OA-FLOW-TW.md",
"source_sha256": "9f2b58cfe9dae71eeee70ec557b2356366c1587215665fb5e36dea2d26b48397"
},
"OA-FLOW-L33": {
"source": "src/OA-FLOW-L33.md",
"source_sha256": "1c167e195c4378a4088c51872a48d07ed4555d1f1df61c14c6848a0b56e96deb"
},
"OA-FLOW-L19": {
"source": "src/OA-FLOW-L19.md",
"source_sha256": "b2d60cd724fcab7b6be0dde15b1a614dcd217d33690af2c7be500d76cb04c317"
},
"OA-FLOW-CORE": {
"source": "src/OA-FLOW-CORE.md",
"source_sha256": "708b935bb687b3d9ed4fe5e9416877bdfd6cca004bca769982289c9c7c721372"
},
"OA-FLOW-ET": {
"source": "src/OA-FLOW-ET.md",
"source_sha256": "9c4fbd9f9f78d78e21e764ab82dec3786186940340aae763be94a941815a9fbd"
},
"OA-FLOW-HCR": {
"source": "src/OA-FLOW-HCR.md",
"source_sha256": "fb9f5e3968506e66160f87ba3f156753e42f022113a9eb944e085368955254ae"
},
"OA-FLOW-GCC": {
"source": "src/OA-FLOW-GCC.md",
"source_sha256": "774c940afec269d96959ad20f3027800dd756921ad7964c7268f9ede1ef99b42"
},
"OA-FLOW-L112": {
"source": "src/OA-FLOW-L112.md",
"source_sha256": "d12f2176c2a2dd8bc87b09856d619776805361dcec368b7535942682d692f7b2"
},
"OA-FLOW-L113": {
"source": "src/OA-FLOW-L113.md",
"source_sha256": "4d28e4ebb06480dacec0b126f93bee5c706bd2e70221ec7e098dc8c61870130f"
},
"OA-FLOW-L114": {
"source": "src/OA-FLOW-L114.md",
"source_sha256": "3e19a82f44c64b9990182eb5b0c9e65a9e34ac6cbae8902a3ad04b103ffd6c43"
},
"OA-FLOW-L117": {
"source": "src/OA-FLOW-L117.md",
"source_sha256": "2aff888431ce13839d22280549ca14111e37b069be99905c8bcec6d88eb657be"
},
"OA-FLOW-L118": {
"source": "src/OA-FLOW-L118.md",
"source_sha256": "409d7a44325d29ad6da369bec4f848394841d549b0ed85279dcf44aa25762026"
},
"OA-FLOW-L115": {
"source": "src/OA-FLOW-L115.md",
"source_sha256": "c732eb93f5c043ae3658fa4a6bd36af95a0ad9551f816af02f53d2a986b657f9"
},
"OA-FLOW-L116": {
"source": "src/OA-FLOW-L116.md",
"source_sha256": "6d2a08b459fd22cf96d4750927490a1597b68e466c2a92e56002ca55076b8f6a"
},
"OA-FLOW-FCT": {
"source": "src/OA-FLOW-FCT.md",
"source_sha256": "1c45cc3775a48e125d70328b8296bbc9f153bbede48c23a06b24b158c07f3eda"
},
"OA-FLOW-L18": {
"source": "src/OA-FLOW-L18.md",
"source_sha256": "c64b0b2a9bd2eff20850a598baa22e2323ac6d7390638b3b31872d199011c14a"
},
"OA-FLOW-L30": {
"source": "src/OA-FLOW-L30.md",
"source_sha256": "8e3f7bed671a4fdd9ad2c35b354b7ec0879337609e371f91879f6df9b8d4f05d"
},
"OA-FLOW-CDEC": {
"source": "src/OA-FLOW-CDEC.md",
"source_sha256": "c955d3203895350740ac10c54c84c6389855fda0e86c70fd758c7cc00fa6098e"
},
"OA-FLOW-SCW": {
"source": "src/OA-FLOW-SCW.md",
"source_sha256": "77ed34bfc31f73865eb77d7aeaa75c78eb2969cde600f4655be2541703abc9a9"
},
"OA-FLOW-GRD": {
"source": "src/OA-FLOW-GRD.md",
"source_sha256": "176aa35a1d664f9245ad2daa9290d268b5f912e13a2ecae52da9c331cd3a3c31"
},
"OA-FLOW-SEC": {
"source": "src/OA-FLOW-SEC.md",
"source_sha256": "cc30f80c7db32447903d84c8614723fa92c0022dbaab9b0f00cca5a624784529"
},
"OA-FLOW-HLB": {
"source": "src/OA-FLOW-HLB.md",
"source_sha256": "4bef1b5179a65a7f27e3aab6ea2139e12e80271ddfe01b871e4b0cde07ba458a"
},
"OA-FLOW-REL": {
"source": "src/OA-FLOW-REL.md",
"source_sha256": "4a90ddcb4db7daa12cc696be1c7c3aa316be3247cff1e59d720dc136996e268f"
},
"OA-FLOW-LFT": {
"source": "src/OA-FLOW-LFT.md",
"source_sha256": "335dcdacad4849e48947b0f326bcc6a9b136b7857b65316a284233bc0ed5afe7"
},
"OA-FLOW-L119": {
"source": "src/OA-FLOW-L119.md",
"source_sha256": "42fde4858f29e6cf8efb9f1b1639bdbda6678817bcc2b0e0b8b02ca556ae7b12"
},
"OA-FLOW-L120": {
"source": "src/OA-FLOW-L120.md",
"source_sha256": "714e54a20a824825b3e520953c8461a5b127905df4efcf834c9faba518eb919b"
},
"OA-FLOW-MIV": {
"source": "src/OA-FLOW-MIV.md",
"source_sha256": "45e87c1dbbc116663324ff2bf0be560e56c2977ec5b9af33280e2e780afd2d80"
},
"OA-FLOW-CIM": {
"source": "src/OA-FLOW-CIM.md",
"source_sha256": "7a3e617d2985e9ae8fd54d1bc0171ba113bfbaec198417b0325cbd553c07a402"
},
"OA-FLOW-RCC": {
"source": "src/OA-FLOW-RCC.md",
"source_sha256": "1e91ea3a57b9896008ef422de4351db3ba25bc2d97f0087e4a6bdb315f6b51c3"
},
"OA-FLOW-DWC": {
"source": "src/OA-FLOW-DWC.md",
"source_sha256": "ca0771968d6f96d9666d8a4c749e1b4bce63c34f74b13c1981058c304bd04e44"
},
"OA-FLOW-CST": {
"source": "src/OA-FLOW-CST.md",
"source_sha256": "38d1a56d8601eceab1761df4ad523e13bf5a61b873343df805b73418e0c18596"
},
"OA-FLOW-OES": {
"source": "src/OA-FLOW-OES.md",
"source_sha256": "f201b40f05888c3cc76f71b56e81a5615d9a05ee68f0d8e607fd156f664b6f53"
},
"OA-FLOW-DS": {
"source": "src/OA-FLOW-DS.md",
"source_sha256": "eb951f5a8a63130511d2d49bf21b9b14ff4fd2eb7d82790ef37b1503221bbc17"
},
"OA-FLOW-TIC": {
"source": "src/OA-FLOW-TIC.md",
"source_sha256": "851f05c0e7c3623e28ce65f863102b68de925b6cefcd0647ec517d18e05af767"
},
"OA-FLOW-CGF": {
"source": "src/OA-FLOW-CGF.md",
"source_sha256": "aa683a097979a5608fc9dacd94d3fd0a155e5b6051a1c0c8664e7527bebe2a88"
},
"OA-FLOW-TCC": {
"source": "src/OA-FLOW-TCC.md",
"source_sha256": "4597e0426583b81d6b3d927383f272acc126f7d6abd9055e189d9fc06fe6c2b7"
},
"OA-FLOW-TID": {
"source": "src/OA-FLOW-TID.md",
"source_sha256": "9f681aa7ad72cc8d2df4af1d254673ad55939d8f35a05b762a591eba22559a99"
},
"OA-FLOW-CBD": {
"source": "src/OA-FLOW-CBD.md",
"source_sha256": "ff6e3bf7ff3af6583ed89dd4b9719d16b544167ed3a10b79f8245d3e7ea79d69"
},
"OA-FLOW-BFC": {
"source": "src/OA-FLOW-BFC.md",
"source_sha256": "3594662c1ac2fd1e2820aec5c91d2350460741786bc8334e538ca05b749e50e6"
},
"OA-FLOW-DDP": {
"source": "src/OA-FLOW-DDP.md",
"source_sha256": "5b231ebc4f2e04d9ba76930ed8d6190093c1b9070e63955622f3f21654c9badb"
},
"OA-FLOW-PDM": {
"source": "src/OA-FLOW-PDM.md",
"source_sha256": "4cd63d4994e6bad31522c2e60dae4b29ec953a61bb7b4a481cfa36d5ca12c2c9"
},
"OA-FLOW-TF": {
"source": "src/OA-FLOW-TF.md",
"source_sha256": "c8fa68ac735eedc29600cd4ca005414432fce01446a9a478fd4d5fd160dfdbad"
},
"OA-FLOW-SR": {
"source": "src/OA-FLOW-SR.md",
"source_sha256": "b868cfa83cfa7c925df5a1aac20b824f4e312cc23b57d5e083a4f06ea4e5b4ae"
},
"OA-FLOW-ZDC": {
"source": "src/OA-FLOW-ZDC.md",
"source_sha256": "7136d488d023a539642e99c2ec177f6d42a103f1affdd50212b5ffc2e0b1eff2"
},
"OA-FLOW-IE": {
"source": "src/OA-FLOW-IE.md",
"source_sha256": "34062c6a5adbc5d4214749f4f613d2636d8c76a9533caa4ea1b42c78d8096346"
},
"OA-FLOW-IC": {
"source": "src/OA-FLOW-IC.md",
"source_sha256": "57e86f7c317ddefe007ba00721bb39fff57e0e854e3875d62f263fdc9dd6b64d"
},
"OA-FLOW-HA": {
"source": "src/OA-FLOW-HA.md",
"source_sha256": "a41e16550e473c1fcd0dcaa4042d8dee7ca4135a6ccedcfc058ef2c49f6d3949"
},
"OA-FLOW-SMC": {
"source": "src/OA-FLOW-SMC.md",
"source_sha256": "0ae96d04a3fbec690651303490067cc9169d1c0795029072321a8eca80e6eaae"
},
"OA-FLOW-KRT": {
"source": "src/OA-FLOW-KRT.md",
"source_sha256": "4fa982fe6225dcc5d392a2a6371ee1d7a50d0ca64486c605fefd12e3563ac00e"
},
"OA-FLOW-L121": {
"source": "src/OA-FLOW-L121.md",
"source_sha256": "04aedc0cdfb565cfce6779e69a51f14e8a916708f884d5841b78affdc6714c34"
},
"OA-FLOW-L122": {
"source": "src/OA-FLOW-L122.md",
"source_sha256": "15a335885147fda321ed8b05d738711c2a590ef9461191940b4d5446abc3d399"
},
"OA-FLOW-L123": {
"source": "src/OA-FLOW-L123.md",
"source_sha256": "334a96d6ca9719a8db834f0bddf77ef510ab2dd803ab000824fd69f2eabfcbda"
},
"OA-FLOW-L124": {
"source": "src/OA-FLOW-L124.md",
"source_sha256": "3c53aa556d52d0acadb49d92d38f156b4522c226a146c2ceea6e3a6c7d0a5654"
},
"OA-FLOW-L125": {
"source": "src/OA-FLOW-L125.md",
"source_sha256": "b8693dcbb341c9669247021d1c63a57d767870a8d161d28a636939821aa385c9"
},
"OA-FLOW-L126": {
"source": "src/OA-FLOW-L126.md",
"source_sha256": "01c4e943b0b161d3febc126f4d6dc380ef2646ead7888cb07c111728ce15635f"
},
"OA-FLOW-L127": {
"source": "src/OA-FLOW-L127.md",
"source_sha256": "a8e1b419caa8f4c81840dfc70514a4f10fa0a3cc4118702b5f85e62bc2cba7d9"
},
"OA-FLOW-L128": {
"source": "src/OA-FLOW-L128.md",
"source_sha256": "80dd03b1dedfc3d3115f06c502589d716a23ec0ae10f2b1261afb685a2ea8db5"
},
"OA-FLOW-SAR": {
"source": "src/OA-FLOW-SAR.md",
"source_sha256": "9146f71f98c7b8b247b6fed827f6857aaec664a5195d5ac8b12fed883fca9a32"
},
"OA-FLOW-LAC": {
"source": "src/OA-FLOW-LAC.md",
"source_sha256": "b1c4c2bd8398132d7a6d61c07281922bf4d61efc0f54e8f832a949dee8219cc0"
},
"OA-FLOW-TP": {
"source": "src/OA-FLOW-TP.md",
"source_sha256": "49aa8fba053c24a625b3f7b9d36e6f2d9e091a7b85447eb98e233ac1421cf5d0"
},
"OA-FLOW-CMAS": {
"source": "src/OA-FLOW-CMAS.md",
"source_sha256": "265e47828666924aaecd5110c7c0c702232f082b94254c611aed7eb0dd99e941"
},
"OA-FLOW-SMULT": {
"source": "src/OA-FLOW-SMULT.md",
"source_sha256": "e96cd0946916e02a285badc6bf222dd6df018f177623f7f3dc343555774392e7"
},
"OA-FLOW-NCOEF": {
"source": "src/OA-FLOW-NCOEF.md",
"source_sha256": "92fee26802b1ee7a8004197e0fd76dab644e34931a0cf425d6ebb7021ebb0b26"
},
"OA-FLOW-CBND": {
"source": "src/OA-FLOW-CBND.md",
"source_sha256": "e695bcce533f1d6bc8dfd7e3f51d22a4df21b239f5c7fce1d1d737b5401411d1"
},
"OA-FLOW-MCOM": {
"source": "src/OA-FLOW-MCOM.md",
"source_sha256": "875f75dc9290c3c68168e993456be0dad8a6b37ef065c3d532eb00bec310d9b9"
},
"OA-FLOW-L129": {
"source": "src/OA-FLOW-L129.md",
"source_sha256": "64ece422419dbdee22b1ffa2b719f9140d08bfc225f126ba2c1218728b91fb8e"
},
"OA-FLOW-L130": {
"source": "src/OA-FLOW-L130.md",
"source_sha256": "8d770e1e87af15ad623f178f318159e88dd819c4a4942ef597088bc667164b4f"
},
"OA-FLOW-L131": {
"source": "src/OA-FLOW-L131.md",
"source_sha256": "1541ca9ccc5178afe1bbef35054dc2cb2309cacef06d18a557ffff11b77b7ef4"
},
"OA-FLOW-L132": {
"source": "src/OA-FLOW-L132.md",
"source_sha256": "3e02f1e0f6e2efdedcbdcb473e8fadab387f088f40621a116957adf4a6c52a96"
},
"OA-FLOW-L133": {
"source": "src/OA-FLOW-L133.md",
"source_sha256": "e2489242d5a6f4db3103723a16a7e1de5efe87db882e011bdaee9989f38b2115"
},
"OA-FLOW-WORD": {
"source": "src/OA-FLOW-WORD.md",
"source_sha256": "eb341ec6b65d454f1beac912ff79aedf8fd6c006c416d44e8e9c9bfc51edd8a5"
},
"OA-FLOW-UMET": {
"source": "src/OA-FLOW-UMET.md",
"source_sha256": "b1ee2affe1d24f6615e631650d54db3589fbce414fdd9654695f6107d4bdc036"
},
"OA-FLOW-IAP": {
"source": "src/OA-FLOW-IAP.md",
"source_sha256": "48f3819dd88962b18d8b133847b2c0611367d6c99f82b677917af504612cbe84"
},
"OA-FLOW-TAIL": {
"source": "src/OA-FLOW-TAIL.md",
"source_sha256": "ae86869f1f1c39fadaf3cd93c08bd16e679788a5212699ef2ca6732c741b5475"
},
"OA-FLOW-SORB": {
"source": "src/OA-FLOW-SORB.md",
"source_sha256": "5d76b84c5b002a28006e07a4dd45981495d806193a3dd671b373176e02f9fa7f"
},
"OA-FLOW-CINV": {
"source": "src/OA-FLOW-CINV.md",
"source_sha256": "038faaf70a611eef3fcd409be5f9bdede51993a1d9a1e8a104155822232552f8"
},
"OA-FLOW-CEXP": {
"source": "src/OA-FLOW-CEXP.md",
"source_sha256": "ad1d3c16920faa1d51801a040ed5701825ccdb7caa26b7bb72bdc8c0740f357e"
},
"OA-FLOW-CSAS": {
"source": "src/OA-FLOW-CSAS.md",
"source_sha256": "7cfa27b0b07ef39b160d02c95939b11f4ad7203b163ccf774e247ff02470edea"
},
"OA-FLOW-CAPP": {
"source": "src/OA-FLOW-CAPP.md",
"source_sha256": "f84e33839177af3b6cc6acf9f61926763531986420b30dbb71927627bf9931d8"
},
"OA-FLOW-CPER": {
"source": "src/OA-FLOW-CPER.md",
"source_sha256": "c74214671a23448fc91af2955244f3886ca351dcc322175469fbf9003fda6388"
},
"OA-FLOW-CRNG": {
"source": "src/OA-FLOW-CRNG.md",
"source_sha256": "11e6e0ffa7d8ad2adb70314cb21dc92cae9e0621e713bf811bf94b081cc0bd71"
},
"OA-FLOW-MGO": {
"source": "src/OA-FLOW-MGO.md",
"source_sha256": "9de7ebfd18cdb9859cf737e51edd217e77d3c9e8a2f0bbe125b043f9a15d16a1"
},
"OA-FLOW-MGI": {
"source": "src/OA-FLOW-MGI.md",
"source_sha256": "68f09ad6d809166f3ebff9321ce01b8c3b7113b609505aa0c23b2f17b929dcfd"
},
"OA-FLOW-MHD": {
"source": "src/OA-FLOW-MHD.md",
"source_sha256": "3664b022ea134d2f7869795139ccf22755b25af1e8b5f2fb523105ff88a09ef8"
},
"OA-FLOW-WCO": {
"source": "src/OA-FLOW-WCO.md",
"source_sha256": "e3425b7d0736760c235d9b29d217d7f0de9e131febf6c714ea1451ed2aa2b651"
},
"OA-FLOW-CSEC": {
"source": "src/OA-FLOW-CSEC.md",
"source_sha256": "8791e310307734f567bcb509f9432f44f372cb097fa609f30bd3f8371c34ff09"
},
"OA-FLOW-WCH": {
"source": "src/OA-FLOW-WCH.md",
"source_sha256": "b2ae102e01895187d47feeb641be2af04aef3098b866e4136e1dd91bfc4834e2"
},
"OA-FLOW-NSQ": {
"source": "src/OA-FLOW-NSQ.md",
"source_sha256": "02f8c3114c456ec46f06ea4cbe0443266b9bb0cbb1d7396298a3e791005e30af"
},
"OA-FLOW-PSQ": {
"source": "src/OA-FLOW-PSQ.md",
"source_sha256": "4740082719eb7a47be31b057eb264bb0a368c1bd78b3d4053f391b02851544aa"
},
"OA-FLOW-BEX": {
"source": "src/OA-FLOW-BEX.md",
"source_sha256": "562016033fac166b6da614125c389ead9c06cba975622abcabadb5beeba55ee8"
},
"OA-FLOW-BRL": {
"source": "src/OA-FLOW-BRL.md",
"source_sha256": "30ae336e0b2b41148cee5ee4f5eada7c52eeda8f31a56d74e6360797fd9c6df8"
},
"OA-FLOW-CPP": {
"source": "src/OA-FLOW-CPP.md",
"source_sha256": "33a54884b368bb5acea270e302cacc8c2a6883995f68b3831f013ea8bc03e317"
},
"OA-FLOW-TRP": {
"source": "src/OA-FLOW-TRP.md",
"source_sha256": "0ba73ff34da7c3e2e72f27b54a93ef689d0a4a25d9cbf852bc4fa3d7d1f81588"
},
"OA-FLOW-NB": {
"source": "src/OA-FLOW-NB.md",
"source_sha256": "7dfc646cfbd902c2fa59273e7dc27a32573639f876c1da9dbb32408e09e66203"
},
"OA-FLOW-TG": {
"source": "src/OA-FLOW-TG.md",
"source_sha256": "3f654d08f6247d62309b804d285bec2dd7ad56ba9ec78ec6b8a2389dacd80c77"
},
"OA-FLOW-ARP": {
"source": "src/OA-FLOW-ARP.md",
"source_sha256": "de9a02f0e434a15ef90be2dc00ab174629456ba2d327c1cede1d2336e8f9b95d"
},
"OA-FLOW-NNR": {
"source": "src/OA-FLOW-NNR.md",
"source_sha256": "a095a2cd8f1384456bd1b0f0549c37a4beb2bbe00be5f168118a1c7b052378bd"
},
"OA-FLOW-L97": {
"source": "src/OA-FLOW-L97.md",
"source_sha256": "05e07b1120b5bf0b61ce848aa216bc6b6366cb3f29ecce35c4c20ced8e3a4e15"
},
"OA-FLOW-L98": {
"source": "src/OA-FLOW-L98.md",
"source_sha256": "a9323d88f72e4eea49b84a19eaed84bb4dab5af9e7b9bf90a99ea9bf1e00d24c"
},
"OA-FLOW-L99": {
"source": "src/OA-FLOW-L99.md",
"source_sha256": "5288902aad3740e08ff40a074e9001113773a65bc7af66148d94597eabf99250"
},
"OA-FLOW-L100": {
"source": "src/OA-FLOW-L100.md",
"source_sha256": "d9968e792d8d0af011cec1fc1905e6ad6a7e5d41c385abbb9940cf6c05bc0e9c"
},
"OA-FLOW-L101": {
"source": "src/OA-FLOW-L101.md",
"source_sha256": "8ddb869f398ddc5f9d61f99c8f070fda24ad6a4b26f778c7f2f2ca4028583cb7"
},
"OA-FLOW-L102": {
"source": "src/OA-FLOW-L102.md",
"source_sha256": "e425c70b13b0b2cbb732b6a7e0e25ec129e409dcd8e2a47b27c7705bef8ada89"
},
"OA-FLOW-L103": {
"source": "src/OA-FLOW-L103.md",
"source_sha256": "d369a8bb7d3e0a271c2d6e1ca479e86ae42633612b759900fe861708a03f0eb7"
},
"OA-FLOW-L104": {
"source": "src/OA-FLOW-L104.md",
"source_sha256": "b9eb162bddcaf02dc3b9b4d98f153725bb24aada717fd9f5c3454ed971183ec3"
},
"OA-FLOW-L106": {
"source": "src/OA-FLOW-L106.md",
"source_sha256": "2b735f186aa2b60e473d59e7ba056ea04a5f2e961fd8b99cdb177ec3127dd97a"
},
"OA-FLOW-L109": {
"source": "src/OA-FLOW-L109.md",
"source_sha256": "41472e4dc958c7c04c7ed7d52d61ea99914d0d3b27d4292843bf8cb9dfdd7ff5"
},
"OA-FLOW-L110": {
"source": "src/OA-FLOW-L110.md",
"source_sha256": "733d69717e6aa638427bc7bb5d9599778249c32d92849408f016cfa7f9f64b15"
},
"OA-FLOW-L134": {
"source": "src/OA-FLOW-L134.md",
"source_sha256": "f204750ccb1bad3e92a1f0cf88dc75e2cd16950ec1eb37170fe4e23a2303fb3f"
},
"OA-FLOW-L135": {
"source": "src/OA-FLOW-L135.md",
"source_sha256": "612ca484bcc412e5dcd20486f4ab49db93d4a6ad2f6d88e59001242e69d19fe5"
},
"OA-FLOW-L136": {
"source": "src/OA-FLOW-L136.md",
"source_sha256": "6f3119388b8367f887160158cdec0a8c125b9789c4ed0392860b48ea77775ae6"
},
"OA-FLOW-L86": {
"source": "src/OA-FLOW-L86.md",
"source_sha256": "b18600c16606211ee24f0f9aba58c46eb7c4b71805e9fc8770e763ef6572f22f"
},
"OA-FLOW-L137": {
"source": "src/OA-FLOW-L137.md",
"source_sha256": "1f435c1971597abdd7d2c42b6fb7f98f7a1cf1ddf77cf92b6a64924a32a335b5"
}
},
"current_scalar_proof_bindings": [
{
"id": "OA-FLOW.SC.0",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-00",
"start_line": 8,
"end_line": 27,
"range_sha256": "a5fe9d52348394e56a90b606809b21774c96806f3bda5754bbb7c06812e81a09",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.SC.1",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-01",
"start_line": 29,
"end_line": 49,
"range_sha256": "292ec35e155772c6704a1503b7dbd39bc192ec5b66cca3ed5fdb0122df2aaa41",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"PRIMITIVE-SETS-AND-EXTENDED-NONNEGATIVE-SUMS"
]
},
{
"id": "OA-FLOW.SC.2",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-02",
"start_line": 51,
"end_line": 79,
"range_sha256": "c2466cd07ec73269d920c2630eae6a14ca0be7cbadf45d369f06f8f45d045b37",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.1",
"OA-FLOW.CF.1",
"PRIMITIVE-CHOICE"
]
},
{
"id": "OA-FLOW.SC.3",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-03",
"start_line": 81,
"end_line": 109,
"range_sha256": "2b5422dab16d15b9708daeff09a581527de5123f6fb4e4411df4f006089b7797",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.2"
]
},
{
"id": "OA-FLOW.SC.4",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-04",
"start_line": 111,
"end_line": 143,
"range_sha256": "48e2d3f6ba07c752efa95f6b3930d6e12f3a02fb63664ced05b176b448ce06c0",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3"
]
},
{
"id": "OA-FLOW.SC.5",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-05",
"start_line": 161,
"end_line": 195,
"range_sha256": "4936a40727ec878aa8d7dc622316315ada5c91691c730f6331b477141a1e8084",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.SC.6",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-06",
"start_line": 197,
"end_line": 232,
"range_sha256": "1626e0ccff1308960a057b262178840589329e4977659f89cc46be20028b34ce",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.3",
"OA-FLOW.SC.4"
]
},
{
"id": "OA-FLOW.SC.7",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-07",
"start_line": 234,
"end_line": 272,
"range_sha256": "b61a44358323c5a7d6ae21ab4ff459bec17d8285935200b7db9eb170254f7930",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.4",
"OA-FLOW.SC.5",
"OA-FLOW.SC.6"
]
},
{
"id": "OA-FLOW.SC.8",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-08",
"start_line": 274,
"end_line": 299,
"range_sha256": "73c30f3d665316b2049f631c48a57b2646f90b66dd2d1abeda65db5a159fcd5a",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.2",
"OA-FLOW.SC.3",
"OA-FLOW.SC.5",
"OA-FLOW.CF.1"
]
},
{
"id": "OA-FLOW.SC.9",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-09",
"start_line": 301,
"end_line": 320,
"range_sha256": "7645c2af9a3a260a86058ed35120247439c68b78edabe3de5d81fd732f2aef66",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.3",
"OA-FLOW.SC.5"
]
},
{
"id": "OA-FLOW.SC.10",
"unit": "OA-FLOW-SC",
"source_file": "src/OA-FLOW-SC.md",
"source_sha256": "a56d077a309f7f1f801110940793644d9c6a7128db0bd128a9744c2cf3f0b637",
"reader": "OA-FLOW-SC.html#sc-10",
"start_line": 322,
"end_line": 335,
"range_sha256": "42c55b9d3d9c9563d45e336089287c083431bd455859a5d16aaa2fe8ea5d02d8",
"range_identity_convention": "UTF-8 text with LF line endings and one final LF",
"prerequisites": [
"OA-FLOW.SC.0",
"OA-FLOW.SC.5",
"OA-FLOW.SC.7",
"OA-FLOW.SC.8"
]
}
]
}