{
  "schema": "AN04-restored-linear-reduction-proof-map/v1",
  "proofs": [
    {
      "id": "LR:R0",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Proposition 1.1.",
      "dependencies": [
        "G2:G21",
        "G2:G51"
      ],
      "scope": "Arbitrary real quadratic phase: critical rank, pure radical, Lagrangian image and nondegenerate quotient, including empty blocks"
    },
    {
      "id": "LR:R1",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Lemma 1.2.",
      "dependencies": [
        "LR:R0",
        "GS:Z7",
        "TZ:Z6",
        "U001:Q6",
        "U001:U001-A4",
        "U001:U001-A5",
        "U001:U001-A6"
      ],
      "scope": "Compact radical testing, independence of projection and complement, exact density-valued Gaussian and fixed-count normalization"
    },
    {
      "id": "LR:R2",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "## 2. The density map",
      "dependencies": [
        "G2:G23",
        "U001:U001-A5"
      ],
      "scope": "Canonical positive reduction map from two exact sequences and the perfect symplectic quotient pairing"
    },
    {
      "id": "LR:R3",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "Such coordinates exist with",
      "dependencies": [
        "G2:G21",
        "G2:G23"
      ],
      "scope": "Adapted coordinates preserving the distinguished vertical plane, with explicit completion fixing all prescribed horizontal vectors"
    },
    {
      "id": "LR:R4",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Lemma 3.1.",
      "dependencies": [
        "LR:R0",
        "LR:R1",
        "LR:R2",
        "LR:R3",
        "GS:Z7",
        "GS:Z8",
        "MC:G1"
      ],
      "scope": "Restriction/integration quadratic operation, critical image and canonical identification of its radical with the geometric intersection"
    },
    {
      "id": "LR:R5",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Proof of independence.**",
      "dependencies": [
        "LR:R4",
        "GS:Z3",
        "GS:Z4",
        "GS:Z9",
        "MC:G1",
        "MC:G4",
        "U001:Q6"
      ],
      "scope": "Phase changes, square elimination, unequal variable counts, adapted shears and block-triangular base changes"
    },
    {
      "id": "LR:R6",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "For smoothness, work near",
      "dependencies": [
        "LR:R5",
        "G2:G22",
        "GS:Z7",
        "TG:Z2",
        "ML:M0a",
        "U001:P8-smooth"
      ],
      "scope": "Smooth constant-excess families through phase-Hessian rank changes: fixed-size input phases, smooth kernel quotients and transverse full-Hessian frames"
    },
    {
      "id": "LR:R7",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "## 5. Compute the factor",
      "dependencies": [
        "LR:R2",
        "LR:R4",
        "LR:R5",
        "G2:G22",
        "GS:Z7"
      ],
      "scope": "Contained-constraint normalization, critical density and all three dimension terms"
    },
    {
      "id": "LR:R8",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "## 6. Compute the factor",
      "dependencies": [
        "LR:R2",
        "LR:R3",
        "LR:R4",
        "LR:R5",
        "G2:G22",
        "G2:G23",
        "GS:Z7",
        "U001:Q6"
      ],
      "scope": "Zero-intersection adapted common complement, invertible graph block, exact Schur complement, density determinant and Fresnel phase"
    },
    {
      "id": "LR:R9",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "The agreement with direct reduction",
      "dependencies": [
        "LR:R1",
        "LR:R2",
        "LR:R3",
        "LR:R5",
        "LR:R7",
        "LR:R8"
      ],
      "scope": "Staged reduction for an arbitrary isotropic flag: compatible coordinates, nested radicals, equal quotient phases, telescoping prefactors and density identifications"
    },
    {
      "id": "LR:R10",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Theorem 7.1",
      "dependencies": [
        "LR:R6",
        "LR:R7",
        "LR:R8",
        "LR:R9",
        "G2:G71",
        "CC:C1",
        "CC:C4",
        "MC:G4"
      ],
      "scope": "Canonical linear symbol composition, reflected middle form, symplectic density contraction and exact excess factor"
    },
    {
      "id": "LR:R11",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "## 8. A graph factor",
      "dependencies": [
        "LR:R10",
        "MC:G5",
        "U001:Q4",
        "U001:Q5"
      ],
      "scope": "Graph density transport, symplectic volume unit, identity Gaussian and Maslov unit"
    },
    {
      "id": "LR:E1",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "LR:R0",
        "LR:R1",
        "U001:Q6"
      ],
      "scope": "Exercise 9.1 and solution"
    },
    {
      "id": "LR:E2",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "LR:R7",
        "LR:R8"
      ],
      "scope": "Exercise 9.2 and solution"
    },
    {
      "id": "LR:E3",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "LR:R8"
      ],
      "scope": "Exercise 9.3 and solution"
    },
    {
      "id": "LR:E4",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "LR:R9"
      ],
      "scope": "Exercise 9.4 and solution"
    },
    {
      "id": "LR:E5",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "LR:R10",
        "LR:R1"
      ],
      "scope": "Exercise 9.5 and solution"
    },
    {
      "id": "LR:E6",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "f8c1fde3a196a4fe59c8cfea9ffa285ea56516e6902023b779b5d12ffd085918",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "LR:R11"
      ],
      "scope": "Exercise 9.6 and solution"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json"
  ],
  "transitive_U001_ids": [
    "A0",
    "CLOSED-BALL-CONTRACTION",
    "D0",
    "E0",
    "EXPONENTIAL-TRIGONOMETRIC",
    "F0-ALG",
    "F0-CALC",
    "F0-COMP",
    "F0-DIFF",
    "FTC-TAYLOR-COMPACT-PARAMETERS",
    "I0",
    "J0",
    "L10.1.12",
    "L10.1.13",
    "L10.1.14",
    "L10.1.15",
    "L10.1.2",
    "L10.1.5",
    "L10.1.6",
    "L10.2.2",
    "L10.3.10",
    "L10.3.2",
    "L10.3.7",
    "L10.4.1",
    "L10.4.2",
    "L10.4.3",
    "L10.5.1",
    "L10.5.5",
    "L10.5.9",
    "L10.7.2",
    "L11.4.2",
    "L2.1.10-inc",
    "L2.1.17",
    "L2.2.1",
    "L2.2.11",
    "L2.2.3",
    "L2.2.5",
    "L2.3.2",
    "L2.3.4-upper",
    "L2.3.5",
    "L2.3.8",
    "L2.4.5",
    "L2.6.5",
    "L3.1.7",
    "L3.3.7",
    "L3.3.8",
    "L4.2.2",
    "L4.2.3",
    "L4.2.4",
    "L4.2.6",
    "L5.1.10",
    "L5.1.11",
    "L5.1.13",
    "L5.1.2",
    "L5.1.7",
    "L5.1.8",
    "L5.2.1",
    "L5.2.2",
    "L5.2.4-positive",
    "L5.2.6",
    "L5.2.7",
    "L5.3.1",
    "L5.3.3",
    "L5.3.5",
    "L5.4.1",
    "L5.4.2",
    "L5.5.2-gt1",
    "L5.5.3",
    "L5.5.4",
    "L5.5.5",
    "L7.1.4",
    "L7.1.5",
    "L7.2.19",
    "L7.2.22",
    "L7.2.6",
    "L7.2.9-open",
    "L7.3.11",
    "L7.3.12",
    "L7.3.9",
    "L7.4.10",
    "L7.4.11",
    "L7.4.2",
    "L7.4.4",
    "L7.4.9",
    "L7.5.11",
    "L7.5.12",
    "L7.5.2",
    "L7.5.5",
    "L7.5.6",
    "L7.6.2",
    "L8.1.13",
    "L8.1.14",
    "L8.1.16",
    "L8.1.17",
    "L8.1.18",
    "L8.2.4",
    "L8.2.5",
    "L8.2.6",
    "L8.2.7",
    "L8.2.8",
    "L8.2.9",
    "L8.3.2",
    "L8.3.3",
    "L8.3.5",
    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
    "L8.4.6",
    "L8.5.1",
    "L8.5.6",
    "L8.6.2",
    "L9.1.1",
    "M1",
    "M2",
    "P1",
    "P10.1",
    "P10.2",
    "P10.2-typo",
    "P10.3",
    "P10.4",
    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
    "P12.2-upper",
    "P12.3",
    "P12.4",
    "P12.5-endpoints",
    "P12.5-extension",
    "P12.6",
    "P12.7",
    "P13.1",
    "P13.2-mertens",
    "P13.2-series",
    "P14.2-inverse",
    "P14.2-powers",
    "P14.3",
    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P17.1",
    "P17.2",
    "P17.3",
    "P17.4",
    "P17.4-grid",
    "P17.5-order",
    "P17.5-upper",
    "P18.1",
    "P18.2",
    "P18.3",
    "P18.4",
    "P18.5",
    "P19.1",
    "P19.2",
    "P19.3",
    "P19.4",
    "P2",
    "P20.1",
    "P20.2",
    "P20.3",
    "P20.4",
    "P20.5",
    "P21.1",
    "P21.2",
    "P21.3",
    "P21.4",
    "P21.5",
    "P3",
    "P4",
    "P5",
    "P6.0",
    "P6.1",
    "P6.2",
    "P6.3-inf",
    "P6.3-subtract",
    "P6.3-tail",
    "P6.4-ball",
    "P6.4-complete",
    "P6.5",
    "P6.6",
    "P7",
    "P8-root",
    "P8-smooth",
    "P9.1",
    "P9.2",
    "P9.2-operator",
    "P9.3",
    "P9.4",
    "P9.5",
    "Q1",
    "Q2",
    "Q3",
    "Q4",
    "Q5",
    "Q6",
    "Q9",
    "R0",
    "U001-2.1",
    "U001-3.1",
    "U001-3.2",
    "U001-4.1",
    "U001-5.1",
    "U001-6.1",
    "U001-7.2",
    "U001-A1",
    "U001-A2",
    "U001-A3",
    "U001-A4",
    "U001-A5",
    "U001-A6",
    "U001-DEF"
  ],
  "transitive_earlier_package_ids": [
    "CC:C0",
    "CC:C1",
    "CC:C4",
    "CH:T0",
    "CH:T1",
    "CH:T2",
    "CH:T3",
    "CH:W1",
    "CH:W2",
    "CH:W3",
    "CH:W4",
    "CH:W5",
    "G2:F0",
    "G2:F1",
    "G2:F2",
    "G2:F3",
    "G2:F4",
    "G2:G0",
    "G2:G1",
    "G2:G21",
    "G2:G22",
    "G2:G23",
    "G2:G31",
    "G2:G41",
    "G2:G51",
    "G2:G71",
    "G2:NF0",
    "G2:NF1",
    "G2:NF2",
    "G2:NF3",
    "G2:NF4",
    "G2:NF5",
    "G2:NF6",
    "G2:NF7",
    "GEO:C0",
    "GEO:C1",
    "GEO:C2",
    "GEO:C3",
    "GEO:C4",
    "GEO:C5",
    "GR:G12-forward",
    "GR:G12-reverse",
    "GR:G5",
    "GR:G8",
    "GS:Z0",
    "GS:Z2",
    "GS:Z3",
    "GS:Z4",
    "GS:Z5",
    "GS:Z6",
    "GS:Z7",
    "GS:Z8",
    "GS:Z9",
    "IR:R0",
    "IR:R1",
    "IR:R2",
    "IR:R3",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "K:K7",
    "MC:G0",
    "MC:G1",
    "MC:G2",
    "MC:G3",
    "MC:G4",
    "MC:G5",
    "ML:M0",
    "ML:M0a",
    "ML:M1",
    "ML:M2",
    "ML:M3",
    "ML:M4",
    "ML:M5",
    "ML:M6",
    "OSC:O1",
    "OSC:O3",
    "OSC:O4",
    "P1:B1",
    "P1:B2",
    "P1:B3",
    "P1:B4",
    "P1:B5",
    "P1:B6",
    "P2:F1",
    "P2:F2",
    "P2:F3",
    "P2:F4",
    "P2:F5",
    "P2:ML1",
    "P2:ML2",
    "P2:ML3",
    "P2:ML4",
    "P2:ML6",
    "P2:ML7",
    "P2:OP0",
    "P2:OP1",
    "P2:OP2",
    "P2:OP3",
    "P2:OP4",
    "P2:OP5",
    "P2:OP6",
    "P2:QG1",
    "P2:QG2",
    "P2:QG3",
    "P2:QG4",
    "P2:QG5",
    "P2:QG6",
    "P2:QG7",
    "P2:QG8",
    "P3:L1",
    "P3:L2",
    "P3:L3",
    "P3:M0",
    "P3:M1",
    "P3:M2",
    "P3:M3",
    "P3:M4",
    "P3:M5",
    "P3:M6",
    "P3:M7",
    "P3:M8",
    "PH:F0",
    "PH:F1",
    "PH:F2",
    "PH:F3",
    "PH:F4",
    "PH:F5",
    "PH:F6",
    "PS:PS0",
    "PS:PS1",
    "TG:T0",
    "TG:T6",
    "TG:Z2",
    "TZ:Z0",
    "TZ:Z6"
  ],
  "external_dependency_policy": {
    "record": "../20261004-free-stationary-phase/external-proof-dependencies.json",
    "notice": "The elementary proof providers have exact earlier-programme routes in C10/C20, with public PDF destinations and editable TeX passages. The author English edition remains available. Human source text and figures stay external to this CC0 export and retain their own terms.",
    "mathematical_proofs_rewritten": false
  }
}