{
  "schema": "AN04-restored-first-order-cauchy-proof-map/v1",
  "proofs": [
    {
      "id": "FC:M1",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "## M1.",
      "dependencies": [
        "P2:OP1",
        "P2:OP3",
        "P3:L3"
      ],
      "scope": "Exact full and partial classes, Schwartz action and adjoint, all joint time parameter derivatives"
    },
    {
      "id": "FC:M2",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "## M2.",
      "dependencies": [
        "FC:M1",
        "P2:OP3",
        "P3:L1",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Actual partial-times-full smoothing ideal in both orders, rapid independent parameter bounds and full operator identity"
    },
    {
      "id": "FC:M3",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "## M3.",
      "dependencies": [
        "FC:M2",
        "U001:U001-A4",
        "P2:OP3",
        "U001:P14.2-powers"
      ],
      "scope": "Strict conic cutoff, exact multiplier product, BA full ordinary expansion with every finite remainder"
    },
    {
      "id": "FC:M4",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "## M4.",
      "dependencies": [
        "FC:M3",
        "FC:M2",
        "P2:OP3"
      ],
      "scope": "Smoothing-cone adjoint split, AB full expansion, finite Leibniz/binomial cancellation and all remainders"
    },
    {
      "id": "FC:M5",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "We record explicitly",
      "dependencies": [
        "P2:OP4",
        "CH:W4",
        "CH:T0",
        "K:K3"
      ],
      "scope": "Actual global tempered off-diagonal Schwartz tests, every derivative, global pseudolocality and elliptic regularity"
    },
    {
      "id": "FC:M6",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "For use here and in U030",
      "dependencies": [
        "P3:L1",
        "P3:L3",
        "P3:M3",
        "P3:M4",
        "P3:M6"
      ],
      "scope": "Complete all-dimension Fourier Sobolev embedding with exact integrability condition and parameter bounds"
    },
    {
      "id": "FC:M7",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "Now let \\(u\\in",
      "dependencies": [
        "FC:M4",
        "FC:M5",
        "CE:G1",
        "P3:M4"
      ],
      "scope": "Tempered compact-time solution, actual characteristic operator and separated forcing error"
    },
    {
      "id": "FC:M8",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "ced32981f0aa2f29f1f67a1dbcc6143ed4544fa5feb5b352d7d78e3d4a3e42e9",
      "proof_locator": "For the transported spatial test",
      "dependencies": [
        "FC:M4",
        "FC:M5",
        "FC:M6",
        "CE:G1",
        "P3:L1",
        "P3:L3",
        "P3:M4"
      ],
      "scope": "Actual ordinary localized commuting test, joint all-Sobolev regularity and elliptic removal"
    },
    {
      "id": "FC:S0",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "## Exact earlier proofs",
      "dependencies": [
        "U001:U001-A4",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "Finite smooth localization construction and actual current retained source bindings"
    },
    {
      "id": "FC:S1",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "### 18.5.",
      "dependencies": [
        "FC:S0",
        "CH:T0",
        "CH:W1",
        "CH:W2",
        "P3:L1",
        "P3:L3",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Full original smooth-map transverse pullback integral and every finite distribution estimate"
    },
    {
      "id": "FC:S2",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "Here is the needed independence",
      "dependencies": [
        "FC:S1",
        "P3:L1",
        "P3:M3",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full Gaussian approximation, all separated-support derivatives, cutoff independence and local gluing"
    },
    {
      "id": "FC:S3",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "For completeness fix a nonzero",
      "dependencies": [
        "FC:S2",
        "CH:W1",
        "CH:W2",
        "U001:F0-COMP",
        "P3:M3"
      ],
      "scope": "Full exact pullback wavefront inclusion, both regular-frequency regions and all powers"
    },
    {
      "id": "FC:S4",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "For a submersion",
      "dependencies": [
        "FC:S3",
        "RP:W2",
        "CH:W3",
        "U001:P3"
      ],
      "scope": "Exact submersion converse from actual inverse chart and constant-factor projection, all zero dimensions"
    },
    {
      "id": "FC:S5",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "### 18.6.",
      "dependencies": [
        "FC:S1",
        "FC:S2",
        "FC:S3",
        "CH:W1",
        "P3:M3",
        "P3:M4",
        "P3:L1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full no-normal parameter family, every derivative in strong test topology and reconstruction of original distribution"
    },
    {
      "id": "FC:S6",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "## S1.",
      "dependencies": [
        "FC:S1",
        "FC:S3",
        "TG:U1",
        "CH:W1",
        "U001:F0-COMP",
        "P3:M3"
      ],
      "scope": "Fixed-wavefront sequential continuity, uniform finite order and compact Fourier convergence, closed output cone"
    },
    {
      "id": "FC:S7",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "The smooth Gaussian approximations",
      "dependencies": [
        "FC:S2",
        "FC:S6",
        "CH:W1",
        "P3:M3",
        "P3:L1"
      ],
      "scope": "Actual smooth fixed-wavefront approximation and unique continuous extension"
    },
    {
      "id": "FC:S8",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "cd52912a9414ec6802bd4bde40fd45b4abaedcc35afe65b9d3fed406e4cbbb8c",
      "proof_locator": "## S2.",
      "dependencies": [
        "FC:S5",
        "CE:G1",
        "U001:U001-A4",
        "P3:M4"
      ],
      "scope": "Actual equality of smooth pullback and continuous Sobolev trace by continuous scalar testing"
    },
    {
      "id": "FC:A0",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "## 1.",
      "dependencies": [
        "CE:G1",
        "CE:G9",
        "CE:G10",
        "CE:H6",
        "CE:H12",
        "P2:OP3"
      ],
      "scope": "Exact global scalar hypotheses, full strong continuity, dual pairing and source scope",
      "complete_receiving_proof_section_lines": [
        7,
        35
      ],
      "complete_receiving_proof_section_sha256": "05211137075098569152bdc1eab9543db825430fbdb06a98549089eca66371bb"
    },
    {
      "id": "FC:E1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Begin with",
      "dependencies": [
        "FC:A0",
        "CE:G9",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete scalar weighted squared-norm and maximum quadratic proof including zero norm",
      "complete_receiving_proof_section_lines": [
        36,
        73
      ],
      "complete_receiving_proof_section_sha256": "64812575ae578ac281ef314e96834d0b37542dd30e293dd3f2895389e51bfe65"
    },
    {
      "id": "FC:E2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Extend the weighted forcing",
      "dependencies": [
        "FC:E1",
        "CE:H2",
        "P3:M4"
      ],
      "scope": "All p energy bound, exact factor2/lambda-half and infinity endpoint with common threshold",
      "complete_receiving_proof_section_lines": [
        36,
        73
      ],
      "complete_receiving_proof_section_sha256": "64812575ae578ac281ef314e96834d0b37542dd30e293dd3f2895389e51bfe65"
    },
    {
      "id": "FC:E3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "For general \\(s\\), put",
      "dependencies": [
        "FC:E2",
        "CE:G1",
        "CE:G9",
        "CE:G10",
        "P2:OP3"
      ],
      "scope": "All-real Sobolev conjugation, uniform zero-order remainder and actual lower form bound",
      "complete_receiving_proof_section_lines": [
        36,
        73
      ],
      "complete_receiving_proof_section_sha256": "64812575ae578ac281ef314e96834d0b37542dd30e293dd3f2895389e51bfe65"
    },
    {
      "id": "FC:C1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "**Uniqueness in the claimed class.**",
      "dependencies": [
        "FC:E3",
        "CE:G10",
        "CE:H10"
      ],
      "scope": "Correct lower-order energy uniqueness with actual C1 representative",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "**The reversed adjoint estimate.**",
      "dependencies": [
        "FC:E3",
        "P2:OP3",
        "CE:G9",
        "CE:G10"
      ],
      "scope": "Adjoint plus sign, exact zero terminal test domain and reverse dual-order estimate",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Define on the range",
      "dependencies": [
        "FC:C2",
        "CE:H1",
        "CE:H2",
        "CE:H12"
      ],
      "scope": "Well-defined bounded conjugate-linear functional and full complex Hahn–Banach extension",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C4",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "**The Hilbert representation",
      "dependencies": [
        "FC:C3",
        "CE:H5",
        "CE:H11",
        "CE:H12",
        "CT:T1"
      ],
      "scope": "Actual Hilbert representation, L-infinity norm test and L1 density extension",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C5",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "The test identity is now",
      "dependencies": [
        "FC:C4",
        "CE:H6",
        "CE:H7",
        "CE:H9",
        "CE:H10"
      ],
      "scope": "Strong measurable action, actual vector primitive, unique continuous representative and exact initial trace",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C6",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "**Gain the regularity needed",
      "dependencies": [
        "FC:C5",
        "CE:G10",
        "CE:H10",
        "FC:E3"
      ],
      "scope": "Smooth-data construction two orders higher achieves actual energy domain",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C7",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "For arbitrary data,",
      "dependencies": [
        "FC:C6",
        "FC:C1",
        "CE:H6",
        "CE:H10",
        "CE:H5"
      ],
      "scope": "Full uniform solution limit and completeness, exact equation trace and every energy endpoint",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:C8",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "For homogeneous forcing",
      "dependencies": [
        "FC:C7",
        "FC:C1",
        "FC:M6",
        "CE:G10"
      ],
      "scope": "All spatial orders, correct one-time-derivative scope and smooth-time bootstrap",
      "complete_receiving_proof_section_lines": [
        74,
        118
      ],
      "complete_receiving_proof_section_sha256": "9bcefcfa614d54a31c561d39b6055de2bd80a1a6f5047df3963c6dab1cef2db6"
    },
    {
      "id": "FC:F1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "## 4.",
      "dependencies": [
        "FC:A0",
        "FC:C7",
        "P2:OP3"
      ],
      "scope": "Exact uniform classical family and both time-direction lower bounds",
      "complete_receiving_proof_section_lines": [
        119,
        147
      ],
      "complete_receiving_proof_section_sha256": "e89f71dfc54c07116a3390852dbedeb04035f812b32d956c017047a5eac5ee65"
    },
    {
      "id": "FC:F2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Here is the needed time-continuous",
      "dependencies": [
        "FC:F1",
        "G2:NF1",
        "G2:NF2",
        "G2:NF7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-COMP"
      ],
      "scope": "Full time-continuous state-smooth contraction and differentiated integral proof, complete finite-time frequency/base bounds",
      "complete_receiving_proof_section_lines": [
        119,
        147
      ],
      "complete_receiving_proof_section_sha256": "e89f71dfc54c07116a3390852dbedeb04035f812b32d956c017047a5eac5ee65"
    },
    {
      "id": "FC:F3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Homogeneity and uniqueness give",
      "dependencies": [
        "FC:F2",
        "G2:G0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Homogeneous two-time flow, all normalized derivative estimates and full symplectic variational identity",
      "complete_receiving_proof_section_lines": [
        119,
        147
      ],
      "complete_receiving_proof_section_sha256": "e89f71dfc54c07116a3390852dbedeb04035f812b32d956c017047a5eac5ee65"
    },
    {
      "id": "FC:Q1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Choose a compactly based",
      "dependencies": [
        "CH:W1",
        "CH:W4",
        "FC:M5",
        "P2:OP4",
        "U001:U001-A4"
      ],
      "scope": "Actual initial regular compact-base classical conic test with H-infinity output",
      "complete_receiving_proof_section_lines": [
        148,
        190
      ],
      "complete_receiving_proof_section_sha256": "7f719cf84bcca51ec18fef3f8bbf9a4e836c3a253afa6f9784649cb1ec487192"
    },
    {
      "id": "FC:Q2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "Here \\(b=a/i\\)",
      "dependencies": [
        "FC:Q1",
        "FC:F3",
        "P2:OP3"
      ],
      "scope": "Full scalar commutator expansion, first sign, triangular recurrence and degree count",
      "complete_receiving_proof_section_lines": [
        148,
        190
      ],
      "complete_receiving_proof_section_sha256": "7f719cf84bcca51ec18fef3f8bbf9a4e836c3a253afa6f9784649cb1ec487192"
    },
    {
      "id": "FC:Q3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a25b0db6177473807d8d6b099c799e732896f3a7e19768c4ca58bbfb009030e3",
      "proof_locator": "The complete solution is",
      "dependencies": [
        "FC:Q2",
        "FC:F2",
        "FC:F3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every transported homogeneous coefficient, actual integral and full compact-support/derivative estimates",
      "complete_receiving_proof_section_lines": [
        148,
        190
      ],
      "complete_receiving_proof_section_sha256": "7f719cf84bcca51ec18fef3f8bbf9a4e836c3a253afa6f9784649cb1ec487192"
    },
    {
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