{
  "schema": "AN04-restored-folded-forms-proof-map/v1",
  "proofs": [
    {
      "id": "FF:L0",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "## 1.",
      "dependencies": [
        "SN:F2",
        "G2:F0",
        "U001:P9.1",
        "U001:P3"
      ],
      "scope": "Alternating-form rank/wedge proof, smooth ambient/restricted radicals, exact intersection and transverse kernel"
    },
    {
      "id": "FF:M1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "## 2.",
      "dependencies": [
        "FF:L0",
        "SN:F0",
        "G2:F1",
        "G2:G31",
        "G2:NF7"
      ],
      "scope": "Characteristic flow quotient, symplectic section and actual smooth transverse ambient-kernel collar"
    },
    {
      "id": "FF:M2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "Suppose in addition that a smooth involution",
      "dependencies": [
        "FF:M1",
        "FD:R2",
        "G2:NF7"
      ],
      "scope": "Reflection line in ambient radical, antisymmetric extension and exact flow reflection coordinates"
    },
    {
      "id": "FF:O1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "FF:M1",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P3",
        "U001:P14.2-powers"
      ],
      "scope": "First-jet normalization, mixed pair changes, positive normal scaling and smooth second-order remainder"
    },
    {
      "id": "FF:O2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "We produce a relative primitive",
      "dependencies": [
        "FF:O1",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Explicit closed relative primitive with one extra vanishing order on both signs of the fold"
    },
    {
      "id": "FF:O3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "Let \\(\\sigma_t=",
      "dependencies": [
        "FF:O2",
        "U001:P2"
      ],
      "scope": "Smooth factorization of the singular inverse, uniform invertibility, full smooth Moser field and zero first jet"
    },
    {
      "id": "FF:O4",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "The common-time flow argument",
      "dependencies": [
        "FF:O3",
        "G2:F4",
        "G2:NF4",
        "G2:F2"
      ],
      "scope": "Actual time-one Moser flow, variational first jet and full ordinary folded Darboux equality"
    },
    {
      "id": "FF:O5",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "For the involution assertion",
      "dependencies": [
        "FF:O4",
        "FF:M2",
        "G2:F0",
        "G2:F1",
        "G2:NF7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Smooth contraction primitive at parameter zero, symmetry of the solved field and exact reflection-preserving normal form"
    },
    {
      "id": "FF:H1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
        "FF:L0",
        "FF:M1",
        "SN:F1",
        "CR:H3",
        "SI:H1"
      ],
      "scope": "Nonradial restricted characteristic geometry, commuting homogeneous section and marked homogeneous spectator chart"
    },
    {
      "id": "FF:H2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "Next choose the transverse ambient-kernel field",
      "dependencies": [
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        "FF:M2",
        "SN:F1",
        "SI:H1",
        "FF:O1",
        "U001:P14.2-powers"
      ],
      "scope": "Weighted kernel field and equivariant collar, exact half-degree normal coordinate, mixed coefficient degrees and parity"
    },
    {
      "id": "FF:H3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "The model \\(\\sigma_0\\), the remainder",
      "dependencies": [
        "FF:H2",
        "FF:O2",
        "FF:O3",
        "FF:O4",
        "FF:O5",
        "SI:H1"
      ],
      "scope": "Homogeneous primitive, invariant smooth Moser field, full conic time-one map and homogeneous folded Darboux with reflection"
    },
    {
      "id": "FF:D1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Lemma 5.1",
      "dependencies": [
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        "SN:F1",
        "G2:G1",
        "G2:G31",
        "U001:P3"
      ],
      "scope": "Function-preserving ordinary Darboux pair by actual commuting Hamilton flows and orthogonal section"
    },
    {
      "id": "FF:T1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Theorem 5.2",
      "dependencies": [
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        "FD:H1",
        "U001:P3"
      ],
      "scope": "Actual fold over a symplectic target with exact prescribed source square and full differential-coordinate check"
    },
    {
      "id": "FF:T2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "For the converse, smooth invariant descent",
      "dependencies": [
        "FF:T1",
        "FD:D3",
        "G2:F0",
        "U001:P3"
      ],
      "scope": "Attained-side target chart, full boundary form equality and nondegeneracy of the descended coordinate differential"
    },
    {
      "id": "FF:T3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "The arbitrary descended extensions",
      "dependencies": [
        "FF:T2",
        "G2:G1",
        "SN:F0",
        "G2:NF7"
      ],
      "scope": "Full target Hamilton collar, matching complete boundary first jets, propagated canonical brackets and exact converse"
    },
    {
      "id": "FF:HD",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Lemma 6.1",
      "dependencies": [
        "FD:H0",
        "FD:H1",
        "FD:D3",
        "SI:H1",
        "U001:P14.2-powers"
      ],
      "scope": "Homogeneous invariant descent on matched radial slices, retained fold Hessian and exact extension along rays"
    },
    {
      "id": "FF:HT",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "Suppose the source coordinates in the converse",
      "dependencies": [
        "FF:HD",
        "FF:T3",
        "SN:F1",
        "SI:H1"
      ],
      "scope": "Full homogeneous target extension, exact Hamilton-flow scaling and retained degrees on both sides"
    },
    {
      "id": "FF:HF",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "There is a homogeneous forward version too.",
      "dependencies": [
        "FF:HD",
        "FD:R1",
        "SN:H1",
        "CR:H3",
        "FD:D1",
        "FF:T1",
        "SI:H1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Nonradial homogeneous forward fold theorem, exact retained zero momentum, invariant slice and positive odd half-degree square root"
    },
    {
      "id": "FF:Q1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Proposition 7.1",
      "dependencies": [
        "FF:L0",
        "FF:M2",
        "FD:D1",
        "FD:D2",
        "G2:F0"
      ],
      "scope": "Closed attained-side quotient form from parity, square scaling, pullback Jacobian and nonzero top coefficient"
    },
    {
      "id": "FF:Q2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "For the ordinary extension",
      "dependencies": [
        "FF:Q1",
        "G2:F3",
        "FD:E2",
        "U001:P2"
      ],
      "scope": "Full closed nondegenerate ordinary quotient extension through a proved half-neighborhood primitive"
    },
    {
      "id": "FF:Q3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "For the homogeneous assertion",
      "dependencies": [
        "FF:Q1",
        "FF:H3",
        "FD:R2",
        "FD:E2",
        "G2:F1",
        "SI:H1",
        "U001:P14.2-powers"
      ],
      "scope": "Homogeneous quotient coordinates and normalized primitive extension, retained closedness/nondegeneracy on a full cone"
    },
    {
      "id": "FF:X1",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "FF:L0",
        "FF:O4"
      ],
      "scope": "Exercise 9.1 and solution"
    },
    {
      "id": "FF:X2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "FF:L0",
        "G2:F0"
      ],
      "scope": "Exercise 9.2 and solution"
    },
    {
      "id": "FF:X3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "FF:O1"
      ],
      "scope": "Exercise 9.3 and solution"
    },
    {
      "id": "FF:X4",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "FF:O2",
        "FF:O3",
        "FF:O4",
        "U001:P3"
      ],
      "scope": "Exercise 9.4 and solution"
    },
    {
      "id": "FF:X5",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "FF:O5",
        "U001:P14.2-powers"
      ],
      "scope": "Exercise 9.5 and solution"
    },
    {
      "id": "FF:X6",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "FF:T3",
        "U001:P14.3"
      ],
      "scope": "Exercise 9.6 and solution"
    },
    {
      "id": "FF:X7",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.7 ",
      "dependencies": [
        "FF:Q2",
        "G2:F0",
        "U001:P14.3"
      ],
      "scope": "Exercise 9.7 and solution"
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    {
      "id": "FF:X8",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.8 ",
      "dependencies": [
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        "U001:P14.2-powers"
      ],
      "scope": "Exercise 9.8 and solution"
    },
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      "id": "FF:X9",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.9 ",
      "dependencies": [
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        "FF:H3"
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      "scope": "Exercise 9.9 and solution"
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      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "a55cb5630890282ad2ef5b45a697187308528cb94bd1914e5262ec3c1edcb66c",
      "proof_locator": "**Exercise 9.10 ",
      "dependencies": [
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        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
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  "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
  }
}