{
  "schema": "AN04-restored-folded-forms-proof-map/v1",
  "proofs": [
    {
      "id": "FF:L0",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "Next choose the transverse ambient-kernel field",
      "dependencies": [
        "FF:H1",
        "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Lemma 5.1",
      "dependencies": [
        "SN:F0",
        "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Theorem 5.2",
      "dependencies": [
        "FF:D1",
        "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "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": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "FF:L0",
        "FF:O4"
      ],
      "scope": "Original Exercise 9.1 and complete unchanged solution"
    },
    {
      "id": "FF:X2",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "FF:L0",
        "G2:F0"
      ],
      "scope": "Original Exercise 9.2 and complete unchanged solution"
    },
    {
      "id": "FF:X3",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "FF:O1"
      ],
      "scope": "Original Exercise 9.3 and complete unchanged solution"
    },
    {
      "id": "FF:X4",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "FF:O2",
        "FF:O3",
        "FF:O4",
        "U001:P3"
      ],
      "scope": "Original Exercise 9.4 and complete unchanged solution"
    },
    {
      "id": "FF:X5",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "FF:O5",
        "U001:P14.2-powers"
      ],
      "scope": "Original Exercise 9.5 and complete unchanged solution"
    },
    {
      "id": "FF:X6",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "FF:T3",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 9.6 and complete unchanged solution"
    },
    {
      "id": "FF:X7",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.7 ",
      "dependencies": [
        "FF:Q2",
        "G2:F0",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 9.7 and complete unchanged solution"
    },
    {
      "id": "FF:X8",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.8 ",
      "dependencies": [
        "FF:H3",
        "U001:P14.2-powers"
      ],
      "scope": "Original Exercise 9.8 and complete unchanged solution"
    },
    {
      "id": "FF:X9",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.9 ",
      "dependencies": [
        "FF:H1",
        "FF:H3"
      ],
      "scope": "Original Exercise 9.9 and complete unchanged solution"
    },
    {
      "id": "FF:X10",
      "source": "folded-forms-and-symplectic-targets.md",
      "source_sha256": "951ae2e263b8f1457c625edf3b0bbbaa83b1a711d38a3efdbf0e3d6421380e2c",
      "proof_locator": "**Exercise 9.10 ",
      "dependencies": [
        "FF:T1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Original Exercise 9.10 and complete unchanged solution"
    }
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    "P7",
    "P8-root",
    "P8-smooth",
    "P9.1",
    "P9.2",
    "P9.2-operator",
    "P9.3",
    "P9.4",
    "P9.5",
    "Q1",
    "Q2",
    "Q3",
    "Q4",
    "Q5",
    "R0",
    "U001-A4",
    "U001-A5",
    "U001-DEF"
  ],
  "transitive_earlier_package_ids": [
    "CH:T0",
    "CH:T1",
    "CH:T2",
    "CH:T3",
    "CH:W1",
    "CH:W4",
    "CH:W5",
    "CR:H1",
    "CR:H2",
    "CR:H3",
    "FD:D1",
    "FD:D2",
    "FD:D3",
    "FD:E0",
    "FD:E1",
    "FD:E2",
    "FD:H0",
    "FD:H1",
    "FD:R1",
    "FD:R2",
    "G2:F0",
    "G2:F1",
    "G2:F2",
    "G2:F3",
    "G2:F4",
    "G2:G0",
    "G2:G1",
    "G2:G21",
    "G2:G23",
    "G2:G31",
    "G2:G41",
    "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",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "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",
    "SI:C1",
    "SI:C2",
    "SI:C3",
    "SI:F1",
    "SI:H1",
    "SI:I0",
    "SI:I1",
    "SI:I2",
    "SI:I3",
    "SI:J1",
    "SI:J2",
    "SI:J3",
    "SI:J4",
    "SI:M0",
    "SI:S1",
    "SI:W1",
    "SN:F0",
    "SN:F1",
    "SN:F2",
    "SN:H1",
    "SN:L1"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}