{
  "schema": "AN04-restored-subprincipal-transport-proof-map/v1",
  "proofs": [
    {
      "id": "ST:S1",
      "source": "subprincipal-coordinate-invariance.md",
      "source_sha256": "a3ee16450fcce62e330243c3663724e4c7c48845e39049c1a323fc9f7830a54a",
      "proof_locator": "## S1.",
      "dependencies": [
        "CH:T1",
        "CH:T2",
        "P2:OP4",
        "U001:P9.1",
        "U001:P9.5",
        "U001:P4"
      ],
      "scope": "Explicit first coordinate correction from the exact amplitude, both Jacobians, determinant derivatives, cancellation and full S^(m-2) derivative bounds"
    },
    {
      "id": "ST:S2",
      "source": "subprincipal-coordinate-invariance.md",
      "source_sha256": "a3ee16450fcce62e330243c3663724e4c7c48845e39049c1a323fc9f7830a54a",
      "proof_locator": "## 3.",
      "dependencies": [
        "ST:S1",
        "U001:P4"
      ],
      "scope": "Retained G11-G12: scalar subprincipal expression, all chain-rule indices, mixed derivatives and determinant trace"
    },
    {
      "id": "ST:S3",
      "source": "subprincipal-coordinate-invariance.md",
      "source_sha256": "a3ee16450fcce62e330243c3663724e4c7c48845e39049c1a323fc9f7830a54a",
      "proof_locator": "The density line bundle",
      "dependencies": [
        "ST:S2",
        "P2:OP3",
        "U001:P14.1",
        "U001:P14.2-powers",
        "U001:P15.1",
        "U001:P15.2"
      ],
      "scope": "Retained complex density transitions, scalar multiplication conjugation, G13 and exact half-density cancellation"
    },
    {
      "id": "ST:S4",
      "source": "subprincipal-coordinate-invariance.md",
      "source_sha256": "a3ee16450fcce62e330243c3663724e4c7c48845e39049c1a323fc9f7830a54a",
      "proof_locator": "There is a version without homogeneity.",
      "dependencies": [
        "ST:S1",
        "ST:S2",
        "ST:S3",
        "PS:PS5",
        "HT:H2"
      ],
      "scope": "Retained G14: invariant two-term ordinary symbol class, local remainder and scalar patching"
    },
    {
      "id": "ST:T0",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "For a complex number",
      "dependencies": [
        "ST:S3",
        "U001:U001-A5"
      ],
      "scope": "Complex density powers, positive transition cocycle and intrinsic pullback"
    },
    {
      "id": "ST:T1",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "Let \\(V=\\sum_j",
      "dependencies": [
        "ST:T0",
        "G2:NF7",
        "GS:Z6",
        "U001:P9.1",
        "U001:P9.5"
      ],
      "scope": "Smooth flow definition, exact determinant derivative, arbitrary density weight, complex linear extension and flat Maslov transitions"
    },
    {
      "id": "ST:T2",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "For a real field and compactly supported",
      "dependencies": [
        "ST:T1",
        "U001:U001-A6",
        "PS:PS5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Skew half-density pairing with finite partition and explicit cancellation of cutoff derivatives"
    },
    {
      "id": "ST:T3",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Lemma 2.1.",
      "dependencies": [
        "G2:G0",
        "G2:G21",
        "G2:G71"
      ],
      "scope": "Hamilton tangency from the exact symplectic sign, complexification and input reflection"
    },
    {
      "id": "ST:T4",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Proof of existence.**",
      "dependencies": [
        "G2:G21",
        "G2:G23"
      ],
      "scope": "Diagonal Lagrangian complement by isotropic line reduction, all dimensions and transversality"
    },
    {
      "id": "ST:T5",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Proof of almost-everywhere transversality.**",
      "dependencies": [
        "ST:T4",
        "U001:P9.1",
        "P3:M2",
        "P3:M4"
      ],
      "scope": "Nonzero determinant polynomial, explicit finite root count, inductive null slices and product integration"
    },
    {
      "id": "ST:T6",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "## 5. Separate base coordinates",
      "dependencies": [
        "ST:T4",
        "U001:P3",
        "G2:G41",
        "G2:G61"
      ],
      "scope": "Separate quadratic base jets using both nonzero covectors, conic frequency graph and exact homogeneous generating function"
    },
    {
      "id": "ST:T7",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "Let \\(n=n_X+n_Y\\).",
      "dependencies": [
        "ST:T6",
        "IR:R2",
        "IR:R3",
        "IR:R7",
        "GS:Z11",
        "OSC:O4",
        "HT:H6",
        "HT:H8"
      ],
      "scope": "Frequency-only amplitude and exact normalization, flat symbol frame, base reduction, supported asymptotic sum and smooth remainder"
    },
    {
      "id": "ST:T8",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "## 6. Integration by parts",
      "dependencies": [
        "ST:T7",
        "ST:T3",
        "ST:S3",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "RC:K8",
        "P2:OP1",
        "P2:OP5",
        "CH:W4",
        "OSC:O1",
        "HT:H8"
      ],
      "scope": "Actual local PA with nonproper A, proper intermediate supports, smooth errors, Taylor division, all symbol derivatives and justified one-order gain"
    },
    {
      "id": "ST:T9",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "## 7. Half the divergence",
      "dependencies": [
        "ST:T8",
        "ST:T1",
        "ST:S3",
        "GS:Z10",
        "GS:Z11",
        "HT:H2",
        "HT:H3",
        "PS:PS5"
      ],
      "scope": "Critical-graph Hamilton field, total versus fixed-base derivatives, exact subprincipal sign, intrinsic patching, all derivative estimates and quotient independence"
    },
    {
      "id": "ST:T10",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "ST:T0",
        "ST:T1",
        "ST:T3",
        "ST:T5",
        "ST:T7",
        "ST:T8",
        "ST:T9",
        "ST:S4"
      ],
      "scope": "Full scalar ordinary transport theorem, exact orders, relation, half-density and proper-support hypotheses"
    },
    {
      "id": "ST:T11",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "## 8. A lifting model",
      "dependencies": [
        "ST:T10",
        "OSC:O6",
        "RC:C10",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Actual compact-circle lifting kernel, conormal order, delta normalization, symmetrized derivative and exact density coefficient"
    },
    {
      "id": "ST:T12",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "On an interval where",
      "dependencies": [
        "ST:T11",
        "U001:P14.2-powers",
        "U001:P15.1",
        "U001:P15.2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete local scalar transport solution, integrating factor and uniqueness without dividing by a solution"
    },
    {
      "id": "ST:E1",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "ST:T1",
        "U001:P15.1"
      ],
      "scope": "Original Exercise 9.1 with its complete unchanged solution"
    },
    {
      "id": "ST:E2",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "ST:T4",
        "ST:T5"
      ],
      "scope": "Original Exercise 9.2 with its complete unchanged solution"
    },
    {
      "id": "ST:E3",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "ST:T6"
      ],
      "scope": "Original Exercise 9.3 with its complete unchanged solution"
    },
    {
      "id": "ST:E4",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "ST:T11",
        "U001:P15.2"
      ],
      "scope": "Original Exercise 9.4 with its complete unchanged solution"
    },
    {
      "id": "ST:E5",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "ST:T11",
        "ST:T12",
        "U001:P16.2"
      ],
      "scope": "Original Exercise 9.5 with its complete unchanged solution"
    },
    {
      "id": "ST:E6",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "6b54ee6a0c2a46a4a37dfe26bfed327641221ae739199f64dd693b6f50df934a",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "ST:T12",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Original Exercise 9.6 with its complete unchanged solution"
    }
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    "OSC:O3",
    "OSC:O4",
    "OSC:O6",
    "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",
    "PH:F6a",
    "PH:F7",
    "PS:PS0",
    "PS:PS1",
    "PS:PS2",
    "PS:PS3",
    "PS:PS4",
    "PS:PS5",
    "PS:PS6",
    "RC:C0",
    "RC:C1",
    "RC:C10",
    "RC:C2",
    "RC:C3",
    "RC:C5",
    "RC:C6",
    "RC:C7",
    "RC:C8",
    "RC:C9",
    "RC:K0",
    "RC:K1",
    "RC:K2",
    "RC:K3",
    "RC:K4",
    "RC:K5",
    "RC:K6",
    "RC:K7",
    "RC:K8",
    "TG:B0",
    "TG:B1",
    "TG:B2",
    "TG:B3",
    "TG:T0",
    "TG:T5",
    "TG:T6",
    "TG:U1",
    "TG:Z2",
    "TZ:E5",
    "TZ:Z0",
    "TZ:Z4",
    "TZ:Z5",
    "TZ:Z6"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}