{
  "schema": "AN04-restored-subprincipal-transport-proof-map/v1",
  "proofs": [
    {
      "id": "ST:S1",
      "source": "subprincipal-coordinate-invariance.md",
      "source_sha256": "2dca22c64d423d83b92ee3ac58c2271ce0f7e5389d2c1377eb35a071c73c9877",
      "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": "2dca22c64d423d83b92ee3ac58c2271ce0f7e5389d2c1377eb35a071c73c9877",
      "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": "2dca22c64d423d83b92ee3ac58c2271ce0f7e5389d2c1377eb35a071c73c9877",
      "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": "2dca22c64d423d83b92ee3ac58c2271ce0f7e5389d2c1377eb35a071c73c9877",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "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": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "ST:T1",
        "U001:P15.1"
      ],
      "scope": "Exercise 9.1 with its solution"
    },
    {
      "id": "ST:E2",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "ST:T4",
        "ST:T5"
      ],
      "scope": "Exercise 9.2 with its solution"
    },
    {
      "id": "ST:E3",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "ST:T6"
      ],
      "scope": "Exercise 9.3 with its solution"
    },
    {
      "id": "ST:E4",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "ST:T11",
        "U001:P15.2"
      ],
      "scope": "Exercise 9.4 with its solution"
    },
    {
      "id": "ST:E5",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "ST:T11",
        "ST:T12",
        "U001:P16.2"
      ],
      "scope": "Exercise 9.5 with its solution"
    },
    {
      "id": "ST:E6",
      "source": "hamilton-fields-and-subprincipal-transport.md",
      "source_sha256": "819a3804294c65dcf33822e937ca82679d09c19ee209c8172fb9a3efc9c55130",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "ST:T12",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Exercise 9.6 with its solution"
    }
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    "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"
  ],
  "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
  }
}