{
  "schema": "AN04-restored-phase-space-proof-map/v1",
  "proofs": [
    {
      "id": "G2:NF0",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "## Exact prerequisite bindings for this extract",
      "dependencies": [
        "U001:P1",
        "U001:P2",
        "U001:P3",
        "U001:P9.3",
        "U001:F0-COMP",
        "U001:F0-CALC",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact finite norm/calculus/compactness inputs and explicit interval compact-subdivision and uniqueness facts"
    },
    {
      "id": "G2:NF1",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.1. The original integral equation and its complete local solution",
      "dependencies": [
        "G2:NF0"
      ],
      "scope": "17.1. The original integral equation and its complete local solution"
    },
    {
      "id": "G2:NF2",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.2. Ordered linear transport on the original two-sided interval",
      "dependencies": [
        "G2:NF1"
      ],
      "scope": "17.2. Ordered linear transport on the original two-sided interval"
    },
    {
      "id": "G2:NF3",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.3. All parameter derivatives of the linear inverse",
      "dependencies": [
        "G2:NF2"
      ],
      "scope": "17.3. All parameter derivatives of the linear inverse"
    },
    {
      "id": "G2:NF4",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.4. The first actual nonlinear parameter derivative",
      "dependencies": [
        "G2:NF1",
        "G2:NF2",
        "G2:NF3"
      ],
      "scope": "17.4. The first actual nonlinear parameter derivative"
    },
    {
      "id": "G2:NF5",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.5. Every nonlinear parameter derivative, including its original blocks",
      "dependencies": [
        "G2:NF3",
        "G2:NF4"
      ],
      "scope": "17.5. Every nonlinear parameter derivative, including its original blocks"
    },
    {
      "id": "G2:NF6",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.6. Variable initial times without replacing the original time coordinate",
      "dependencies": [
        "G2:NF2",
        "G2:NF5",
        "U001:P3"
      ],
      "scope": "17.6. Variable initial times without replacing the original time coordinate"
    },
    {
      "id": "G2:NF7",
      "source": "finite-coordinate-flows.md",
      "source_sha256": "34c2533ad3119903db5ef432403de3e4e9f1f10a9bf39c3020b355d0e4116a23",
      "proof_locator": "### 17.7. Continuation, the exact composition law and coordinate transport",
      "dependencies": [
        "G2:NF1",
        "G2:NF5",
        "G2:NF6",
        "G2:NF0"
      ],
      "scope": "17.7. Continuation, the exact composition law and coordinate transport"
    },
    {
      "id": "G2:F0",
      "source": "differential-forms-and-flow-pullbacks.md",
      "source_sha256": "a8784813cf57bd7bdc2ce3bc558b0e50d78e5f5ac164155767f225743b2f5fe8",
      "proof_locator": "## F0.",
      "dependencies": [
        "U001:P1",
        "U001:P2",
        "U001:P3",
        "U001:P9.3",
        "U001:F0-COMP",
        "U001:F0-CALC",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "GEO:C0"
      ],
      "scope": "Exterior algebra, d squared, signed product rules, pullback naturality and contraction"
    },
    {
      "id": "G2:F1",
      "source": "differential-forms-and-flow-pullbacks.md",
      "source_sha256": "a8784813cf57bd7bdc2ce3bc558b0e50d78e5f5ac164155767f225743b2f5fe8",
      "proof_locator": "## F1.",
      "dependencies": [
        "G2:F0",
        "G2:NF7"
      ],
      "scope": "Cartan formula, Lie derivative commuting with d and Lie-contraction commutator"
    },
    {
      "id": "G2:F2",
      "source": "differential-forms-and-flow-pullbacks.md",
      "source_sha256": "a8784813cf57bd7bdc2ce3bc558b0e50d78e5f5ac164155767f225743b2f5fe8",
      "proof_locator": "## F2.",
      "dependencies": [
        "G2:F0",
        "G2:F1",
        "G2:NF7"
      ],
      "scope": "Full time-dependent pullback differentiation"
    },
    {
      "id": "G2:F3",
      "source": "differential-forms-and-flow-pullbacks.md",
      "source_sha256": "a8784813cf57bd7bdc2ce3bc558b0e50d78e5f5ac164155767f225743b2f5fe8",
      "proof_locator": "## F3.",
      "dependencies": [
        "G2:F0",
        "G2:F1",
        "G2:F2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-COMP"
      ],
      "scope": "Radial homotopy in every degree, closed one-form primitive and quadratic vanishing"
    },
    {
      "id": "G2:F4",
      "source": "differential-forms-and-flow-pullbacks.md",
      "source_sha256": "a8784813cf57bd7bdc2ce3bc558b0e50d78e5f5ac164155767f225743b2f5fe8",
      "proof_locator": "## F4.",
      "dependencies": [
        "G2:NF7",
        "G2:F3",
        "U001:P2",
        "U001:F0-COMP"
      ],
      "scope": "Uniform existence through time one and the smooth Darboux deformation field"
    },
    {
      "id": "G2:G0",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "## 1. Signs fixed by the cotangent pairing",
      "dependencies": [
        "GEO:C0",
        "U001:P3",
        "PH:F1",
        "G2:F0"
      ],
      "scope": "Definitions, cotangent pairing, Hamiltonian signs and exact finite/constant-rank inputs"
    },
    {
      "id": "G2:G1",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Proposition 1.1",
      "dependencies": [
        "G2:G0",
        "G2:F1",
        "G2:F2",
        "G2:NF7"
      ],
      "scope": "Hamiltonian identities, Jacobi, flow preservation and cotangent-lift pairing"
    },
    {
      "id": "G2:G21",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Lemma 2.1",
      "dependencies": [
        "G2:G0"
      ],
      "scope": "Symplectic complements, rank-nullity and symplectic basis"
    },
    {
      "id": "G2:G22",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Lemma 2.2",
      "dependencies": [
        "G2:G21",
        "GEO:C1"
      ],
      "scope": "Common Lagrangian complement, basis extension and the nonzero determinant polynomial"
    },
    {
      "id": "G2:G23",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Proposition 2.3",
      "dependencies": [
        "G2:G21",
        "GEO:C0"
      ],
      "scope": "Linear symplectic reduction, kernel and exact quotient dimension"
    },
    {
      "id": "G2:G31",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "G2:G21",
        "G2:F3",
        "G2:F4",
        "G2:F2",
        "U001:P3"
      ],
      "scope": "Darboux, symplectic volume, closed-graph primitives and local zero section"
    },
    {
      "id": "G2:G41",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "## 4. Homogeneity and the contact slice",
      "dependencies": [
        "G2:G21",
        "G2:G31",
        "G2:G1",
        "G2:F1",
        "G2:NF7",
        "U001:P3"
      ],
      "scope": "Dilation one-form, nonzero radial field, local slice, conic vanishing and contact form"
    },
    {
      "id": "G2:G51",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Theorem 5.1",
      "dependencies": [
        "G2:G0",
        "G2:G41",
        "G2:F0",
        "PH:F1",
        "U001:P4"
      ],
      "scope": "Clean critical-map rank, exact kernel, local conic Lagrangian and excess"
    },
    {
      "id": "G2:G61",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Theorem 6.1",
      "dependencies": [
        "G2:G51",
        "G2:G41",
        "GEO:C0",
        "U001:P3"
      ],
      "scope": "Mixed generating function, vertical annihilator and minimal variable count"
    },
    {
      "id": "G2:G62",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Theorem 6.2",
      "dependencies": [
        "G2:G22",
        "G2:G41",
        "G2:G1",
        "U001:P3",
        "GEO:C3",
        "GEO:C4"
      ],
      "scope": "Base-induced frequency coordinates, homogeneous primitive and uniqueness"
    },
    {
      "id": "G2:G71",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "## 7. Kernel relations and clean composition",
      "dependencies": [
        "G2:G23",
        "G2:G51",
        "G2:F0",
        "PH:F1"
      ],
      "scope": "Input sign, clean canonical composition, quotient rank and local branch scope"
    },
    {
      "id": "G2:G72",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Proposition 7.2",
      "dependencies": [
        "G2:G71",
        "G2:G51",
        "U001:P3",
        "U001:P8-smooth"
      ],
      "scope": "Clean addition of phases and homogeneous coordinate conversion"
    },
    {
      "id": "G2:M8",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "## 8. Models and exercises with solutions",
      "dependencies": [
        "G2:G61",
        "G2:G51"
      ],
      "scope": "Conormal and redundant clean-phase models"
    },
    {
      "id": "G2:E81",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Exercise 8.1",
      "dependencies": [
        "G2:G71"
      ],
      "scope": "Exercise 8.1 and its complete solution"
    },
    {
      "id": "G2:E82",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Exercise 8.2",
      "dependencies": [
        "G2:G51"
      ],
      "scope": "Exercise 8.2 and its complete solution"
    },
    {
      "id": "G2:E83",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Exercise 8.3",
      "dependencies": [
        "G2:G1",
        "G2:G71"
      ],
      "scope": "Exercise 8.3 and its complete solution"
    },
    {
      "id": "G2:E84",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Exercise 8.4",
      "dependencies": [
        "G2:G71"
      ],
      "scope": "Exercise 8.4 and its complete solution"
    },
    {
      "id": "G2:E85",
      "source": "phase-space-and-generating-families.md",
      "source_sha256": "64229251bcbced3c13c6d092cdf881d0a34356e4bbd763682abde9e599237a72",
      "proof_locator": "**Exercise 8.5",
      "dependencies": [
        "G2:G61",
        "G2:G62",
        "U001:P8-smooth"
      ],
      "scope": "Exercise 8.5 and its complete solution"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json"
  ],
  "transitive_U001_ids": [
    "A0",
    "CLOSED-BALL-CONTRACTION",
    "D0",
    "E0",
    "EXPONENTIAL-TRIGONOMETRIC",
    "F0-ALG",
    "F0-CALC",
    "F0-COMP",
    "F0-DIFF",
    "FTC-TAYLOR-COMPACT-PARAMETERS",
    "I0",
    "J0",
    "L10.1.12",
    "L10.1.13",
    "L10.1.14",
    "L10.1.15",
    "L10.1.2",
    "L10.1.5",
    "L10.1.6",
    "L10.2.2",
    "L10.3.10",
    "L10.3.2",
    "L10.3.7",
    "L10.4.1",
    "L10.4.2",
    "L10.4.3",
    "L10.5.1",
    "L10.5.5",
    "L10.5.9",
    "L10.7.2",
    "L11.4.2",
    "L2.1.10-inc",
    "L2.1.17",
    "L2.2.1",
    "L2.2.11",
    "L2.2.3",
    "L2.2.5",
    "L2.3.2",
    "L2.3.4-upper",
    "L2.3.5",
    "L2.3.8",
    "L2.4.5",
    "L2.6.5",
    "L3.1.7",
    "L3.3.7",
    "L3.3.8",
    "L4.2.2",
    "L4.2.3",
    "L4.2.4",
    "L4.2.6",
    "L5.1.10",
    "L5.1.11",
    "L5.1.13",
    "L5.1.2",
    "L5.1.7",
    "L5.1.8",
    "L5.2.1",
    "L5.2.2",
    "L5.2.4-positive",
    "L5.2.6",
    "L5.2.7",
    "L5.3.1",
    "L5.3.3",
    "L5.3.5",
    "L5.4.1",
    "L5.4.2",
    "L5.5.2-gt1",
    "L5.5.3",
    "L5.5.4",
    "L5.5.5",
    "L7.1.4",
    "L7.1.5",
    "L7.2.19",
    "L7.2.22",
    "L7.2.6",
    "L7.2.9-open",
    "L7.3.11",
    "L7.3.12",
    "L7.3.9",
    "L7.4.10",
    "L7.4.11",
    "L7.4.2",
    "L7.4.4",
    "L7.4.9",
    "L7.5.11",
    "L7.5.12",
    "L7.5.2",
    "L7.5.5",
    "L7.5.6",
    "L7.6.2",
    "L8.1.13",
    "L8.1.14",
    "L8.1.16",
    "L8.1.17",
    "L8.1.18",
    "L8.2.4",
    "L8.2.5",
    "L8.2.6",
    "L8.2.7",
    "L8.2.8",
    "L8.2.9",
    "L8.3.2",
    "L8.3.3",
    "L8.3.5",
    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
    "L8.4.6",
    "L8.5.1",
    "L8.5.6",
    "L8.6.2",
    "L9.1.1",
    "P1",
    "P10.1",
    "P10.2",
    "P10.2-typo",
    "P10.3",
    "P10.4",
    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
    "P12.2-upper",
    "P12.3",
    "P12.4",
    "P12.5-endpoints",
    "P12.5-extension",
    "P12.6",
    "P12.7",
    "P13.1",
    "P13.2-mertens",
    "P13.2-series",
    "P14.2-inverse",
    "P14.2-powers",
    "P14.3",
    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P17.1",
    "P17.2",
    "P17.3",
    "P17.4",
    "P17.4-grid",
    "P17.5-order",
    "P17.5-upper",
    "P18.1",
    "P18.2",
    "P18.3",
    "P18.4",
    "P18.5",
    "P19.1",
    "P19.2",
    "P19.3",
    "P19.4",
    "P2",
    "P20.1",
    "P20.2",
    "P20.3",
    "P20.4",
    "P20.5",
    "P21.1",
    "P21.2",
    "P21.3",
    "P21.4",
    "P21.5",
    "P3",
    "P4",
    "P5",
    "P6.0",
    "P6.1",
    "P6.2",
    "P6.3-inf",
    "P6.3-subtract",
    "P6.3-tail",
    "P6.4-ball",
    "P6.4-complete",
    "P6.5",
    "P6.6",
    "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",
    "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"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}