{
  "schema": "AN04-restored-prescribed-coordinates-proof-map/v1",
  "proofs": [
    {
      "id": "PC:S0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "## 1.",
      "dependencies": [
        "G2:G1",
        "G2:F0",
        "G2:F1"
      ],
      "scope": "Hamilton signs, Jacobi and the commutator identity"
    },
    {
      "id": "PC:L0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Lemma 1.1",
      "dependencies": [
        "PC:S0",
        "G2:G21"
      ],
      "scope": "Alternating-subspace extension including radical dual correction and symplectic complements"
    },
    {
      "id": "PC:O0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Theorem 2.1",
      "dependencies": [
        "PC:L0"
      ],
      "scope": "Ordinary prescribed-function theorem and compatible marked tangent completion"
    },
    {
      "id": "PC:O1",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "Here is the required smooth equation solver.",
      "dependencies": [
        "PC:O0",
        "SN:F0",
        "SN:F1",
        "U001:P3",
        "U001:F0-DIFF"
      ],
      "scope": "Joint commuting-flow chart, exact constant field equations and affine section jets"
    },
    {
      "id": "PC:O2",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "To add \\(q_J\\)",
      "dependencies": [
        "PC:O1",
        "PC:S0"
      ],
      "scope": "Finite ordinary coordinate induction, exact old functions and full Poisson matrix"
    },
    {
      "id": "PC:O3",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "For example, on \\(T^*\\mathbb R^2\\)",
      "dependencies": [
        "PC:O2",
        "U001:F0-DIFF"
      ],
      "scope": "Nonlinear canonical shear, exact primitive difference and failure of homogeneous degrees"
    },
    {
      "id": "PC:H0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "A conic symplectic manifold has",
      "dependencies": [
        "PC:S0",
        "G2:F1",
        "G2:G41"
      ],
      "scope": "Radial primitive, Cartan formula and Hamilton weights"
    },
    {
      "id": "PC:H1",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Proof, first step:",
      "dependencies": [
        "PC:H0",
        "PC:L0"
      ],
      "scope": "Compatible marked linear extension including the radial vector and exceptional nonzero momentum"
    },
    {
      "id": "PC:H2",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Second step:",
      "dependencies": [
        "PC:H1",
        "SN:F0",
        "SN:F1",
        "U001:P3",
        "U001:P15.1"
      ],
      "scope": "Actual semidirect flow chart, dilation conjugation and positive radial sign"
    },
    {
      "id": "PC:H3",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "To add a position \\(q_i\\)",
      "dependencies": [
        "PC:H2",
        "PC:O1"
      ],
      "scope": "Exact homogeneous position and momentum solutions, all marked jets and finite induction"
    },
    {
      "id": "PC:H4",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Third step:",
      "dependencies": [
        "PC:H3",
        "PC:O2",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exceptional last position, one-variable radial defect and exact integral correction"
    },
    {
      "id": "PC:H5",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Fourth step:",
      "dependencies": [
        "PC:H4",
        "G2:G41",
        "SN:F0",
        "U001:P3"
      ],
      "scope": "Local-to-conic extension, single-ray sections, bracket degrees and full homogeneous symplectomorphism"
    },
    {
      "id": "PC:H6",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "The exceptional nonzero value is necessary",
      "dependencies": [
        "PC:H5",
        "PC:H0"
      ],
      "scope": "Necessary marked momentum, omitted position and exact primitive preservation"
    },
    {
      "id": "PC:I0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "Let \\(V\\) be a conic embedded",
      "dependencies": [
        "PC:H0",
        "G2:F0",
        "G2:G21"
      ],
      "scope": "Both directions of isotropy versus vanishing pulled-back primitive as a germ"
    },
    {
      "id": "PC:I1",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "Degree-one defining functions with independent",
      "dependencies": [
        "PC:I0",
        "G2:G41",
        "U001:P3"
      ],
      "scope": "Positive-ray defining functions spanning the full conormal"
    },
    {
      "id": "PC:I2",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Case \\(k<n\\)",
      "dependencies": [
        "PC:I1",
        "PC:H5",
        "G2:G21",
        "U001:P3"
      ],
      "scope": "Removal of an actual canonical normal pair and full homogeneous product extension"
    },
    {
      "id": "PC:I3",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Case \\(k=n>1\\)",
      "dependencies": [
        "PC:I1",
        "PC:H5",
        "SN:F0",
        "U001:P3"
      ],
      "scope": "Lagrangian tangent flow, cylinder equality and reduced conic Lagrangian"
    },
    {
      "id": "PC:I4",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Base \\(n=k=1\\)",
      "dependencies": [
        "PC:I0",
        "PC:H5",
        "SN:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Positive-momentum one-dimensional base and complete ambient-dimension induction"
    },
    {
      "id": "PC:I5",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "It remains to turn (4.3) into (4.2)",
      "dependencies": [
        "PC:I2",
        "PC:I3",
        "PC:I4",
        "U001:F0-DIFF"
      ],
      "scope": "Exact homogeneous rotation, inverse, primitive, spectators and both model inclusions"
    },
    {
      "id": "PC:C0",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "## 5.",
      "dependencies": [
        "PC:I0",
        "U001:P3"
      ],
      "scope": "Nonzero conormal geometry in adapted base coordinates"
    },
    {
      "id": "PC:C1",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Proof.** Here is the required constant-rank map",
      "dependencies": [
        "U001:P3",
        "U001:F0-DIFF",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete constant-rank map proof by invertible minor, derivative block and line-segment calculus"
    },
    {
      "id": "PC:C2",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "The primitive vanishes on \\(T\\Lambda\\)",
      "dependencies": [
        "PC:C0",
        "PC:C1",
        "PC:I0",
        "U001:P3"
      ],
      "scope": "Conormal germ equality, equal-dimensional open inclusion and strict rank bound"
    },
    {
      "id": "PC:C3",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "The constant-rank locus is open and dense",
      "dependencies": [
        "PC:C2",
        "U001:F0-DIFF"
      ],
      "scope": "Maximal-minor argument on every open patch and exact density qualification"
    },
    {
      "id": "PC:C4",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "Here is an original deformed-cusp example.",
      "dependencies": [
        "PC:C2",
        "PC:C3",
        "U001:P3",
        "U001:F0-DIFF"
      ],
      "scope": "Smooth lifted embedding, both projection ranks, exact primitive and nonconormal cusp germ"
    },
    {
      "id": "PC:X1",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.1 ",
      "dependencies": [
        "PC:L0"
      ],
      "scope": "Original Exercise 6.1 with complete unchanged solution"
    },
    {
      "id": "PC:X2",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.2 ",
      "dependencies": [
        "PC:O3"
      ],
      "scope": "Original Exercise 6.2 with complete unchanged solution"
    },
    {
      "id": "PC:X3",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.3 ",
      "dependencies": [
        "PC:O2"
      ],
      "scope": "Original Exercise 6.3 with complete unchanged solution"
    },
    {
      "id": "PC:X4",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.4 ",
      "dependencies": [
        "PC:H2"
      ],
      "scope": "Original Exercise 6.4 with complete unchanged solution"
    },
    {
      "id": "PC:X5",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.5 ",
      "dependencies": [
        "PC:H4"
      ],
      "scope": "Original Exercise 6.5 with complete unchanged solution"
    },
    {
      "id": "PC:X6",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.6 ",
      "dependencies": [
        "PC:H6"
      ],
      "scope": "Original Exercise 6.6 with complete unchanged solution"
    },
    {
      "id": "PC:X7",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.7 ",
      "dependencies": [
        "PC:I5"
      ],
      "scope": "Original Exercise 6.7 with complete unchanged solution"
    },
    {
      "id": "PC:X8",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.8 ",
      "dependencies": [
        "PC:I0"
      ],
      "scope": "Original Exercise 6.8 with complete unchanged solution"
    },
    {
      "id": "PC:X9",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.9 ",
      "dependencies": [
        "PC:C4"
      ],
      "scope": "Original Exercise 6.9 with complete unchanged solution"
    },
    {
      "id": "PC:X10",
      "source": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
      "source_sha256": "dbb0bd624276617c125f9da9adaa7ca076025ecdaa8e8fef7ab0e86c6e84481c",
      "proof_locator": "**Exercise 6.10 ",
      "dependencies": [
        "PC:C2",
        "PC:C4"
      ],
      "scope": "Original Exercise 6.10 with complete unchanged solution"
    }
  ],
  "earlier_maps": [
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    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json",
    "../20261005-restored-simultaneous-reflections/proof-map.json",
    "../20261005-restored-folded-forms/proof-map.json",
    "../20261005-restored-symplectic-reflections/proof-map.json",
    "../20261005-restored-canonical-folds/proof-map.json",
    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json",
    "../20261005-restored-invariant-folds/proof-map.json"
  ],
  "transitive_U001_ids": [
    "A0",
    "CLOSED-BALL-CONTRACTION",
    "D0",
    "E0",
    "EXPONENTIAL-TRIGONOMETRIC",
    "F0-ALG",
    "F0-CALC",
    "F0-COMP",
    "F0-DIFF",
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    "I0",
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    "L10.1.12",
    "L10.1.13",
    "L10.1.14",
    "L10.1.15",
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    "L10.1.5",
    "L10.1.6",
    "L10.2.2",
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    "L10.4.2",
    "L10.4.3",
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    "L4.2.3",
    "L4.2.4",
    "L4.2.6",
    "L5.1.10",
    "L5.1.11",
    "L5.1.13",
    "L5.1.2",
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    "L5.3.3",
    "L5.3.5",
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    "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",
    "G2:F0",
    "G2:F1",
    "G2:F2",
    "G2:F3",
    "G2:F4",
    "G2:G0",
    "G2:G1",
    "G2:G21",
    "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",
    "SN:F0",
    "SN:F1"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}