{
  "schema": "AN04-restored-submanifolds-proof-map/v1",
  "proofs": [
    {
      "id": "SN:F0",
      "source": "flows-constant-rank-and-leaves.md",
      "source_sha256": "a26dc7f362c96a2ab558436d641c7101fc2c0ffe060866043770fdf02c7faa23",
      "proof_locator": "## F0.",
      "dependencies": [
        "G2:NF1",
        "G2:NF7",
        "U001:P3"
      ],
      "scope": "Tangent fields preserve submanifolds, transverse flow chart and cylindrical submanifold"
    },
    {
      "id": "SN:F1",
      "source": "flows-constant-rank-and-leaves.md",
      "source_sha256": "a26dc7f362c96a2ab558436d641c7101fc2c0ffe060866043770fdf02c7faa23",
      "proof_locator": "## F1.",
      "dependencies": [
        "G2:NF4",
        "G2:NF7",
        "G2:F1",
        "G2:F2",
        "U001:P3"
      ],
      "scope": "Commuting flows, joint coordinates, exact homogeneous pushforward weights and Hamiltonian form preservation"
    },
    {
      "id": "SN:F2",
      "source": "flows-constant-rank-and-leaves.md",
      "source_sha256": "a26dc7f362c96a2ab558436d641c7101fc2c0ffe060866043770fdf02c7faa23",
      "proof_locator": "## F2.",
      "dependencies": [
        "U001:P2",
        "U001:P9.1"
      ],
      "scope": "Explicit smooth constant-rank kernels/images, complements, quotients and symplectic bundle frames"
    },
    {
      "id": "SN:F3",
      "source": "flows-constant-rank-and-leaves.md",
      "source_sha256": "a26dc7f362c96a2ab558436d641c7101fc2c0ffe060866043770fdf02c7faa23",
      "proof_locator": "## F3.",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1"
      ],
      "scope": "Plaque-chain maximal leaves, Hausdorff/countable immersed structures, tangent curves and distribution-preserving diffeomorphisms"
    },
    {
      "id": "SN:L1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "## 1.",
      "dependencies": [
        "G2:G21",
        "G2:G23",
        "G2:G1",
        "U001:P9.1"
      ],
      "scope": "Three-block decomposition, quotient forms/ranks, defining Poisson bracket and Lagrangian-containment criteria"
    },
    {
      "id": "SN:H1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Lemma 2.1",
      "dependencies": [
        "SN:F0",
        "SN:F1",
        "SN:L1",
        "G2:F1",
        "G2:G1",
        "G2:G41",
        "U001:P3"
      ],
      "scope": "Actual homogeneous canonical pair with retained momentum, exact symplectic product, smooth local flows and conic extension"
    },
    {
      "id": "SN:H2",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "We will also use the following",
      "dependencies": [
        "SN:H1",
        "CR:H3",
        "G2:G1"
      ],
      "scope": "Canonical extension of the pair and changed zero-degree base function with exact bracket"
    },
    {
      "id": "SN:N1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "SN:L1",
        "SN:H1",
        "SN:H2",
        "SN:F0",
        "CR:H3",
        "G2:G41",
        "U001:P3"
      ],
      "scope": "Full homogeneous constant-rank submanifold normal form by two dimension-lowering reductions; nonzero canonical-one-form hypothesis"
    },
    {
      "id": "SN:N2",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Theorem 3.2",
      "dependencies": [
        "SN:L1",
        "SN:H1",
        "SN:H2",
        "SN:F0",
        "G2:G31",
        "U001:P3"
      ],
      "scope": "Ordinary normal form including isotropic and zero-dimensional final slices; no radial hypothesis"
    },
    {
      "id": "SN:C1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
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        "SN:N2",
        "SN:F2",
        "SN:F3",
        "G2:G41"
      ],
      "scope": "Actual characteristic foliation and global leafwise radial dichotomy for a conic constant-rank submanifold"
    },
    {
      "id": "SN:C2",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "There is a local symplectic leaf space.",
      "dependencies": [
        "SN:C1",
        "G2:F0",
        "G2:F1"
      ],
      "scope": "Closed restricted form descends to a nondegenerate local quotient, without a global Hausdorff quotient claim"
    },
    {
      "id": "SN:C3",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Proposition 4.2",
      "dependencies": [
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        "SN:F2",
        "SN:C1"
      ],
      "scope": "Smooth symplectic quotient bundle, two orthogonal summands and exact restriction to a leaf normal bundle"
    },
    {
      "id": "SN:M1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "## 5.",
      "dependencies": [
        "SN:H1",
        "SN:N1",
        "SN:C1",
        "SN:C2",
        "G2:G1"
      ],
      "scope": "Exact coisotropic coordinate model, one-form preservation, characteristic and reduced coordinates"
    },
    {
      "id": "SN:R1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Proposition 6.1",
      "dependencies": [
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        "CR:R1",
        "SN:N1",
        "SN:F0",
        "PH:F1",
        "G2:G71",
        "U001:P3"
      ],
      "scope": "Constant-rank conic coisotropic projection germs, characteristic cylinders, reduced canonical graph and separate homogeneous flattening"
    },
    {
      "id": "SN:A1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Theorem 6.2",
      "dependencies": [
        "SN:R1",
        "CR:A2",
        "CR:M1",
        "GE:D9",
        "GE:D11",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "CT:T2",
        "P3:M5",
        "U001:Q3"
      ],
      "scope": "Sharp universal constant-rank order with actual compact modulated counterexample, microlocal inverse transfer and positive-rank bundles"
    },
    {
      "id": "SN:E1",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "SN:L1"
      ],
      "scope": "Original Exercise 7.1 and complete unchanged solution"
    },
    {
      "id": "SN:E2",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "SN:N2",
        "SN:C1"
      ],
      "scope": "Original Exercise 7.2 and complete unchanged solution"
    },
    {
      "id": "SN:E3",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "SN:M1",
        "G2:G1"
      ],
      "scope": "Original Exercise 7.3 and complete unchanged solution"
    },
    {
      "id": "SN:E4",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "SN:N1",
        "SN:H1"
      ],
      "scope": "Original Exercise 7.4 and complete unchanged solution"
    },
    {
      "id": "SN:E5",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "SN:C1",
        "G2:G1"
      ],
      "scope": "Original Exercise 7.5 and complete unchanged solution"
    },
    {
      "id": "SN:E6",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "SN:C3"
      ],
      "scope": "Original Exercise 7.6 and complete unchanged solution"
    },
    {
      "id": "SN:E7",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "SN:R1",
        "CR:M1",
        "P3:M5",
        "P3:L3"
      ],
      "scope": "Original Exercise 7.7 and complete unchanged solution"
    },
    {
      "id": "SN:E8",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "SN:A1",
        "CT:T2",
        "RC:C1"
      ],
      "scope": "Original Exercise 7.8 and complete unchanged solution"
    },
    {
      "id": "SN:E9",
      "source": "homogeneous-submanifold-normal-forms.md",
      "source_sha256": "c0e37b7db4c82f24950bbfbd091848d966b9af4d087feda0f28f0b065a95d7f6",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "SN:C1",
        "SN:M1",
        "G2:G1"
      ],
      "scope": "Original Exercise 7.9 and complete unchanged solution"
    }
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    "LR:R6",
    "LR:R7",
    "LR:R8",
    "LR:R9",
    "MC:G0",
    "MC:G1",
    "MC:G2",
    "MC:G3",
    "MC:G4",
    "MC:G5",
    "MC:G6",
    "MC:G7",
    "ML:M0",
    "ML:M0a",
    "ML:M1",
    "ML:M2",
    "ML:M3",
    "ML:M4",
    "ML:M5",
    "ML:M6",
    "OSC:O1",
    "OSC:O3",
    "OSC:O4",
    "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: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",
    "ST:T4",
    "ST:T6",
    "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",
    "WM:W0",
    "WM:W1",
    "WM:W2"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}