{
  "schema": "AN04-restored-corank-continuity-proof-map/v1",
  "proofs": [
    {
      "id": "CR:L1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Theorem 1.1",
      "dependencies": [
        "G2:G21",
        "G2:G23",
        "G2:G71",
        "U001:P9.1"
      ],
      "scope": "Full signed linear splitting, quotient graph, radical, corank and both pointwise projection ranks"
    },
    {
      "id": "CR:H1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Proof: initial homogeneous coordinates.**",
      "dependencies": [
        "G2:G41",
        "G2:NF7",
        "U001:P3",
        "U001:P15.1"
      ],
      "scope": "Actual homogeneous initial chart with arbitrary radial-compatible tangent isomorphism"
    },
    {
      "id": "CR:H2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**A primitive that vanishes to second order.**",
      "dependencies": [
        "CR:H1",
        "G2:F0",
        "G2:F1",
        "U001:P4"
      ],
      "scope": "Radial primitive, symmetric first coefficient jet, radial null vector, exact homogeneous quadratic correction and second-order vanishing"
    },
    {
      "id": "CR:H3",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Homogeneous Moser flow.**",
      "dependencies": [
        "CR:H2",
        "G2:F2",
        "G2:F4",
        "G2:NF4",
        "G2:NF7",
        "U001:P2"
      ],
      "scope": "One common time-one domain, fixed point and identity differential, commuting dilations and exact homogeneous symplectic inverse chart"
    },
    {
      "id": "CR:R1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "## 3.",
      "dependencies": [
        "CR:L1",
        "G2:G41",
        "G2:G71"
      ],
      "scope": "Equivalent individual radial exclusions, nonzero common one-form and positive-dimensional shared quotient"
    },
    {
      "id": "CR:R2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "CR:R1",
        "CR:L1",
        "CR:H3",
        "G2:G21"
      ],
      "scope": "Radial-preserving quotient sections, adapted symplectic bases and realizable separate tangent normal form"
    },
    {
      "id": "CR:P1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "## 4.",
      "dependencies": [
        "CR:R2",
        "G2:G71",
        "U001:P3",
        "IR:R7",
        "GE:D2",
        "K:K1",
        "HT:H8",
        "P3:M3",
        "P3:M4",
        "P3:L3",
        "U001:Q3"
      ],
      "scope": "Partial generating function, exact input signs, normalization and quarter-corank order; actual smooth cutoff convergence and kernel action"
    },
    {
      "id": "CR:P2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "The tangent information also records",
      "dependencies": [
        "CR:P1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "HT:H2"
      ],
      "scope": "Full vanishing two-jet and homogeneous cubic phase estimate with precise frequency scaling"
    },
    {
      "id": "CR:A1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Proof in normalized coordinates.**",
      "dependencies": [
        "CR:P1",
        "GE:D3",
        "GE:D4",
        "WM:W2",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "P3:M7"
      ],
      "scope": "Uniform compact parameter graph estimate, actual Hilbert integral and exact parameter-volume norm including zero dimensions"
    },
    {
      "id": "CR:A2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Restore the original coordinates.**",
      "dependencies": [
        "CR:A1",
        "GE:D9",
        "GE:D11",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "CH:T2",
        "PS:PS5",
        "WM:W1"
      ],
      "scope": "Undo canonical changes with proper graph inverses, finite localized assembly and rank-changing sufficient bound"
    },
    {
      "id": "CR:A3",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Corollary 5.2",
      "dependencies": [
        "CR:A2",
        "GE:D7",
        "GE:G0",
        "WM:W6",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "P1:B4"
      ],
      "scope": "Every real Sobolev shift from an upper corank bound, actual proper reconstruction and distributional extension"
    },
    {
      "id": "CR:I1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "### The measurable vector integrals",
      "dependencies": [
        "P3:M1",
        "P3:M3",
        "P3:M4",
        "P3:M5",
        "P3:M6",
        "CT:T1"
      ],
      "scope": "Complete strong vector integral from simple approximants, dominated norm limit, actual parameter measurability and integration bound"
    },
    {
      "id": "CR:M1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "## 6.",
      "dependencies": [
        "CR:P1",
        "P3:L3",
        "P3:M3",
        "P3:M4",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.2-powers"
      ],
      "scope": "Actual flat kernel, exact order and nonzero modulation limit proving sharpness for this family"
    },
    {
      "id": "CR:E1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "CR:L1"
      ],
      "scope": "Original Exercise 7.1 and its complete unchanged solution"
    },
    {
      "id": "CR:E2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "CR:P1",
        "P3:M4",
        "OSC:O6"
      ],
      "scope": "Original Exercise 7.2 and its complete unchanged solution"
    },
    {
      "id": "CR:E3",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "CR:R1",
        "CR:L1"
      ],
      "scope": "Original Exercise 7.3 and its complete unchanged solution"
    },
    {
      "id": "CR:E4",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "CR:R1",
        "CR:P1",
        "CR:P2",
        "CR:A2"
      ],
      "scope": "Original Exercise 7.4 and its complete unchanged solution"
    },
    {
      "id": "CR:E5",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "CR:H2",
        "CR:H3",
        "G2:F0"
      ],
      "scope": "Original Exercise 7.5 and its complete unchanged solution"
    },
    {
      "id": "CR:E6",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "CR:I1",
        "P3:M5"
      ],
      "scope": "Original Exercise 7.6 and its complete unchanged solution"
    },
    {
      "id": "CR:E7",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "CR:M1",
        "CR:A1"
      ],
      "scope": "Original Exercise 7.7 and its complete unchanged solution"
    },
    {
      "id": "CR:E8",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "c237b1d723e7f2fce32cfd565681c0a3d00e5e25d7784a1e140cc6f83092b07b",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "CR:A3"
      ],
      "scope": "Original Exercise 7.8 and its complete unchanged solution"
    }
  ],
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    "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",
    "WM:W3",
    "WM:W4",
    "WM:W5",
    "WM:W6"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}