{
  "schema": "AN04-restored-corank-continuity-proof-map/v1",
  "proofs": [
    {
      "id": "CR:L1",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "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": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "CR:L1"
      ],
      "scope": "Exercise 7.1 and its solution"
    },
    {
      "id": "CR:E2",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "CR:P1",
        "P3:M4",
        "OSC:O6"
      ],
      "scope": "Exercise 7.2 and its solution"
    },
    {
      "id": "CR:E3",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "CR:R1",
        "CR:L1"
      ],
      "scope": "Exercise 7.3 and its solution"
    },
    {
      "id": "CR:E4",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "CR:R1",
        "CR:P1",
        "CR:P2",
        "CR:A2"
      ],
      "scope": "Exercise 7.4 and its solution"
    },
    {
      "id": "CR:E5",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "CR:H2",
        "CR:H3",
        "G2:F0"
      ],
      "scope": "Exercise 7.5 and its solution"
    },
    {
      "id": "CR:E6",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "CR:I1",
        "P3:M5"
      ],
      "scope": "Exercise 7.6 and its solution"
    },
    {
      "id": "CR:E7",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "CR:M1",
        "CR:A1"
      ],
      "scope": "Exercise 7.7 and its solution"
    },
    {
      "id": "CR:E8",
      "source": "corank-geometry-and-sufficient-continuity.md",
      "source_sha256": "ebe2636d71258b4da100936363923c7b991bbb5eafc511db78376be7c369a6a0",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "CR:A3"
      ],
      "scope": "Exercise 7.8 and its solution"
    }
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    "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"
  ],
  "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
  }
}