{
  "schema": "AN04-restored-phase-equivalence-proof-map/v1",
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    {
      "id": "PE:Q0",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "## 1. A fiber Hessian",
      "dependencies": [
        "G2:G51",
        "G2:G61",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact inputs, marked nondegenerate phases and fixed embedded conic Lagrangian germ"
    },
    {
      "id": "PE:Q1",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Lemma 1.1",
      "dependencies": [
        "PE:Q0",
        "U001:P5",
        "GEO:C0"
      ],
      "scope": "Vertical nullity, congruence, inertia and radial radical"
    },
    {
      "id": "PE:Q2",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Lemma 2.1",
      "dependencies": [
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        "U001:P2",
        "U001:P3",
        "U001:P8-smooth",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full parameter Morse coordinates, smooth signed LDL factorization and actual inverse"
    },
    {
      "id": "PE:Q3",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Proposition 3.1",
      "dependencies": [
        "PE:Q1",
        "PE:Q2"
      ],
      "scope": "Homogeneous quadratic reduction, critical rank and minimal-phase Hessian"
    },
    {
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      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Lemma 4.1",
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        "G2:F4",
        "U001:P2",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Varying-rank minimal-phase rigidity, full homotopy, exact vertical field and smooth homogeneous inverse"
    },
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      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "## 5. Equal signatures",
      "dependencies": [
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        "PE:Q3",
        "PE:Q4",
        "U001:P8-smooth"
      ],
      "scope": "Common radius, equal-signature equivalence and explicit stable block map with all original-coordinate inverses"
    },
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      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Proposition 6.1",
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        "U001:F0-CALC",
        "U001:F0-COMP"
      ],
      "scope": "Ordinary symbol estimates, exact amplitude Jacobian, pulled-cutoff tail and normalized distribution identity"
    },
    {
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      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "If a stabilization adds",
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        "PE:Q6",
        "OSC:O5"
      ],
      "scope": "Stabilized amplitude order and leading factor, base-dependent radius and full smooth remainder"
    },
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      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Example 7.1",
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        "PE:Q5",
        "PE:Q6"
      ],
      "scope": "Nonlinear radius change, full inverse and determinant, exact radial flow and specified figure slices"
    },
    {
      "id": "PE:M72",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Example 7.2",
      "dependencies": [
        "PE:Q4"
      ],
      "scope": "Changing-rank minimal graph and exact cancellation field"
    },
    {
      "id": "PE:E1",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.1 ",
      "dependencies": [
        "PE:Q1"
      ],
      "scope": "Original Exercise 8.1 and complete solution"
    },
    {
      "id": "PE:E2",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.2 ",
      "dependencies": [
        "PE:Q2"
      ],
      "scope": "Original Exercise 8.2 and complete solution"
    },
    {
      "id": "PE:E3",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.3 ",
      "dependencies": [
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      "scope": "Original Exercise 8.3 and complete solution"
    },
    {
      "id": "PE:E4",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.4 ",
      "dependencies": [
        "PE:Q4"
      ],
      "scope": "Original Exercise 8.4 and complete solution"
    },
    {
      "id": "PE:E5",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.5 ",
      "dependencies": [
        "PE:M72"
      ],
      "scope": "Original Exercise 8.5 and complete solution"
    },
    {
      "id": "PE:E6",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.6 ",
      "dependencies": [
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      ],
      "scope": "Original Exercise 8.6 and complete solution"
    },
    {
      "id": "PE:E7",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.7 ",
      "dependencies": [
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        "PE:Q6"
      ],
      "scope": "Original Exercise 8.7 and complete solution"
    },
    {
      "id": "PE:E8",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.8 ",
      "dependencies": [
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      ],
      "scope": "Original Exercise 8.8 and complete solution"
    },
    {
      "id": "PE:E9",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.9 ",
      "dependencies": [
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      ],
      "scope": "Original Exercise 8.9 and complete solution"
    },
    {
      "id": "PE:E10",
      "source": "homogeneous-phase-equivalence-and-stabilization.md",
      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.10 ",
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        "PE:M71"
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      "scope": "Original Exercise 8.10 and complete solution"
    },
    {
      "id": "PE:E11",
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      "source_sha256": "3d7da59081c1966508bb88f1466cd48a7e425b4941e4899a816dfede8feab83f",
      "proof_locator": "**Exercise 8.11 ",
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      "proof_locator": "**Exercise 8.12 ",
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