{
  "schema": "AN04-restored-canonical-folds-proof-map/v1",
  "proofs": [
    {
      "id": "CF:N0",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "For the local branch statement,",
      "dependencies": [
        "U001:P3",
        "U001:P9.1"
      ],
      "scope": "Local embedded-branch reduction by an invertible coordinate minor and smooth graph inverse"
    },
    {
      "id": "CF:G1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "## 1.",
      "dependencies": [
        "CF:N0",
        "FD:H1",
        "FD:H2",
        "G2:F0"
      ],
      "scope": "Common closed pulled-back form and exact equality of the two simple critical hypersurface germs"
    },
    {
      "id": "CF:G2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "The restriction of a fold to its critical",
      "dependencies": [
        "CF:G1",
        "FF:L0",
        "U001:P9.1"
      ],
      "scope": "Explicit hyperplane radical proof and both folded-form rank conditions from an actual critical-image diffeomorphism"
    },
    {
      "id": "CF:G3",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "Each is a line transverse",
      "dependencies": [
        "CF:G2",
        "FD:R1",
        "FF:M2"
      ],
      "scope": "Independent actual projection kernels span the ambient radical; unique sheet exchanges preserve the form with the correct reflection lines"
    },
    {
      "id": "CF:O1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Theorem 2.1",
      "dependencies": [
        "CF:G3",
        "SF:O5",
        "FF:T3",
        "U001:P3"
      ],
      "scope": "Complete ordinary source normalization and both full symplectic target converses with exact half-shift and square"
    },
    {
      "id": "CF:O2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "Eliminate \\(u=2(x_1-y_1)\\)",
      "dependencies": [
        "CF:O1",
        "U001:P3"
      ],
      "scope": "Whole ordinary relation equality in both directions, all source parameters recovered and exact coefficient two"
    },
    {
      "id": "CF:O3",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "There is an ordinary generating family",
      "dependencies": [
        "CF:O2",
        "G2:F0",
        "U001:F0-DIFF"
      ],
      "scope": "Full ordinary generating family, independent auxiliary constraints, immersion and exact signed canonical primitive including n=1"
    },
    {
      "id": "CF:H0",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "## 3.",
      "dependencies": [
        "CF:G1",
        "G2:F1",
        "SI:H1",
        "U001:P3"
      ],
      "scope": "Canonical one-form from Cartan, equality under projected radial contraction, and inherited full conic chart on the relation"
    },
    {
      "id": "CF:H1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "On the fold, (3.3) says",
      "dependencies": [
        "CF:H0",
        "CF:G3",
        "SF:H1"
      ],
      "scope": "Exact equivalence of nonzero pulled-back one-form and radial independence from the two-kernel plane; n>=2"
    },
    {
      "id": "CF:H2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "The sheet exchanges are homogeneous.",
      "dependencies": [
        "CF:H0",
        "FD:R1"
      ],
      "scope": "Dilation conjugacy and uniqueness prove actual homogeneous sheet exchanges"
    },
    {
      "id": "CF:H3",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
        "CF:H1",
        "CF:H2",
        "SF:C3",
        "U001:P3"
      ],
      "scope": "Full homogeneous simultaneous source normal form with degree-zero signed ratio and positive last momentum"
    },
    {
      "id": "CF:H4",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "To apply the target converse precisely,",
      "dependencies": [
        "CF:H3",
        "FF:HT",
        "U001:P14.2-powers"
      ],
      "scope": "Actual half-degree normal coordinate, source parity and complete homogeneous output target chart"
    },
    {
      "id": "CF:H5",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "For \\(\\pi_Y\\), choose the source positions",
      "dependencies": [
        "CF:H4",
        "FF:HT",
        "G2:F0",
        "U001:P3"
      ],
      "scope": "Complete homogeneous input source chart, exact mixed-term cancellation and full target extension across the unattained side"
    },
    {
      "id": "CF:H6",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "Combining (4.5) and (4.8),",
      "dependencies": [
        "CF:H5",
        "CF:N0",
        "U001:P3"
      ],
      "scope": "Whole homogeneous model and eliminated equations, smooth product inverse and every local sheet retained"
    },
    {
      "id": "CF:P1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "## 5.",
      "dependencies": [
        "CF:H6",
        "G2:F0",
        "U001:F0-DIFF",
        "U001:P3"
      ],
      "scope": "Full smooth homogeneous phase with n+1 degree-one auxiliary variables and exact frequency derivatives at fixed tau"
    },
    {
      "id": "CF:P2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "The differentials of all",
      "dependencies": [
        "CF:P1",
        "U001:P3",
        "U001:P9.1"
      ],
      "scope": "Independent critical gradients including at the fold, nonzero full phase differential and smooth critical manifold"
    },
    {
      "id": "CF:P3",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "on it gives the embedding",
      "dependencies": [
        "CF:P2",
        "CF:H6",
        "G2:G61"
      ],
      "scope": "Exact full phase-to-relation embedding, both covector signs, nonzero covectors and zero critical value"
    },
    {
      "id": "CF:M1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "## 6.",
      "dependencies": [
        "CF:H6",
        "G2:F0",
        "FD:H2",
        "FF:L0"
      ],
      "scope": "Full model one-form equality and Lagrangian property; both exact Jacobians, transverse kernel Hessians, top wedge and radial data"
    },
    {
      "id": "CF:M2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "For a fixed output covector",
      "dependencies": [
        "CF:M1",
        "CF:H3",
        "U001:P14.2-powers"
      ],
      "scope": "Both signed sheets, coalescence and attained-side ranges; exact normal-slice diagram and actual projection exchanges"
    },
    {
      "id": "CF:E1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "## 7.",
      "dependencies": [
        "CF:M1",
        "CF:H0",
        "CF:H1",
        "FD:H2",
        "U001:P14.2-powers",
        "U001:F0-DIFF"
      ],
      "scope": "Genuine embedded homogeneous relation with two folds and nonzero target covectors but zero common pulled-back primitive; invariant obstruction"
    },
    {
      "id": "CF:X1",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.1 ",
      "dependencies": [
        "CF:G1"
      ],
      "scope": "Exercise 8.1 and solution"
    },
    {
      "id": "CF:X2",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.2 ",
      "dependencies": [
        "CF:G2",
        "CF:G3"
      ],
      "scope": "Exercise 8.2 and solution"
    },
    {
      "id": "CF:X3",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.3 ",
      "dependencies": [
        "CF:O1",
        "CF:O2",
        "CF:O3"
      ],
      "scope": "Exercise 8.3 and solution"
    },
    {
      "id": "CF:X4",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.4 ",
      "dependencies": [
        "CF:H0",
        "CF:H1"
      ],
      "scope": "Exercise 8.4 and solution"
    },
    {
      "id": "CF:X5",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.5 ",
      "dependencies": [
        "CF:H4",
        "CF:H5"
      ],
      "scope": "Exercise 8.5 and solution"
    },
    {
      "id": "CF:X6",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.6 ",
      "dependencies": [
        "CF:P1",
        "CF:P2"
      ],
      "scope": "Exercise 8.6 and solution"
    },
    {
      "id": "CF:X7",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.7 ",
      "dependencies": [
        "CF:M1"
      ],
      "scope": "Exercise 8.7 and solution"
    },
    {
      "id": "CF:X8",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.8 ",
      "dependencies": [
        "CF:M2"
      ],
      "scope": "Exercise 8.8 and solution"
    },
    {
      "id": "CF:X9",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.9 ",
      "dependencies": [
        "CF:E1"
      ],
      "scope": "Exercise 8.9 and solution"
    },
    {
      "id": "CF:X10",
      "source": "canonical-relations-with-two-folds.md",
      "source_sha256": "cc03c859de851557359273f4b909c761f99bc9290f0b00bdf001d3892ab9f91d",
      "proof_locator": "**Exercise 8.10 ",
      "dependencies": [
        "CF:P3",
        "CF:M1",
        "G2:F0"
      ],
      "scope": "Exercise 8.10 and solution"
    }
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    "mathematical_proofs_rewritten": false
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}