{
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    {
      "id": "FD:H0",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "## 1.",
      "dependencies": [
        "U001:F0-DIFF",
        "U001:P9.1"
      ],
      "scope": "Intrinsic kernel-to-cokernel Hessian and fold dimension criterion"
    },
    {
      "id": "FD:H1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Theorem 1.2",
      "dependencies": [
        "FD:H0",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Actual local fold coordinates by critical graph and exact Taylor factor, including dimension one"
    },
    {
      "id": "FD:H2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Corollary 1.3",
      "dependencies": [
        "FD:H1",
        "U001:P2"
      ],
      "scope": "Coordinate-independent determinant derivative test and transverse kernel"
    },
    {
      "id": "FD:R1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Proposition 2.1",
      "dependencies": [
        "FD:H1",
        "U001:F0-DIFF"
      ],
      "scope": "Unique nonidentity sheet exchange, fixed hypersurface and reflection line"
    },
    {
      "id": "FD:R2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Theorem 2.2",
      "dependencies": [
        "U001:P9.1",
        "U001:P3"
      ],
      "scope": "Exact parity coordinates for an involution with hypersurface fixed set"
    },
    {
      "id": "FD:D1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Odd division with smooth parity and finite mixed-derivative control"
    },
    {
      "id": "FD:D2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Lemma 3.2",
      "dependencies": [
        "FD:D1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Even square descent, actual boundary derivatives and precise Taylor coefficients"
    },
    {
      "id": "FD:E0",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "Put \\(b_k=-2^k\\).",
      "dependencies": [
        "U001:F0-CALC"
      ],
      "scope": "Complete finite polynomial interpolation and moment identity from explicit factorization"
    },
    {
      "id": "FD:E1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "FD:E0",
        "U001:L5.4.1",
        "U001:P13.2-series",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Infinite rapid coefficients, positive product bound and explicit finite-head/uniform-tail passage to all moments"
    },
    {
      "id": "FD:E2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Theorem 4.2",
      "dependencies": [
        "FD:E1",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Joint smooth half-line extension, all matching jets, finite seminorms and parameter support preservation"
    },
    {
      "id": "FD:D3",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Theorem 5.1",
      "dependencies": [
        "FD:H1",
        "FD:R1",
        "FD:D1",
        "FD:D2",
        "FD:E2"
      ],
      "scope": "Necessary and sufficient invariant descent and full-neighborhood even/odd decomposition"
    },
    {
      "id": "FD:P0",
      "source": "homogeneous-fold-phase-and-order.md",
      "source_sha256": "597eff27e2624ce000e1d3d4656923aa246e1c43c8b41bd47a06dd787b243d20",
      "proof_locator": "## P0.",
      "dependencies": [
        "G2:G61",
        "U001:F0-DIFF"
      ],
      "scope": "Explicit homogeneous fold phase, independence at the fold, exact critical relation and immersion"
    },
    {
      "id": "FD:P1",
      "source": "homogeneous-fold-phase-and-order.md",
      "source_sha256": "597eff27e2624ce000e1d3d4656923aa246e1c43c8b41bd47a06dd787b243d20",
      "proof_locator": "## P1.",
      "dependencies": [
        "U001:P14.2-powers",
        "U001:F0-DIFF"
      ],
      "scope": "Both directions of radial Jacobian symbol conversion with every mixed derivative"
    },
    {
      "id": "FD:P2",
      "source": "homogeneous-fold-phase-and-order.md",
      "source_sha256": "597eff27e2624ce000e1d3d4656923aa246e1c43c8b41bd47a06dd787b243d20",
      "proof_locator": "## P2.",
      "dependencies": [
        "FD:P0",
        "FD:P1",
        "OSC:O1",
        "OSC:O3",
        "U001:Q3",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Actual distributional change of variables, compact-parameter cutoff limits, Fourier normalization and order conversion"
    },
    {
      "id": "FD:S1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Proposition 6.1",
      "dependencies": [
        "FD:D3",
        "U001:P14.2-powers"
      ],
      "scope": "Uniform ordinary square coefficients, fixed-kernel frequency differentiation and homogeneous-ratio chain rule"
    },
    {
      "id": "FD:S2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Theorem 6.2",
      "dependencies": [
        "FD:S1",
        "FD:P2",
        "CS:N2"
      ],
      "scope": "Exact critical amplitude realization, both fractional orders, sign and converse; earlier necessary corank-two threshold context"
    },
    {
      "id": "FD:X1",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "FD:H0",
        "FD:H1"
      ],
      "scope": "Original Exercise 7.1 and complete unchanged solution"
    },
    {
      "id": "FD:X2",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "FD:H1",
        "FD:H2",
        "FD:R1"
      ],
      "scope": "Original Exercise 7.2 and complete unchanged solution"
    },
    {
      "id": "FD:X3",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "FD:D3",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 7.3 and complete unchanged solution"
    },
    {
      "id": "FD:X4",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "FD:D2",
        "FD:E2",
        "U001:P14.3"
      ],
      "scope": "Original Exercise 7.4 and complete unchanged solution"
    },
    {
      "id": "FD:X5",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "FD:D1",
        "FD:D2",
        "U001:P16.1"
      ],
      "scope": "Original Exercise 7.5 and complete unchanged solution"
    },
    {
      "id": "FD:X6",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "FD:E0",
        "FD:E1"
      ],
      "scope": "Original Exercise 7.6 and complete unchanged solution"
    },
    {
      "id": "FD:X7",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "FD:S1",
        "U001:P14.2-powers"
      ],
      "scope": "Original Exercise 7.7 and complete unchanged solution"
    },
    {
      "id": "FD:X8",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "FD:S2"
      ],
      "scope": "Original Exercise 7.8 and complete unchanged solution"
    },
    {
      "id": "FD:X9",
      "source": "folds-reflections-and-uniform-descent.md",
      "source_sha256": "102b70d731244d43df12251d8424201bf48ca381d7f3189e57f8d0fd776b9a47",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "FD:R2",
        "U001:P9.1"
      ],
      "scope": "Original Exercise 7.9 and complete unchanged solution"
    }
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    "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",
    "SN:A1",
    "SN:F0",
    "SN:F1",
    "SN:H1",
    "SN:H2",
    "SN:L1",
    "SN:N1",
    "SN:R1",
    "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
}