{
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    {
      "id": "SI:F1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "## 2.",
      "dependencies": [
        "FD:R2",
        "SN:F0",
        "G2:NF1",
        "G2:NF7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1",
        "U001:P3"
      ],
      "scope": "First normal form from explicit transverse-flow integrals, invertible coordinate differential and weighted matching"
    },
    {
      "id": "SI:W1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Weighted-remainder verification.**",
      "dependencies": [
        "FD:D1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-DIFF"
      ],
      "scope": "Preservation of unequal smooth remainder divisibility under both compositions and parity-preserving inverse conjugacy"
    },
    {
      "id": "SI:J1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
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        "SI:W1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Even weighted coefficients vanish; odd errors eliminated with exact two homological equations and inverse remainder order"
    },
    {
      "id": "SI:J2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Finite-jet justification.**",
      "dependencies": [
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        "SI:W1",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Compatible stabilization of every finite normal jet, its composition/inverse and common tangential domain"
    },
    {
      "id": "SI:J3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Lemma 3.2",
      "dependencies": [
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        "U001:P13.2-series",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full local parameter Borel realization, parity, all derivative bounds, FTC termwise differentiation and exact jets"
    },
    {
      "id": "SI:J4",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "Apply the lemma to every coordinate component",
      "dependencies": [
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        "SI:J3",
        "U001:P3"
      ],
      "scope": "Actual parity-preserving diffeomorphism and equality of all normal and mixed jets after conjugation"
    },
    {
      "id": "SI:I0",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
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        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Compact extension preserving flat jets and uniform arbitrary-power bounds in all derivatives"
    },
    {
      "id": "SI:I1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "We first control a number of iterates",
      "dependencies": [
        "SI:I0",
        "U001:F0-DIFF"
      ],
      "scope": "Exact summed nilpotent-shear recurrence and simultaneous normal/derivative bootstrap for O(1/abs(s)) iterates"
    },
    {
      "id": "SI:I2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "For initial \\(|t|\\le1\\)",
      "dependencies": [
        "SI:I1"
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      "scope": "Uniform finite escape, monotone sign, no return and locally uniform finite stopping time off the fixed hypersurface"
    },
    {
      "id": "SI:I3",
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      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "Define, for \\(s\\ne0\\)",
      "dependencies": [
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        "SI:I0",
        "U001:F0-DIFF"
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      "proof_locator": "## 5.",
      "dependencies": [
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      "scope": "Invariant-function identity and exact parity averages for both involutions"
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      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "For each spectator coordinate",
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        "SI:J4",
        "FD:D1",
        "U001:P3",
        "U001:F0-DIFF"
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      "scope": "Actual spectator/normal correction, preservation of flatness under composition and inverse, and exact joint parity"
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      "proof_locator": "The remaining defect",
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        "U001:P3"
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      "proof_locator": "**Theorem 1.1",
      "dependencies": [
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        "FD:R2",
        "U001:P9.1"
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      "scope": "Full ordinary simultaneous theorem, exact reflection eigenlines, intrinsic tangential line and opposite ordered shear products"
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      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Corollary 6.1",
      "dependencies": [
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      "scope": "Invariant transverse slice retaining both reflection lines and their common fixed hypersurface"
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    {
      "id": "SI:H1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Theorem 6.2",
      "dependencies": [
        "SI:S1",
        "SI:M0",
        "U001:P3",
        "U001:L5.4.1",
        "U001:F0-DIFF"
      ],
      "scope": "Full homogeneous theorem under the equivariant-cone definition: unique saturated slice product, exact radial coordinate and both homogeneous conjugacies"
    },
    {
      "id": "SI:X1",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "SI:M0"
      ],
      "scope": "Exercise 7.1 and solution"
    },
    {
      "id": "SI:X2",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.2 ",
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        "U001:L5.4.1",
        "U001:P15.1"
      ],
      "scope": "Exercise 7.2 and solution"
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    {
      "id": "SI:X3",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
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        "SI:M0",
        "U001:P3"
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      "scope": "Exercise 7.3 and solution"
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    {
      "id": "SI:X4",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "SI:J1"
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      "scope": "Exercise 7.4 and solution"
    },
    {
      "id": "SI:X5",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
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        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exercise 7.5 and solution"
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    {
      "id": "SI:X6",
      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
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        "U001:P14.3"
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      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
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        "U001:P14.3"
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      "source": "simultaneous-reflections-and-flat-corrections.md",
      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.8 ",
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      "source_sha256": "798a8f2f8d251dc01c812aa4eefeea84c9bdbe3b8f24b1dca83860a0c77d5c5b",
      "proof_locator": "**Exercise 7.9 ",
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