{
  "schema": "AN04-restored-symplectic-reflections-proof-map/v1",
  "proofs": [
    {
      "id": "SF:R1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 1.",
      "dependencies": [
        "FF:L0",
        "FF:M2",
        "SI:M0",
        "FD:R2"
      ],
      "scope": "Distinct reflection lines span the ambient radical; common invariants annihilate the characteristic line; exact simultaneous tangential direction"
    },
    {
      "id": "SF:H0",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 2.",
      "dependencies": [
        "FF:O5",
        "FD:D1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full smooth Hamiltonian extension for even characteristic-constant functions, tangent fold values and preserved symmetries"
    },
    {
      "id": "SF:F1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "We will repeatedly solve equations",
      "dependencies": [
        "SN:F0",
        "SN:F1",
        "G2:NF7",
        "U001:P3"
      ],
      "scope": "Actual commuting-flow chart, compatible constant equations and exact uniqueness under symmetry and dilation weights"
    },
    {
      "id": "SF:F2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "For a smooth canonical pair",
      "dependencies": [
        "SF:F1",
        "SF:H0",
        "G2:G1",
        "G2:F1",
        "G2:F2"
      ],
      "scope": "Full canonical-plane orthogonal splitting, exact signs, preservation and tangency through the degenerate hypersurface"
    },
    {
      "id": "SF:S0",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 3.",
      "dependencies": [
        "SF:R1",
        "SF:H0"
      ],
      "scope": "Common invariant extensions of characteristic-constant spectator data with exact full boundary differentials"
    },
    {
      "id": "SF:S1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "We construct \\(p_2,q_2,",
      "dependencies": [
        "SF:S0",
        "SF:F2",
        "SF:F1",
        "SF:H0"
      ],
      "scope": "Inductive invariant momenta by full flow propagation, preserved old fold values and earlier zero cross brackets"
    },
    {
      "id": "SF:S2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "On that section, solve",
      "dependencies": [
        "SF:S1",
        "SF:F1",
        "SF:F2",
        "G2:G1"
      ],
      "scope": "Conjugate invariant positions from transverse data; preservation of the new nonzero equation under earlier flows; complete spectator induction"
    },
    {
      "id": "SF:O1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
        "SF:S2",
        "SI:M0",
        "SF:F1",
        "FD:D1"
      ],
      "scope": "Odd normal coordinate with a simple zero, preserved through all spectator flows and commuting with every spectator"
    },
    {
      "id": "SF:O2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "Write \\(p_1=s\\,a\\)",
      "dependencies": [
        "SF:O1",
        "SF:H0",
        "G2:G1"
      ],
      "scope": "Full smooth characteristic field, actual first-position flow equation, parity and nonzero characteristic derivative"
    },
    {
      "id": "SF:O3",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "Off \\(\\Gamma\\) the brackets are exactly",
      "dependencies": [
        "SF:O2",
        "SF:F2",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Full ordinary folded coordinate differential, exact bracket inversion and smooth extension including dimension two"
    },
    {
      "id": "SF:O4",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "The second exchange fixes every spectator",
      "dependencies": [
        "SF:O3",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Residual exchange is a single smooth odd shift with nonzero derivative"
    },
    {
      "id": "SF:O5",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "Temporarily write the remaining position",
      "dependencies": [
        "SF:O4",
        "FD:D1",
        "U001:P14.2-powers",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact preservation and conjugacy, integrated cubic equation, smooth signed root and actual odd inverse with full nonzero Jacobian"
    },
    {
      "id": "SF:H1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 5.",
      "dependencies": [
        "SF:R1",
        "FF:H3",
        "SI:H1",
        "SF:S2",
        "SF:F1",
        "U001:P14.2-powers"
      ],
      "scope": "Nonradial homogeneous folded chart, exact common invariant seeds, half-degree odd seed and homogeneous spectator induction retaining positive last momentum"
    },
    {
      "id": "SF:H2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "a four-dimensional section;",
      "dependencies": [
        "SF:H1",
        "SF:F1",
        "SF:F2",
        "FD:D1"
      ],
      "scope": "Actual common transverse section and half-degree normal coordinate commuting with earlier pairs and last momentum"
    },
    {
      "id": "SF:H3",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "As in (4.3),",
      "dependencies": [
        "SF:H2",
        "SF:O2",
        "SF:F1"
      ],
      "scope": "Invariant degree-zero characteristic field and full first position from two transverse commuting fields"
    },
    {
      "id": "SF:H4",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "The remaining position needs",
      "dependencies": [
        "SF:H3",
        "SF:H0",
        "G2:G1",
        "SF:F1"
      ],
      "scope": "Full smooth transverse field, its exact normal value, singular-commutator cancellation, weights and parity"
    },
    {
      "id": "SF:H5",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "The fields \\(Y,Z,H_{p_n}\\) are independent",
      "dependencies": [
        "SF:H4",
        "SF:F1",
        "SF:H1"
      ],
      "scope": "Transverse conic one-dimensional section, nonzero last-momentum differential and actual last position solving all three equations"
    },
    {
      "id": "SF:H6",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "All brackets are now those of",
      "dependencies": [
        "SF:H5",
        "SF:F2",
        "U001:P2",
        "U001:P3",
        "U001:P14.2-powers"
      ],
      "scope": "Full homogeneous coordinate independence, all cross brackets, exact form and half-to-full degree normal change"
    },
    {
      "id": "SF:H7",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "The conic domain can be made explicit.",
      "dependencies": [
        "SF:H6",
        "SF:F1",
        "SI:H1"
      ],
      "scope": "Full positive-saturation coordinate diffeomorphism using the positive last momentum and unique radial slice product"
    },
    {
      "id": "SF:C1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 6.",
      "dependencies": [
        "SF:H7",
        "G2:F0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Residual homogeneous exchange depends on one ratio; exact differential constraint, oddness and first derivative"
    },
    {
      "id": "SF:C2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Theorem 6.1",
      "dependencies": [
        "SF:C1",
        "G2:F0"
      ],
      "scope": "Full homogeneous normalization map and necessity/sufficiency of every form and primitive preservation equation"
    },
    {
      "id": "SF:C3",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "If \\(R=t\\,r(t)\\),",
      "dependencies": [
        "SF:C2",
        "SF:O5",
        "FD:D1",
        "U001:P3",
        "U001:P14.2-powers"
      ],
      "scope": "Smooth even quotients, exact signed cubic inverse, nonzero full Jacobian and both shift conjugacies"
    },
    {
      "id": "SF:M1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "## 7.",
      "dependencies": [
        "SF:C3",
        "FF:L0",
        "G2:F0"
      ],
      "scope": "Exact top wedge, reflection/radial independence, primitive, ordered product, shift diagram and genuine excluded radial example"
    },
    {
      "id": "SF:X1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.1 ",
      "dependencies": [
        "SF:R1",
        "SF:F1"
      ],
      "scope": "Exercise 8.1 and solution"
    },
    {
      "id": "SF:X2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.2 ",
      "dependencies": [
        "SF:H0"
      ],
      "scope": "Exercise 8.2 and solution"
    },
    {
      "id": "SF:X3",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.3 ",
      "dependencies": [
        "SF:F2"
      ],
      "scope": "Exercise 8.3 and solution"
    },
    {
      "id": "SF:X4",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.4 ",
      "dependencies": [
        "SF:O5"
      ],
      "scope": "Exercise 8.4 and solution"
    },
    {
      "id": "SF:X5",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.5 ",
      "dependencies": [
        "SF:O5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exercise 8.5 and solution"
    },
    {
      "id": "SF:X6",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.6 ",
      "dependencies": [
        "SF:H4",
        "SF:H5"
      ],
      "scope": "Exercise 8.6 and solution"
    },
    {
      "id": "SF:X7",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.7 ",
      "dependencies": [
        "SF:C1",
        "G2:F0"
      ],
      "scope": "Exercise 8.7 and solution"
    },
    {
      "id": "SF:X8",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.8 ",
      "dependencies": [
        "SF:C3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exercise 8.8 and solution"
    },
    {
      "id": "SF:X9",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.9 ",
      "dependencies": [
        "SF:M1"
      ],
      "scope": "Exercise 8.9 and solution"
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    {
      "id": "SF:X10",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "96fa5ef6c2e6fa9d0865c4f0498cbeffdf008cf7c703c6efaf5bc378c613f870",
      "proof_locator": "**Exercise 8.10 ",
      "dependencies": [
        "SF:M1"
      ],
      "scope": "Exercise 8.10 and solution"
    }
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