{
  "schema": "AN04-restored-symplectic-reflections-proof-map/v1",
  "proofs": [
    {
      "id": "SF:R1",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "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": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.1 ",
      "dependencies": [
        "SF:R1",
        "SF:F1"
      ],
      "scope": "Original Exercise 8.1 and complete unchanged solution"
    },
    {
      "id": "SF:X2",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.2 ",
      "dependencies": [
        "SF:H0"
      ],
      "scope": "Original Exercise 8.2 and complete unchanged solution"
    },
    {
      "id": "SF:X3",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.3 ",
      "dependencies": [
        "SF:F2"
      ],
      "scope": "Original Exercise 8.3 and complete unchanged solution"
    },
    {
      "id": "SF:X4",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.4 ",
      "dependencies": [
        "SF:O5"
      ],
      "scope": "Original Exercise 8.4 and complete unchanged solution"
    },
    {
      "id": "SF:X5",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.5 ",
      "dependencies": [
        "SF:O5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Original Exercise 8.5 and complete unchanged solution"
    },
    {
      "id": "SF:X6",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.6 ",
      "dependencies": [
        "SF:H4",
        "SF:H5"
      ],
      "scope": "Original Exercise 8.6 and complete unchanged solution"
    },
    {
      "id": "SF:X7",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.7 ",
      "dependencies": [
        "SF:C1",
        "G2:F0"
      ],
      "scope": "Original Exercise 8.7 and complete unchanged solution"
    },
    {
      "id": "SF:X8",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.8 ",
      "dependencies": [
        "SF:C3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Original Exercise 8.8 and complete unchanged solution"
    },
    {
      "id": "SF:X9",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.9 ",
      "dependencies": [
        "SF:M1"
      ],
      "scope": "Original Exercise 8.9 and complete unchanged solution"
    },
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      "id": "SF:X10",
      "source": "two-reflections-in-folded-symplectic-coordinates.md",
      "source_sha256": "cc32cbb21dafd414055230c527a31383bafebb0076a3ec97153b37df98ca202f",
      "proof_locator": "**Exercise 8.10 ",
      "dependencies": [
        "SF:M1"
      ],
      "scope": "Original Exercise 8.10 and complete unchanged solution"
    }
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    "P9.1",
    "P9.2",
    "P9.2-operator",
    "P9.3",
    "P9.4",
    "P9.5",
    "Q1",
    "Q2",
    "Q3",
    "Q4",
    "Q5",
    "R0",
    "U001-A4",
    "U001-A5",
    "U001-DEF"
  ],
  "transitive_earlier_package_ids": [
    "CH:T0",
    "CH:T1",
    "CH:T2",
    "CH:T3",
    "CH:W1",
    "CH:W4",
    "CH:W5",
    "CR:H1",
    "CR:H2",
    "CR:H3",
    "FD:D1",
    "FD:R2",
    "FF:H1",
    "FF:H2",
    "FF:H3",
    "FF:L0",
    "FF:M1",
    "FF:M2",
    "FF:O1",
    "FF:O2",
    "FF:O3",
    "FF:O4",
    "FF:O5",
    "G2:F0",
    "G2:F1",
    "G2:F2",
    "G2:F3",
    "G2:F4",
    "G2:G0",
    "G2:G1",
    "G2:G21",
    "G2:G31",
    "G2:G41",
    "G2:NF0",
    "G2:NF1",
    "G2:NF2",
    "G2:NF3",
    "G2:NF4",
    "G2:NF5",
    "G2:NF6",
    "G2:NF7",
    "GEO:C0",
    "GEO:C1",
    "GEO:C2",
    "GEO:C3",
    "GEO:C4",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "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",
    "SI:C1",
    "SI:C2",
    "SI:C3",
    "SI:F1",
    "SI:H1",
    "SI:I0",
    "SI:I1",
    "SI:I2",
    "SI:I3",
    "SI:J1",
    "SI:J2",
    "SI:J3",
    "SI:J4",
    "SI:M0",
    "SI:S1",
    "SI:W1",
    "SN:F0",
    "SN:F1",
    "SN:F2"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}