{
  "schema": "AN04-restored-linear-reduction-proof-map/v1",
  "proofs": [
    {
      "id": "LR:R0",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Proposition 1.1.",
      "dependencies": [
        "G2:G21",
        "G2:G51"
      ],
      "scope": "Arbitrary real quadratic phase: critical rank, pure radical, Lagrangian image and nondegenerate quotient, including empty blocks"
    },
    {
      "id": "LR:R1",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Lemma 1.2.",
      "dependencies": [
        "LR:R0",
        "GS:Z7",
        "TZ:Z6",
        "U001:Q6",
        "U001:U001-A4",
        "U001:U001-A5",
        "U001:U001-A6"
      ],
      "scope": "Compact radical testing, independence of projection and complement, exact density-valued Gaussian and fixed-count normalization"
    },
    {
      "id": "LR:R2",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "## 2. The density map",
      "dependencies": [
        "G2:G23",
        "U001:U001-A5"
      ],
      "scope": "Canonical positive reduction map from two exact sequences and the perfect symplectic quotient pairing"
    },
    {
      "id": "LR:R3",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "Such coordinates exist with",
      "dependencies": [
        "G2:G21",
        "G2:G23"
      ],
      "scope": "Adapted coordinates preserving the distinguished vertical plane, with explicit completion fixing all prescribed horizontal vectors"
    },
    {
      "id": "LR:R4",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Lemma 3.1.",
      "dependencies": [
        "LR:R0",
        "LR:R1",
        "LR:R2",
        "LR:R3",
        "GS:Z7",
        "GS:Z8",
        "MC:G1"
      ],
      "scope": "Restriction/integration quadratic operation, critical image and canonical identification of its radical with the geometric intersection"
    },
    {
      "id": "LR:R5",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Proof of independence.**",
      "dependencies": [
        "LR:R4",
        "GS:Z3",
        "GS:Z4",
        "GS:Z9",
        "MC:G1",
        "MC:G4",
        "U001:Q6"
      ],
      "scope": "Phase changes, square elimination, unequal variable counts, adapted shears and block-triangular base changes"
    },
    {
      "id": "LR:R6",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "For smoothness, work near",
      "dependencies": [
        "LR:R5",
        "G2:G22",
        "GS:Z7",
        "TG:Z2",
        "ML:M0a",
        "U001:P8-smooth"
      ],
      "scope": "Smooth constant-excess families through phase-Hessian rank changes: fixed-size input phases, smooth kernel quotients and transverse full-Hessian frames"
    },
    {
      "id": "LR:R7",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "## 5. Compute the factor",
      "dependencies": [
        "LR:R2",
        "LR:R4",
        "LR:R5",
        "G2:G22",
        "GS:Z7"
      ],
      "scope": "Contained-constraint normalization, critical density and all three dimension terms"
    },
    {
      "id": "LR:R8",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "## 6. Compute the factor",
      "dependencies": [
        "LR:R2",
        "LR:R3",
        "LR:R4",
        "LR:R5",
        "G2:G22",
        "G2:G23",
        "GS:Z7",
        "U001:Q6"
      ],
      "scope": "Zero-intersection adapted common complement, invertible graph block, exact Schur complement, density determinant and Fresnel phase"
    },
    {
      "id": "LR:R9",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "The agreement with direct reduction",
      "dependencies": [
        "LR:R1",
        "LR:R2",
        "LR:R3",
        "LR:R5",
        "LR:R7",
        "LR:R8"
      ],
      "scope": "Staged reduction for an arbitrary isotropic flag: compatible coordinates, nested radicals, equal quotient phases, telescoping prefactors and density identifications"
    },
    {
      "id": "LR:R10",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Theorem 7.1",
      "dependencies": [
        "LR:R6",
        "LR:R7",
        "LR:R8",
        "LR:R9",
        "G2:G71",
        "CC:C1",
        "CC:C4",
        "MC:G4"
      ],
      "scope": "Canonical linear symbol composition, reflected middle form, symplectic density contraction and exact excess factor"
    },
    {
      "id": "LR:R11",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "## 8. A graph factor",
      "dependencies": [
        "LR:R10",
        "MC:G5",
        "U001:Q4",
        "U001:Q5"
      ],
      "scope": "Graph density transport, symplectic volume unit, identity Gaussian and Maslov unit"
    },
    {
      "id": "LR:E1",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "LR:R0",
        "LR:R1",
        "U001:Q6"
      ],
      "scope": "Original Exercise 9.1 and complete unchanged solution"
    },
    {
      "id": "LR:E2",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "LR:R7",
        "LR:R8"
      ],
      "scope": "Original Exercise 9.2 and complete unchanged solution"
    },
    {
      "id": "LR:E3",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "LR:R8"
      ],
      "scope": "Original Exercise 9.3 and complete unchanged solution"
    },
    {
      "id": "LR:E4",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "LR:R9"
      ],
      "scope": "Original Exercise 9.4 and complete unchanged solution"
    },
    {
      "id": "LR:E5",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "LR:R10",
        "LR:R1"
      ],
      "scope": "Original Exercise 9.5 and complete unchanged solution"
    },
    {
      "id": "LR:E6",
      "source": "reduction-and-linear-symbol-composition.md",
      "source_sha256": "fe34dd2470ddb4f03842ac56a86292adb12d40ffb2a94131d9628bde80b282b1",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "LR:R11"
      ],
      "scope": "Original Exercise 9.6 and complete unchanged solution"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json"
  ],
  "transitive_U001_ids": [
    "A0",
    "CLOSED-BALL-CONTRACTION",
    "D0",
    "E0",
    "EXPONENTIAL-TRIGONOMETRIC",
    "F0-ALG",
    "F0-CALC",
    "F0-COMP",
    "F0-DIFF",
    "FTC-TAYLOR-COMPACT-PARAMETERS",
    "I0",
    "J0",
    "L10.1.12",
    "L10.1.13",
    "L10.1.14",
    "L10.1.15",
    "L10.1.2",
    "L10.1.5",
    "L10.1.6",
    "L10.2.2",
    "L10.3.10",
    "L10.3.2",
    "L10.3.7",
    "L10.4.1",
    "L10.4.2",
    "L10.4.3",
    "L10.5.1",
    "L10.5.5",
    "L10.5.9",
    "L10.7.2",
    "L11.4.2",
    "L2.1.10-inc",
    "L2.1.17",
    "L2.2.1",
    "L2.2.11",
    "L2.2.3",
    "L2.2.5",
    "L2.3.2",
    "L2.3.4-upper",
    "L2.3.5",
    "L2.3.8",
    "L2.4.5",
    "L2.6.5",
    "L3.1.7",
    "L3.3.7",
    "L3.3.8",
    "L4.2.2",
    "L4.2.3",
    "L4.2.4",
    "L4.2.6",
    "L5.1.10",
    "L5.1.11",
    "L5.1.13",
    "L5.1.2",
    "L5.1.7",
    "L5.1.8",
    "L5.2.1",
    "L5.2.2",
    "L5.2.4-positive",
    "L5.2.6",
    "L5.2.7",
    "L5.3.1",
    "L5.3.3",
    "L5.3.5",
    "L5.4.1",
    "L5.4.2",
    "L5.5.2-gt1",
    "L5.5.3",
    "L5.5.4",
    "L5.5.5",
    "L7.1.4",
    "L7.1.5",
    "L7.2.19",
    "L7.2.22",
    "L7.2.6",
    "L7.2.9-open",
    "L7.3.11",
    "L7.3.12",
    "L7.3.9",
    "L7.4.10",
    "L7.4.11",
    "L7.4.2",
    "L7.4.4",
    "L7.4.9",
    "L7.5.11",
    "L7.5.12",
    "L7.5.2",
    "L7.5.5",
    "L7.5.6",
    "L7.6.2",
    "L8.1.13",
    "L8.1.14",
    "L8.1.16",
    "L8.1.17",
    "L8.1.18",
    "L8.2.4",
    "L8.2.5",
    "L8.2.6",
    "L8.2.7",
    "L8.2.8",
    "L8.2.9",
    "L8.3.2",
    "L8.3.3",
    "L8.3.5",
    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
    "L8.4.6",
    "L8.5.1",
    "L8.5.6",
    "L8.6.2",
    "L9.1.1",
    "M1",
    "M2",
    "P1",
    "P10.1",
    "P10.2",
    "P10.2-typo",
    "P10.3",
    "P10.4",
    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
    "P12.2-upper",
    "P12.3",
    "P12.4",
    "P12.5-endpoints",
    "P12.5-extension",
    "P12.6",
    "P12.7",
    "P13.1",
    "P13.2-mertens",
    "P13.2-series",
    "P14.2-inverse",
    "P14.2-powers",
    "P14.3",
    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P17.1",
    "P17.2",
    "P17.3",
    "P17.4",
    "P17.4-grid",
    "P17.5-order",
    "P17.5-upper",
    "P18.1",
    "P18.2",
    "P18.3",
    "P18.4",
    "P18.5",
    "P19.1",
    "P19.2",
    "P19.3",
    "P19.4",
    "P2",
    "P20.1",
    "P20.2",
    "P20.3",
    "P20.4",
    "P20.5",
    "P21.1",
    "P21.2",
    "P21.3",
    "P21.4",
    "P21.5",
    "P3",
    "P4",
    "P5",
    "P6.0",
    "P6.1",
    "P6.2",
    "P6.3-inf",
    "P6.3-subtract",
    "P6.3-tail",
    "P6.4-ball",
    "P6.4-complete",
    "P6.5",
    "P6.6",
    "P7",
    "P8-root",
    "P8-smooth",
    "P9.1",
    "P9.2",
    "P9.2-operator",
    "P9.3",
    "P9.4",
    "P9.5",
    "Q1",
    "Q2",
    "Q3",
    "Q4",
    "Q5",
    "Q6",
    "Q9",
    "R0",
    "U001-2.1",
    "U001-3.1",
    "U001-3.2",
    "U001-4.1",
    "U001-5.1",
    "U001-6.1",
    "U001-7.2",
    "U001-A1",
    "U001-A2",
    "U001-A3",
    "U001-A4",
    "U001-A5",
    "U001-A6",
    "U001-DEF"
  ],
  "transitive_earlier_package_ids": [
    "CC:C0",
    "CC:C1",
    "CC:C4",
    "CH:T0",
    "CH:T1",
    "CH:T2",
    "CH:T3",
    "CH:W1",
    "CH:W2",
    "CH:W3",
    "CH:W4",
    "CH:W5",
    "G2:F0",
    "G2:F1",
    "G2:F2",
    "G2:F3",
    "G2:F4",
    "G2:G0",
    "G2:G1",
    "G2:G21",
    "G2:G22",
    "G2:G23",
    "G2:G31",
    "G2:G41",
    "G2:G51",
    "G2:G71",
    "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",
    "GEO:C5",
    "GR:G12-forward",
    "GR:G12-reverse",
    "GR:G5",
    "GR:G8",
    "GS:Z0",
    "GS:Z2",
    "GS:Z3",
    "GS:Z4",
    "GS:Z5",
    "GS:Z6",
    "GS:Z7",
    "GS:Z8",
    "GS:Z9",
    "IR:R0",
    "IR:R1",
    "IR:R2",
    "IR:R3",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "K:K7",
    "MC:G0",
    "MC:G1",
    "MC:G2",
    "MC:G3",
    "MC:G4",
    "MC:G5",
    "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",
    "PS:PS0",
    "PS:PS1",
    "TG:T0",
    "TG:T6",
    "TG:Z2",
    "TZ:Z0",
    "TZ:Z6"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}