{
  "schema": "AN04-restored-analytic-composition-proof-map/v1",
  "proofs": [
    {
      "id": "RC:K0",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "## E0.",
      "dependencies": [
        "U001:F0-COMP",
        "U001:P6.3-inf",
        "U001:P6.0",
        "TG:B2"
      ],
      "scope": "Explicit compact exhaustion, locally convex seminorm criterion and preservation of bounded sets"
    },
    {
      "id": "RC:K1",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.1.",
      "dependencies": [
        "RC:K0",
        "TG:B3",
        "TG:B2"
      ],
      "scope": "Retained test LF topology, exact bounded-support criterion, strong distribution dual and finite local order"
    },
    {
      "id": "RC:K2",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.2.",
      "dependencies": [
        "RC:K1",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P15.2",
        "U001:P13.2-series",
        "U001:Q1",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4"
      ],
      "scope": "Retained full Fejer/Fourier-series proof of product-test density with common compact supports and every derivative"
    },
    {
      "id": "RC:K3",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.3.",
      "dependencies": [
        "RC:K1",
        "TG:B1",
        "TG:B3"
      ],
      "scope": "Retained complete Baire argument giving a joint finite-order bound on every pair of fixed test supports"
    },
    {
      "id": "RC:K4",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.4.",
      "dependencies": [
        "RC:K2",
        "RC:K3",
        "P3:M4"
      ],
      "scope": "Retained kernel construction by absolutely convergent cutoff Fourier series, exact finite bound and product values"
    },
    {
      "id": "RC:K5",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.5.",
      "dependencies": [
        "RC:K4",
        "RC:K2",
        "RC:K1"
      ],
      "scope": "Retained kernel gluing, product-test uniqueness and correspondence with continuous test-to-strong-distribution maps"
    },
    {
      "id": "RC:K6",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "### 13.6.",
      "dependencies": [
        "RC:K5",
        "RC:K2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-COV",
        "U001:U001-A4"
      ],
      "scope": "Retained restriction, derivatives, smooth multiplication, compact support, tensor distributions and diffeomorphic pullback"
    },
    {
      "id": "RC:K7",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "## E1.",
      "dependencies": [
        "RC:K0",
        "RC:K1",
        "RC:K6",
        "CH:T2",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact strong compact-distribution dual, compact smooth approximation, common support/Fourier bounds and transpose continuity"
    },
    {
      "id": "RC:K8",
      "source": "scalar-kernels-and-strong-topology.md",
      "source_sha256": "849c5867a2f85f994aa2b520a1627977ced13425ae5809d8c96dd2e6d6b2415d",
      "proof_locator": "## E2.",
      "dependencies": [
        "RC:K0",
        "RC:K1",
        "RC:K5",
        "RC:K6",
        "RC:K7",
        "PS:PS5",
        "U001:U001-A5"
      ],
      "scope": "Manifold test/smooth/strong-dual topologies, bounded support proof and unchanged finite-bundle/half-density kernel adapter"
    },
    {
      "id": "RC:C0",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Definition 1.1.",
      "dependencies": [
        "RC:K8",
        "GS:Z10",
        "GS:Z11",
        "IR:R7",
        "G2:G0"
      ],
      "scope": "Kernel convention, reflected input, full-punctured-cotangent closure, ordinary FIO order and proper base support"
    },
    {
      "id": "RC:C1",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Theorem 2.1.",
      "dependencies": [
        "RC:C0",
        "RC:K7",
        "RC:K8",
        "CH:T1",
        "CH:T2",
        "CH:W1",
        "PH:F2",
        "AC:A1",
        "U001:Q4"
      ],
      "scope": "Smooth and strong distribution mapping, two-axis Fourier bounds, compact approximation, full wavefront relation and proper support extensions"
    },
    {
      "id": "RC:C2",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "RC:C1",
        "GS:Z14",
        "CH:T3"
      ],
      "scope": "Finite anti-dual bundle adjoint, reflected inverse relation and conjugated Maslov symbol, with support"
    },
    {
      "id": "RC:C3",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Lemma 4.1.",
      "dependencies": [
        "G2:G72",
        "CC:C2",
        "CC:C3"
      ],
      "scope": "Proper connected clean image is closed embedded Lagrangian with compact fibers"
    },
    {
      "id": "RC:C4",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "If \\(Y\\) is compact,",
      "dependencies": [
        "RC:C3",
        "CH:W1",
        "U001:F0-COMP"
      ],
      "scope": "Automatic proper cotangent matching when the middle base is compact and both full-closure axis exclusions hold"
    },
    {
      "id": "RC:C5",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "## 6. Discard",
      "dependencies": [
        "RC:C0",
        "RC:C1",
        "OSC:O1",
        "AC:A2",
        "U001:U001-A4"
      ],
      "scope": "Unequal-frequency smoothing with complete transposed field, negative orders, every output derivative and seminorm continuity"
    },
    {
      "id": "RC:C6",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "## 7. Make",
      "dependencies": [
        "RC:C5",
        "G2:G72",
        "IR:R8",
        "AC:A3",
        "AC:A4"
      ],
      "scope": "Homogeneous middle-variable change, exact Jacobian, clean critical equations, ordinary amplitude order and excess normalization"
    },
    {
      "id": "RC:C7",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "## 8. Identify",
      "dependencies": [
        "RC:C1",
        "RC:C6",
        "AC:A5",
        "AC:A9",
        "RC:K5",
        "PS:PS5"
      ],
      "scope": "Actual operator product via cutoffs converging in higher symbol order, smooth-input limits, global assembly and closed proper support relation"
    },
    {
      "id": "RC:C8",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "## 9. The principal symbol",
      "dependencies": [
        "RC:C6",
        "LR:R10",
        "IR:R8",
        "GS:Z13",
        "PS:PS6",
        "AC:A6",
        "AC:A7",
        "AC:A8",
        "MC:G7"
      ],
      "scope": "Compact fiber symbol with the single excess factor, tangent reduction, all ordinary differentiated estimates and lower-order criterion"
    },
    {
      "id": "RC:C9",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Theorem 5.1",
      "dependencies": [
        "RC:C3",
        "RC:C5",
        "RC:C6",
        "RC:C7",
        "RC:C8"
      ],
      "scope": "Full ordinary clean FIO theorem: supports, order, principal quotient and fixed-phase amplitude continuity"
    },
    {
      "id": "RC:C10",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "## 10. A compact fiber",
      "dependencies": [
        "RC:C9",
        "LR:R11",
        "OSC:O6"
      ],
      "scope": "Compact averaging/lifting model, conormal kernel orders, exact amplitude normalization and symbol constant"
    },
    {
      "id": "RC:E1",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.1 ",
      "dependencies": [
        "RC:C0",
        "RC:C2",
        "RC:C9",
        "OSC:O6"
      ],
      "scope": "Original Exercise 11.1 with its complete unchanged solution"
    },
    {
      "id": "RC:E2",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.2 ",
      "dependencies": [
        "RC:C5"
      ],
      "scope": "Original Exercise 11.2 with its complete unchanged solution"
    },
    {
      "id": "RC:E3",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.3 ",
      "dependencies": [
        "RC:C10"
      ],
      "scope": "Original Exercise 11.3 with its complete unchanged solution"
    },
    {
      "id": "RC:E4",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.4 ",
      "dependencies": [
        "RC:C2",
        "RC:C9",
        "LR:R11"
      ],
      "scope": "Original Exercise 11.4 with its complete unchanged solution"
    },
    {
      "id": "RC:E5",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.5 ",
      "dependencies": [
        "RC:C10",
        "RC:C9",
        "P2:OP2",
        "PS:PS6"
      ],
      "scope": "Original Exercise 11.5 with its complete unchanged solution"
    },
    {
      "id": "RC:E6",
      "source": "clean-composition-of-fourier-integral-operators.md",
      "source_sha256": "f6779c3eed0932cbccf58769f702a66fe3a85b8715626fc236fab95a0dd6024d",
      "proof_locator": "**Exercise 11.6 ",
      "dependencies": [
        "RC:C10",
        "RC:C3"
      ],
      "scope": "Original Exercise 11.6 with its complete unchanged solution"
    }
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    "IR:R5",
    "IR:R6",
    "IR:R7",
    "IR:R8",
    "K:K0",
    "K:K1",
    "K:K1a",
    "K:K2",
    "K:K3",
    "K:K4",
    "K:K5",
    "K:K6",
    "K:K7",
    "LR:R0",
    "LR:R1",
    "LR:R10",
    "LR:R11",
    "LR:R2",
    "LR:R3",
    "LR:R4",
    "LR:R5",
    "LR:R6",
    "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:O2",
    "OSC:O3",
    "OSC:O4",
    "OSC:O6",
    "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",
    "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"
  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}