{
  "schema": "AN04-restored-real-principal-proof-map/v1",
  "proofs": [
    {
      "id": "RP:W0",
      "source": "tensor-wavefront-and-projection.md",
      "source_sha256": "fc8e1d7b207b5d267eb999c85e9497cfb5d85dbfd5de882a7298a691798f6bb0",
      "proof_locator": "For compact localizations",
      "dependencies": [
        "RC:K6",
        "P3:L1",
        "P3:L3",
        "CH:T0"
      ],
      "scope": "Retained AN-03 WF8 full tensor Fourier identity in exact coordinate order"
    },
    {
      "id": "RP:W1",
      "source": "tensor-wavefront-and-projection.md",
      "source_sha256": "fc8e1d7b207b5d267eb999c85e9497cfb5d85dbfd5de882a7298a691798f6bb0",
      "proof_locator": "Write \\(Z_u=",
      "dependencies": [
        "RP:W0",
        "CH:W1",
        "RC:K6",
        "U001:U001-A4"
      ],
      "scope": "Retained WF9 tensor inclusion, full support and cone comparisons, every smaller cutoff"
    },
    {
      "id": "RP:W2",
      "source": "tensor-wavefront-and-projection.md",
      "source_sha256": "fc8e1d7b207b5d267eb999c85e9497cfb5d85dbfd5de882a7298a691798f6bb0",
      "proof_locator": "For the original projection",
      "dependencies": [
        "RP:W1",
        "CH:W1",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P3:M4"
      ],
      "scope": "Retained WF10 exact constant-factor equality with the nonzero integrated-test converse"
    },
    {
      "id": "RP:K0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "For the nonradial model",
      "dependencies": [
        "CH:T0",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Signed forward/backward kernels, exact Heaviside derivative and output fundamental identities"
    },
    {
      "id": "RP:K1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Integration by parts in the input",
      "dependencies": [
        "RP:K0",
        "RC:K6",
        "RC:K7"
      ],
      "scope": "Both input fundamental identities and compact-distribution extension from exact transpose tests"
    },
    {
      "id": "RP:K2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "The formulas have the indicated",
      "dependencies": [
        "RP:K1",
        "CH:T0",
        "RC:K7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Precise time supports, every transposed derivative and local strong distribution continuity"
    },
    {
      "id": "RP:K3",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "For \\(f\\) supported",
      "dependencies": [
        "RP:K0",
        "P3:M4",
        "P3:M6"
      ],
      "scope": "Whole compact-input to local-output L2 bound with exact interval factors and no unproved trace"
    },
    {
      "id": "RP:F0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "## 2.",
      "dependencies": [
        "G2:G0"
      ],
      "scope": "Exact twisted diagonal and directed characteristic-flow relations"
    },
    {
      "id": "RP:F1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "To prove the upper inclusions",
      "dependencies": [
        "RP:F0",
        "RP:W1",
        "RP:W2",
        "CH:W3",
        "P3:L3"
      ],
      "scope": "Full tensor/linear-coordinate upper inclusion and actual output/input covector signs"
    },
    {
      "id": "RP:F2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Both parts occur.",
      "dependencies": [
        "RP:W2",
        "CH:W1",
        "P3:L1",
        "P3:L3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every nonzero conormal and diagonal direction; two integrations give explicit remainder and arbitrary nonproduct cutoffs"
    },
    {
      "id": "RP:F3",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "The difference has a simpler relation",
      "dependencies": [
        "RP:K0",
        "RP:F1",
        "RP:F2",
        "OSC:O1",
        "HT:H0",
        "PS:PS5"
      ],
      "scope": "Actual difference kernel, all two-pi factors, minus-one-half order and complete off-diagonal localization"
    },
    {
      "id": "RP:C0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "## 3.",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Signed smooth primitive on an arbitrary open time interval with every derivative"
    },
    {
      "id": "RP:C1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "For \\(h=v-G\\)",
      "dependencies": [
        "RP:C0",
        "CH:T0",
        "RC:K6",
        "U001:U001-A4",
        "P3:M4"
      ],
      "scope": "Exact zero-integral compact test primitive, transverse distribution and time constancy without traces"
    },
    {
      "id": "RP:C2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "The equality follows from the retained",
      "dependencies": [
        "RP:C1",
        "RP:W2",
        "CH:W1"
      ],
      "scope": "Full product wavefront equality, actual variable permutation and smooth additive error"
    },
    {
      "id": "RP:T0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "## 4.",
      "dependencies": [
        "CH:W1",
        "U001:F0-COMP"
      ],
      "scope": "Single compact-tube forcing-regular cone with uniform base, angular and time margins"
    },
    {
      "id": "RP:T1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Take a smooth order-zero frequency symbol",
      "dependencies": [
        "RP:T0",
        "U001:U001-A4",
        "P2:OP4",
        "P2:OP5"
      ],
      "scope": "Actual proper difference cutoff, frequency/cone cutoffs and exact commuting time operator"
    },
    {
      "id": "RP:T2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Its full symbol is smoothing",
      "dependencies": [
        "RP:T1",
        "P2:OP4",
        "CH:W4",
        "HT:H8"
      ],
      "scope": "Every amplitude coefficient and full differentiated remainder, exact microsupport and identity on the tube"
    },
    {
      "id": "RP:T3",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Choose an open interval",
      "dependencies": [
        "RP:T0",
        "RP:T1",
        "RP:T2",
        "CH:W4",
        "P2:OP4",
        "RC:K7"
      ],
      "scope": "Compact forcing, separated temporal derivative error and actual smooth derivative on the entire retained cylinder"
    },
    {
      "id": "RP:T4",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "The preceding section gives",
      "dependencies": [
        "RP:T3",
        "RP:C2",
        "CH:W4"
      ],
      "scope": "Microlocal equality at every retained covector and full model propagation for microlocally smooth forcing"
    },
    {
      "id": "RP:P0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "**Theorem 5.1",
      "dependencies": [
        "RP:T4",
        "K:K3",
        "CH:W4"
      ],
      "scope": "Full real-homogeneous-principal-symbol propagation statement including arbitrary scalar lower terms"
    },
    {
      "id": "RP:P1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Use a properly supported elliptic",
      "dependencies": [
        "IF:S0",
        "K:K3",
        "CH:W4",
        "G2:G0",
        "G2:NF7"
      ],
      "scope": "Positive global real-order reducers and both inverses, exact Hamilton reparametrization and forcing wavefront equality"
    },
    {
      "id": "RP:P2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "At a characteristic point",
      "dependencies": [
        "FN:R1",
        "PC:H5",
        "GE:D9",
        "GE:D19",
        "RC:C1",
        "CH:T2"
      ],
      "scope": "Full nonradial marked canonical chart and complete lower-term conjugation with the exact original operator identity"
    },
    {
      "id": "RP:P3",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Use nested cutoffs in the matched cones.",
      "dependencies": [
        "RP:P2",
        "GE:D9",
        "GE:D19",
        "RC:C1",
        "RC:C2",
        "CH:T2"
      ],
      "scope": "Both microlocal wavefront equalities with every graph/separated-cone defect and proper distribution action"
    },
    {
      "id": "RP:P4",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "The remaining characteristic points",
      "dependencies": [
        "FN:R0",
        "G2:G41",
        "G2:NF7",
        "CH:W1",
        "RP:P1"
      ],
      "scope": "Actual zero and radial Hamilton curves, conicity and positive reparametrization for all real orders"
    },
    {
      "id": "RP:P5",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "Consequently regularity and singularity",
      "dependencies": [
        "RP:T4",
        "RP:P0",
        "RP:P1",
        "RP:P3",
        "RP:P4",
        "U001:F0-COMP"
      ],
      "scope": "Full connected forcing-regular bicharacteristic intervals, exact symplectic transport and stopping condition"
    },
    {
      "id": "RP:X1",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "**Exercise 6.1 ",
      "dependencies": [
        "RP:K0",
        "RP:K1",
        "RP:F3"
      ],
      "scope": "Original complete solved Exercise6.1"
    },
    {
      "id": "RP:X2",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "**Exercise 6.2 ",
      "dependencies": [
        "RP:F1",
        "RP:F2",
        "RP:W2",
        "RP:T4"
      ],
      "scope": "Original complete solved Exercise6.2"
    },
    {
      "id": "RP:X3",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "**Exercise 6.3 ",
      "dependencies": [
        "RP:P4",
        "RP:F2",
        "FN:R1"
      ],
      "scope": "Original complete solved Exercise6.3"
    },
    {
      "id": "RP:I0",
      "source": "real-principal-type-kernels-and-propagation.md",
      "source_sha256": "28129d3b70fdf5b231cf11511084abb4c8f274eef49c3d9a9629c0d015ed9901",
      "proof_locator": "*The upper panel shows",
      "dependencies": [
        "RP:F0",
        "RP:F1",
        "RP:F2",
        "RP:T4",
        "RP:X2"
      ],
      "scope": "Original exact signed-flow projection and forcing boundary, with omitted covectors explicitly described"
    }
  ],
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    "../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",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json",
    "../20261005-restored-simultaneous-reflections/proof-map.json",
    "../20261005-restored-folded-forms/proof-map.json",
    "../20261005-restored-symplectic-reflections/proof-map.json",
    "../20261005-restored-canonical-folds/proof-map.json",
    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json",
    "../20261005-restored-invariant-folds/proof-map.json",
    "../20261005-restored-prescribed-coordinates/proof-map.json",
    "../20261005-restored-clean-pairs/proof-map.json",
    "../20261005-restored-function-normal-forms/proof-map.json",
    "../20261005-restored-quadratic-forms/proof-map.json",
    "../20261005-restored-positive-ideals/proof-map.json",
    "../20261005-restored-complex-fios/proof-map.json",
    "../20261005-restored-maslov-topology/proof-map.json"
  ],
  "transitive_U001_ids": [
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    "L8.1.16",
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    "L8.1.18",
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    "L8.2.5",
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    "L8.2.8",
    "L8.2.9",
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    "L8.3.6",
    "L8.3.7",
    "L8.3.9",
    "L8.4.1",
    "L8.4.2",
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    "M1",
    "M2",
    "P1",
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    "P10.2",
    "P10.2-typo",
    "P10.3",
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    "P10.5-rules",
    "P10.6",
    "P10.6-zero",
    "P11.1",
    "P11.2-bridge",
    "P11.2-orders",
    "P12.1",
    "P12.2-criterion",
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    "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",
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  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}