{
  "schema": "AN04-restored-sobolev-propagation-proof-map/v1",
  "proofs": [
    {
      "id": "SP:S1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Complete membership and order-shift tests.**",
      "dependencies": [
        "CE:G1",
        "CH:W4",
        "K:K3",
        "P2:OP5",
        "CH:T2"
      ],
      "scope": "Complete both directions of fixed-order H^r test with compact proper support and exact conic inverse"
    },
    {
      "id": "SP:S2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "More generally a proper operator",
      "dependencies": [
        "SP:S1",
        "K:K3",
        "CE:G1",
        "CH:W4"
      ],
      "scope": "Full real-order microlocal mapping and elliptic converse; open conic fixed-order set"
    },
    {
      "id": "SP:S3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "For later examples,",
      "dependencies": [
        "SP:S1",
        "CH:T0",
        "CH:W1",
        "CH:W4",
        "P3:L1",
        "P3:L3",
        "CE:G1",
        "U001:P14.3"
      ],
      "scope": "Both directions of local weighted Fourier criterion, bounded frequencies and smoothing proper changes"
    },
    {
      "id": "SP:S4",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "For a proper graph operator",
      "dependencies": [
        "SP:S1",
        "GE:D7",
        "GE:D9",
        "RC:C1"
      ],
      "scope": "Both Sobolev graph directions with proper supports and actual inverse defect"
    },
    {
      "id": "SP:C1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "For the reverse implication,",
      "dependencies": [
        "SP:S1",
        "RP:W1",
        "RP:W2",
        "P3:M4"
      ],
      "scope": "Constant-factor microlocal L2 reverse without pointwise time traces"
    },
    {
      "id": "SP:C2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "Its twisted wavefront relation",
      "dependencies": [
        "RP:W0",
        "RP:W1",
        "RP:W2",
        "RP:F1",
        "RP:F2",
        "CH:W3",
        "RC:K6",
        "RC:C1"
      ],
      "scope": "Averaging-kernel wavefront with exact zero time covector and no forbidden axis"
    },
    {
      "id": "SP:C3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "The proper wavefront mapping theorem therefore",
      "dependencies": [
        "SP:C2",
        "RC:C1",
        "P3:M4",
        "P3:M5",
        "CE:H2",
        "RC:K6"
      ],
      "scope": "Averaging L2 norm and forward constant-factor implication, all cutoffs"
    },
    {
      "id": "SP:C4",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "On a bounded interval",
      "dependencies": [
        "P3:M4",
        "P3:M5",
        "CE:H2",
        "RC:K6",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual a.e. primitive, representative independence, exact i sign and L2 bound"
    },
    {
      "id": "SP:C5",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "The zero-mean-test argument",
      "dependencies": [
        "SP:C4",
        "RP:C1",
        "SP:C1",
        "SP:C3"
      ],
      "scope": "Distributional time-constant decomposition and both L2 states without initial trace"
    },
    {
      "id": "SP:T1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "The complement of its order-zero",
      "dependencies": [
        "SP:S2",
        "RP:T0",
        "RP:T1",
        "RP:T2",
        "U001:F0-COMP",
        "U001:P14.3"
      ],
      "scope": "Common compact full-microsupport forcing tube, exact commuting proper cutoff"
    },
    {
      "id": "SP:T2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "The commutator is exactly zero",
      "dependencies": [
        "SP:T1",
        "RP:T1",
        "RP:T2",
        "RP:T3",
        "P2:OP5"
      ],
      "scope": "Full commutator identity, compact input and separated temporal term smooth on entire cylinder"
    },
    {
      "id": "SP:T3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "We verify the L2 assertion",
      "dependencies": [
        "SP:T1",
        "SP:S1",
        "K:K4",
        "CH:T2",
        "P2:OP3",
        "P2:OP4",
        "P2:OP5",
        "P2:OP6",
        "U001:P14.3",
        "U001:F0-COMP"
      ],
      "scope": "Finite partition on all normalized microsupport with every differentiated smoothing error"
    },
    {
      "id": "SP:T4",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "on the chosen compact cylinder.",
      "dependencies": [
        "SP:T3",
        "CH:W4",
        "CH:T2",
        "CE:G1",
        "RC:K6"
      ],
      "scope": "Every regular summand and actual smooth compact remainder are L2"
    },
    {
      "id": "SP:T5",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "Near \\(K\\),",
      "dependencies": [
        "SP:T2",
        "SP:T4",
        "SP:C5",
        "RP:T2",
        "CH:W4"
      ],
      "scope": "Model exact L2 propagation and return to original covectors"
    },
    {
      "id": "SP:P1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Theorem 5.1 (Sobolev propagation).**",
      "dependencies": [
        "SP:P4",
        "SP:P5"
      ],
      "scope": "Full original theorem statement, fixed endpoint order and complete real homogeneous scope"
    },
    {
      "id": "SP:P2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Proof.** Choose a positive",
      "dependencies": [
        "SP:S2",
        "IF:S0",
        "RP:P1"
      ],
      "scope": "All real positive elliptic reduction, exact r=s+m-1, positive characteristic reparametrization"
    },
    {
      "id": "SP:P3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "At a nonradial point,",
      "dependencies": [
        "SP:P2",
        "SP:S4",
        "GE:D19",
        "RP:P2",
        "RP:P3",
        "RC:C1",
        "SP:T5"
      ],
      "scope": "Exact all-order conjugation with arbitrary graph order, both defects and BQ order s"
    },
    {
      "id": "SP:P4",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "Zero Hamilton fields",
      "dependencies": [
        "SP:P3",
        "RP:P4",
        "RP:P5",
        "SP:S2"
      ],
      "scope": "Radial/zero cases, both local states and full connected interval at the exact endpoint"
    },
    {
      "id": "SP:P5",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "Off the characteristic set",
      "dependencies": [
        "SP:S2",
        "K:K3"
      ],
      "scope": "Elliptic gain m distinct from characteristic m-1 and no global norm claim"
    },
    {
      "id": "SP:F0",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**The same threshold in every dimension and cone.**",
      "dependencies": [
        "P3:M2",
        "P3:M3",
        "U001:P14.2-powers",
        "U001:F0-COMP"
      ],
      "scope": "Full dyadic box proof of all-dimensional cone threshold, including critical divergence without polar integration"
    },
    {
      "id": "SP:F1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "For clarity, the joint-frequency",
      "dependencies": [
        "P3:L1",
        "CH:W1",
        "P3:M3",
        "U001:P14.2-powers"
      ],
      "scope": "Explicit two-sided joint-frequency bounds and nonzero temporal transform interval in any regular cone"
    },
    {
      "id": "SP:F2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "The lower panel shows",
      "dependencies": [
        "SP:F1",
        "P3:M3",
        "P3:M8",
        "U001:P14.2-powers",
        "U001:L5.4.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact radial integral, all three limits, weighted tail equivalence and one-dimensional cone threshold"
    },
    {
      "id": "SP:F3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "Fix \\(d=1\\),",
      "dependencies": [
        "SP:F1",
        "SP:F2",
        "SP:P4",
        "SP:X2"
      ],
      "scope": "Actual characteristic curve and exact Sobolev-order/time plot, no cotangent or norm mislabelling"
    },
    {
      "id": "SP:X1",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Exercise 6.1",
      "dependencies": [
        "SP:C4",
        "SP:C5",
        "SP:C3",
        "P3:M4"
      ],
      "scope": "complete trace-free slab-averaging solution"
    },
    {
      "id": "SP:X2",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Exercise 6.2",
      "dependencies": [
        "SP:S3",
        "SP:F0",
        "SP:F1",
        "P3:L1",
        "P3:M3",
        "U001:P14.2-powers",
        "U001:L5.4.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "exact delta threshold in every cone and all cutoffs, failure at critical order"
    },
    {
      "id": "SP:X3",
      "source": "sobolev-regularity-along-real-characteristics.md",
      "source_sha256": "79674c6f3a30c1bfe90096532ab4ac3361c289125eb156a090592b0b84c02256",
      "proof_locator": "**Exercise 6.3",
      "dependencies": [
        "SP:P2",
        "SP:P3",
        "GE:D7",
        "SP:S2"
      ],
      "scope": "full graph-order calculation and both m<1/m>1 errors"
    }
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    "notice": "The elementary proof providers have exact earlier-programme routes in C10/C20, with public PDF destinations and editable TeX passages. The author English edition remains available. Human source text and figures stay external to this CC0 export and retain their own terms.",
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