{
  "schema": "AN04-restored-kernel-corrections-proof-map/v1",
  "proofs": [
    {
      "id": "KC:R0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**Proof.** Use the graph cones",
      "dependencies": [
        "GS:Z12",
        "PS:PS5",
        "IR:R1",
        "IR:R3",
        "OSC:O4",
        "CH:W3",
        "U001:F0-COMP"
      ],
      "scope": "One common family of compactly generated frequency-graph boxes, locally finite over the actual base"
    },
    {
      "id": "KC:R1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "In one graph chart, write",
      "dependencies": [
        "KC:R0",
        "GS:Z10",
        "GS:Z11",
        "GS:Z12",
        "OSC:O4",
        "IR:R3",
        "U001:P14.3",
        "U001:Q4"
      ],
      "scope": "Exact coefficient order and Fourier normalization; actual low-frequency cutoff and smooth zero extension before realization"
    },
    {
      "id": "KC:R2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Here is the extra wavefront control",
      "dependencies": [
        "KC:R1",
        "CH:W1",
        "CH:W2",
        "U001:Q3",
        "U001:Q4",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Every separated-cone convolution tail with an explicitly integrable dimension-dependent bound"
    },
    {
      "id": "KC:R3",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "There is no additional wavefront",
      "dependencies": [
        "KC:R2",
        "OSC:O1",
        "OSC:O2",
        "CH:W2",
        "U001:F0-CALC"
      ],
      "scope": "Actual transposed nonstationary operator, all base and angular derivatives and exact off-graph exclusion"
    },
    {
      "id": "KC:R4",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Low-frequency changes have smooth inverse transforms.",
      "dependencies": [
        "KC:R0",
        "KC:R1",
        "KC:R2",
        "KC:R3"
      ],
      "scope": "Full closed-microsupport realization with actual locally finite assembly and one common graph family"
    },
    {
      "id": "KC:S0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Choose a countable collection",
      "dependencies": [
        "KC:R0",
        "PS:PS5",
        "U001:F0-COMP"
      ],
      "scope": "Countable compact angular exhaustion of the open complement, with larger closed neighborhoods still outside the source set"
    },
    {
      "id": "KC:S1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Every demand in (KC6) is attainable.",
      "dependencies": [
        "KC:S0",
        "K:K1",
        "U001:P14.3"
      ],
      "scope": "All added rapid-decay seminorms are attainable with one increasing radius tending at least geometrically to infinity"
    },
    {
      "id": "KC:S2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "The sum is locally finite in frequency.",
      "dependencies": [
        "KC:S1",
        "K:K1"
      ],
      "scope": "Exact ordinary asymptotic remainders with finite low-frequency corrections and every differentiated summable tail"
    },
    {
      "id": "KC:S3",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "For a fixed angular set outside",
      "dependencies": [
        "KC:S0",
        "KC:S1",
        "KC:S2",
        "DT:M0"
      ],
      "scope": "Every off-set radial decay order and derivative survives the infinite sum"
    },
    {
      "id": "KC:S4",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Perform this construction separately",
      "dependencies": [
        "KC:R4",
        "KC:S2",
        "KC:S3"
      ],
      "scope": "Actual locally finite kernel sum, precise distribution remainder order and retained closed wavefront exclusion"
    },
    {
      "id": "KC:T0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "In its coordinates \\((q,t,s,R)\\)",
      "dependencies": [
        "DT:B3",
        "DT:B4",
        "GF:S1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Ordinary nonhomogeneous S0 coefficient and the actual oriented integrating-factor formula"
    },
    {
      "id": "KC:T1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**Proof.** The compact interpolation segments",
      "dependencies": [
        "KC:T0",
        "GF:S1",
        "U001:F0-CALC",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every ordinary S0 exponential derivative, bounded complex factor and exact total radial loss"
    },
    {
      "id": "KC:T2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Differentiating the integral therefore",
      "dependencies": [
        "KC:T1",
        "DT:T0",
        "DT:T2",
        "DT:T3",
        "DT:H2",
        "DT:H3"
      ],
      "scope": "Complete ordinary-symbol transport, exact signs, zero diagonal, global gluing and closed directional microsupport"
    },
    {
      "id": "KC:P0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "For the frequency comparison used here",
      "dependencies": [
        "GF:N3",
        "ST:T8",
        "U001:F0-COMP"
      ],
      "scope": "Positive compact first-frequency lower bound and actual properly supported first-factor kernel action"
    },
    {
      "id": "KC:P1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "For completeness, the scalar product formula",
      "dependencies": [
        "KC:P0",
        "ST:S4",
        "ST:T7",
        "ST:T8",
        "ST:T9",
        "GS:Z11",
        "OSC:O4"
      ],
      "scope": "Full ordinary lower-product transport formula, every derivative/remainder, nonhomogeneous coefficient and exact subprincipal sign"
    },
    {
      "id": "KC:P2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "The coefficient defines an intrinsic scalar class.",
      "dependencies": [
        "KC:P1",
        "KC:R4",
        "PS:PS5",
        "U001:P14.2-powers"
      ],
      "scope": "All-ray elliptic-symbol comparison modulo S^-1, zero extension of weighted pieces and actual global ordinary scalar representative"
    },
    {
      "id": "KC:H0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "The preceding lesson proves that \\(K\\)",
      "dependencies": [
        "DT:H0",
        "DT:H1",
        "DT:H3"
      ],
      "scope": "Full closed reflexive hull idempotence on actual incomplete intervals"
    },
    {
      "id": "KC:I0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "The principal symbol of \\(F\\) has",
      "dependencies": [
        "GS:Z11",
        "GS:Z12",
        "IR:R2",
        "CH:W1",
        "U001:Q3"
      ],
      "scope": "Actual principal representative with every-derivative microsupport inside the kernel wavefront"
    },
    {
      "id": "KC:I1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Solve the zero-diagonal symbol equation",
      "dependencies": [
        "KC:I0",
        "KC:T2",
        "KC:P2",
        "KC:R4",
        "KC:H0",
        "GS:Z12",
        "ST:T10",
        "CH:W4",
        "RC:C1"
      ],
      "scope": "Actual first residual, correct sign and one-order improvement, with properly localized first-factor pseudolocality"
    },
    {
      "id": "KC:I2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Repeat this argument with the actual residual",
      "dependencies": [
        "KC:I1",
        "KC:H0"
      ],
      "scope": "All real starting orders and every signed lower-order induction in the same fixed boxes"
    },
    {
      "id": "KC:I3",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "Use the controlled kernel summation",
      "dependencies": [
        "KC:I2",
        "KC:S4",
        "KC:P1",
        "IR:R4"
      ],
      "scope": "One genuine distribution kernel has every asymptotic remainder and a smooth residual by the complete all-order intersection proof"
    },
    {
      "id": "KC:C0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**Kernel correction theorem.**",
      "dependencies": [
        "KC:I3",
        "KC:H0",
        "DT:H3",
        "IR:R1",
        "CH:W1"
      ],
      "scope": "Full normalized forward and backward kernel correction with exact original order, closed hull and empty-characteristic case"
    },
    {
      "id": "KC:X1",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**1. Idempotence and the two",
      "dependencies": [
        "KC:H0"
      ],
      "scope": "Complete unchanged solved forward/backward hulls, zero travel time and source-diagonal gap"
    },
    {
      "id": "KC:X2",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**2. An infinite sum of smooth",
      "dependencies": [
        "KC:R1",
        "KC:R2",
        "IR:R3",
        "U001:P14.3",
        "U001:Q3",
        "U001:Q4",
        "CH:W1",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Complete unchanged lacunary smooth-kernel sum, exact ordinary order, distribution convergence, positive wavefront and failure of asymptotic estimates"
    },
    {
      "id": "KC:X3",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "**3. A coefficient with no",
      "dependencies": [
        "KC:T0",
        "KC:T1",
        "KC:T2",
        "U001:P14.2-powers",
        "U001:L5.4.1",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Complete unchanged oscillating ordinary coefficient, removable resonances and all mixed radial/time estimates through every zero"
    },
    {
      "id": "KC:V0",
      "source": "global-kernel-corrections-from-directional-symbols.md",
      "source_sha256": "9341fc89af6c9aac15c7b0b3c8510093caba313009f7f24d3df7766470e31af6",
      "proof_locator": "### A finite view of the singular smooth sum",
      "dependencies": [
        "KC:X2",
        "U001:P14.3",
        "U001:L5.4.1",
        "U001:P14.2-powers"
      ],
      "scope": "Exact compactly supported bump, plateau and logarithmic support coordinates; finite numerical illustration distinguished from the proved infinite wavefront"
    }
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