{
  "schema": "AN04-restored-compact-middle-proof-map/v1",
  "proofs": [
    {
      "id": "CM:K0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "All orders may be real.",
      "dependencies": [
        "RC:C0",
        "RC:K8"
      ],
      "scope": "Full reflected-input and half-density conventions, including both forbidden zero endpoint axes"
    },
    {
      "id": "CM:K1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Form the tensor product",
      "dependencies": [
        "CM:K0",
        "RP:W0",
        "RP:W1",
        "FC:S1",
        "FC:S2",
        "U001:F0-CALC"
      ],
      "scope": "Actual matching embedding, exact derivative transpose, every tensor zero-factor case and transverse pullback"
    },
    {
      "id": "CM:K2",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Multiply this pullback by",
      "dependencies": [
        "CM:K1",
        "RC:K0",
        "RC:K1",
        "RC:K5",
        "P3:M3",
        "P3:M4",
        "PS:PS5"
      ],
      "scope": "Actual compact-middle pushforward as a continuous distribution pairing with cutoff independence and coordinate gluing"
    },
    {
      "id": "CM:K3",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "The two half densities at",
      "dependencies": [
        "CM:K2",
        "RC:K8",
        "RC:C0"
      ],
      "scope": "Invariant middle density integration, external half-density frame and preserved matrix factor order"
    },
    {
      "id": "CM:K4",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "This kernel agrees with operator composition",
      "dependencies": [
        "CM:K1",
        "CM:K2",
        "CM:K3",
        "RC:C1",
        "RC:K4",
        "FC:S2",
        "P3:M3"
      ],
      "scope": "Smooth regularization identifies the proposed composition on its actual compact test domain"
    },
    {
      "id": "CM:K5",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Here is an explicit verification",
      "dependencies": [
        "CM:K4",
        "FC:S6",
        "FC:S7",
        "CH:W1",
        "RC:C1",
        "RC:K2",
        "RC:K4",
        "RP:W0",
        "RP:W1",
        "P3:M3",
        "U001:Q2",
        "U001:Q3",
        "U001:Q4"
      ],
      "scope": "Uniform polynomial and axis-cone bounds for every regularized kernel; smooth varying-input convergence, tensor Gaussian factorization and equality on all external tests"
    },
    {
      "id": "CM:W0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "We first prove the fiber-integration",
      "dependencies": [
        "CM:K2",
        "CH:W1",
        "CH:W3",
        "U001:F0-COMP",
        "P3:L1"
      ],
      "scope": "Full proper pushforward wavefront bound from finite compact middle cover and the exact zero-middle-frequency transform"
    },
    {
      "id": "CM:W1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Indeed the joint covector before",
      "dependencies": [
        "CM:K1",
        "CM:W0",
        "FC:S3",
        "RP:W1"
      ],
      "scope": "Exact reflected matching signs and exclusion of terms with only one singular tensor factor"
    },
    {
      "id": "CM:W2",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "There is no omitted closure",
      "dependencies": [
        "CM:W1",
        "RC:C0",
        "U001:F0-COMP"
      ],
      "scope": "Uniform endpoint norm comparability on compact normalized slices; closed matching image and no zero external endpoint"
    },
    {
      "id": "CM:W3",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "One may replace the fixed compact",
      "dependencies": [
        "CM:K2",
        "CM:K5",
        "CM:W0",
        "CM:W1",
        "CM:W2"
      ],
      "scope": "Proper-middle extension and actual compact two-middle-kernel composition"
    },
    {
      "id": "CM:W4",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "If either outside kernel in",
      "dependencies": [
        "CM:W1",
        "CM:W3",
        "CH:W1"
      ],
      "scope": "Both smooth-error ideals with compact middle support, without assuming proper outside kernels"
    },
    {
      "id": "CM:R0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Let \\(P\\in\\Psi^m(X)\\)",
      "dependencies": [
        "GF:N0",
        "GF:N1",
        "GF:N2",
        "GF:N3",
        "GF:Q5"
      ],
      "scope": "All-real positive normalization, exact global compact-return assumptions and ambient closed signed relation with full diagonal"
    },
    {
      "id": "CM:R1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Fix a compact \\(K\\subset X\\).",
      "dependencies": [
        "CM:R0",
        "GF:N1",
        "PS:PS5",
        "U001:P14.3"
      ],
      "scope": "Actual compact return set and a compact smooth plateau on its neighborhood"
    },
    {
      "id": "CM:R2",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Its kernel has compact support.",
      "dependencies": [
        "CM:R1",
        "RC:C1",
        "P2:OP5",
        "U001:F0-COMP"
      ],
      "scope": "Both middle kernels and their commutator have actual compact support by both proper projections"
    },
    {
      "id": "CM:R3",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "For its actual kernel is",
      "dependencies": [
        "CM:R2",
        "P2:OP4",
        "CH:W4"
      ],
      "scope": "Exact commutator microsupport on supp dphi with every ordinary lower term retained"
    },
    {
      "id": "CM:R4",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Let \\(A_1,A_2\\) have wavefront relations",
      "dependencies": [
        "CM:R0",
        "CM:R1",
        "CM:R3",
        "CM:W3",
        "G2:NF7"
      ],
      "scope": "Full same-sign relation matching and compact-return exclusion, including each diagonal alternative and both time directions"
    },
    {
      "id": "CM:V0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "The diagram uses the model",
      "dependencies": [
        "CM:R4",
        "U001:P14.3",
        "U001:P15.1",
        "U001:F0-CALC"
      ],
      "scope": "Exact flat cutoff, positive transition derivative, both closed derivative supports and every affine diagram coordinate"
    },
    {
      "id": "CM:I0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "**Proof.** Use the return cutoff",
      "dependencies": [
        "CM:R2",
        "CM:W3",
        "RC:C1",
        "RC:K4"
      ],
      "scope": "Both exact operator identities on compact test domains with no unrestricted triple association"
    },
    {
      "id": "CM:I1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "The two error products are smooth",
      "dependencies": [
        "CM:I0",
        "CM:R4",
        "CM:W4"
      ],
      "scope": "Exact commutator subtraction with all signs, plateau removal near any compact external pair and full forward/backward uniqueness"
    },
    {
      "id": "CM:I2",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Thus any right parametrix",
      "dependencies": [
        "CM:I1"
      ],
      "scope": "Comparison of like-sided parametrices through an actually existing opposite-sided one; no existence inference"
    },
    {
      "id": "CM:A0",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "Suppose right parametrices with both signs",
      "dependencies": [
        "CM:R0",
        "RC:C2",
        "P2:OP3",
        "G2:NF7"
      ],
      "scope": "Correct opposite-sign right inverse for the adjoint supplies an actual same-sign left inverse"
    },
    {
      "id": "CM:A1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "The last inclusion follows from",
      "dependencies": [
        "CM:A0",
        "CM:I1",
        "RC:C1",
        "CM:W3",
        "CM:W4",
        "P2:OP3"
      ],
      "scope": "Full smooth action by the proper operator and conditional conversion to both identities, with uniqueness"
    },
    {
      "id": "CM:X1",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "**1. A smooth term exposes",
      "dependencies": [
        "CM:I0",
        "CM:I1",
        "P3:L2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete unchanged two exact inverses, failed uncut associativity and nonzero smooth compact-middle commutator"
    },
    {
      "id": "CM:X2",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "**2. Compact support does not",
      "dependencies": [
        "CM:K1",
        "RP:W0",
        "RP:W1",
        "U001:Q2",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Complete unchanged forbidden-normal example, exact Gaussian divergence and failure of a distributional product"
    },
    {
      "id": "CM:X3",
      "source": "comparing-global-parametrices-through-a-compact-middle.md",
      "source_sha256": "b474bd732b94e2d84ffd9e2cdfb7d725703dc5dc34dc395b4659e0797f518c55",
      "proof_locator": "**3. The adjoint must reverse",
      "dependencies": [
        "CM:A0",
        "CM:A1",
        "DT:X2",
        "RP:F2",
        "RP:F3"
      ],
      "scope": "Complete unchanged adjoint sign and nonsmooth opposite-sign difference at its exact ordinary order"
    }
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