{
  "schema": "AN04-restored-graph-egorov-proof-map/v1",
  "proofs": [
    {
      "id": "GE:G0",
      "source": "global-inverses-and-periodic-multipliers.md",
      "source_sha256": "3d20386673b0808475184b5aa8d0f55601eb920fc7c6f0a438db28bf9fe070eb",
      "proof_locator": "## G0.",
      "dependencies": [
        "K:K2",
        "K:K3",
        "K:K4",
        "PS:PS5",
        "CH:T2",
        "P2:OP6"
      ],
      "scope": "Global proper two-sided matrix inverse from local conic inverses; separate bundle cutoffs, locally finite kernel sums, smoothing errors and ordered comparison"
    },
    {
      "id": "GE:G1",
      "source": "global-inverses-and-periodic-multipliers.md",
      "source_sha256": "3d20386673b0808475184b5aa8d0f55601eb920fc7c6f0a438db28bf9fe070eb",
      "proof_locator": "## G1.",
      "dependencies": [
        "RC:K2",
        "P3:M7",
        "P3:M6",
        "CT:T1",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Complete circle basis, Parseval by density, diagonal multiplier norm and exact tail norm"
    },
    {
      "id": "GE:G2",
      "source": "global-inverses-and-periodic-multipliers.md",
      "source_sha256": "3d20386673b0808475184b5aa8d0f55601eb920fc7c6f0a438db28bf9fe070eb",
      "proof_locator": "## G2.",
      "dependencies": [
        "GE:G1",
        "P2:OP4",
        "P3:L3",
        "U001:Q3",
        "U001:Q4",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "RC:K2"
      ],
      "scope": "Full periodization proof of the ordinary circle regularizer, off-zero derivative decay, exact Fourier coefficients, local PDO kernel and compact tail"
    },
    {
      "id": "GE:D0",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "## 1.",
      "dependencies": [
        "G2:G0",
        "G2:F1",
        "GS:Z10",
        "LR:R11",
        "RC:C2",
        "CT:T1"
      ],
      "scope": "Canonical symplectic half volume, reduced order, ordered graph density contraction and finite Hermitian bundle norms"
    },
    {
      "id": "GE:D1",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Proposition 2.1.",
      "dependencies": [
        "GE:D0",
        "ST:T4",
        "ST:T6",
        "G2:G41",
        "U001:P3"
      ],
      "scope": "Mixed graph coordinates from input-only second jets, homogeneous generating function and invertible mixed Hessian"
    },
    {
      "id": "GE:D2",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "The phase \\(\\Phi(x,y,\\eta)",
      "dependencies": [
        "GE:D1",
        "IR:R7",
        "GS:Z11",
        "GS:Z12",
        "OSC:O1",
        "K:K1",
        "HT:H8"
      ],
      "scope": "Exact graph phase normalization, critical density, determinant factor and full left amplitude reduction with all remainders"
    },
    {
      "id": "GE:D3",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Proof on one graph patch.**",
      "dependencies": [
        "GE:D2",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "P2:OP4",
        "P1:B4",
        "P3:M7",
        "WM:W2",
        "CT:T1"
      ],
      "scope": "Actual T*T before bounded extension, finite-seminorm L2 estimate, finite patch assembly and nonidentity cross branches"
    },
    {
      "id": "GE:D4",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "For the left form (2.5)",
      "dependencies": [
        "GE:D2",
        "P3:L3",
        "P2:OP4",
        "P1:B4",
        "CT:T1",
        "U001:P3",
        "U001:P4"
      ],
      "scope": "Direct TT* Fourier change, convex-cone injectivity, full inverse symbol bounds, frequency truncations and uniform norm"
    },
    {
      "id": "GE:D5",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "HT:H2"
      ],
      "scope": "Vanishing symbol gives small fixed jets by nested Taylor estimates; every high-frequency tail seminorm"
    },
    {
      "id": "GE:D6",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Theorem 4.2",
      "dependencies": [
        "GE:D2",
        "GE:D3",
        "GE:D4",
        "GE:D5",
        "CT:T1",
        "CT:T2",
        "P3:L3"
      ],
      "scope": "Localized graph compactness, actual L2 kernels, finite-rank approximation and norm-limit argument"
    },
    {
      "id": "GE:D7",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Theorem 5.1.",
      "dependencies": [
        "GE:D3",
        "RC:C1",
        "RC:C2",
        "RC:C9",
        "WM:W1",
        "WM:W6",
        "P1:B4",
        "P1:B5",
        "K:K3",
        "K:K4",
        "GE:G0"
      ],
      "scope": "All real Sobolev orders, proper reducers, smoothing errors and support-controlled approximation to the original distributional action"
    },
    {
      "id": "GE:D8",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Support lemma.**",
      "dependencies": [
        "RC:K0",
        "PS:PS5",
        "CH:W1",
        "G2:G41"
      ],
      "scope": "Closed proper base relation of a global canonical graph; locally finite forbidden sets and smooth proper cutoff"
    },
    {
      "id": "GE:D9",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Theorem 6.1",
      "dependencies": [
        "GE:D8",
        "GE:G0",
        "GS:Z12",
        "LR:R11",
        "RC:C9",
        "RC:C1",
        "RC:C2",
        "K:K2",
        "K:K3",
        "CH:T2"
      ],
      "scope": "Global and local two-sided elliptic graph inverse, exact inverse symbol bundle and smoothing comparison"
    },
    {
      "id": "GE:D10",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Theorem 7.1",
      "dependencies": [
        "GE:D0",
        "GE:D9",
        "RC:C9",
        "P2:OP4"
      ],
      "scope": "Ordered Egorov for arbitrary A and B, actual beta-p-alpha product and necessary noninverse scalar coefficient"
    },
    {
      "id": "GE:D11",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "It also justifies changes",
      "dependencies": [
        "GE:D3",
        "GE:D9",
        "GE:D10",
        "GS:Z12",
        "LR:R11"
      ],
      "scope": "Quantization and inversion of local canonical changes with exact graph order and L2 invariance"
    },
    {
      "id": "GE:D12",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**One fixed neighborhood.**",
      "dependencies": [
        "K:K1",
        "K:K3",
        "P2:OP3",
        "P2:OP5",
        "CH:T2"
      ],
      "scope": "One fixed finite cylinder and interior cone, full symbol extension, proper representative and microlocal germs"
    },
    {
      "id": "GE:D13",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**The leading matrix and every symbol derivative.**",
      "dependencies": [
        "GE:D12",
        "G2:NF2",
        "G2:NF3",
        "U001:P15.1"
      ],
      "scope": "Actual ordered matrix fundamental solution and right inverse equation, both inverse identities and uniform finite-interval bounds"
    },
    {
      "id": "GE:D14",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "We verify that \\(M,H\\)",
      "dependencies": [
        "GE:D13",
        "G2:NF3",
        "U001:P4"
      ],
      "scope": "Induction for every parameter, frequency and time derivative of M and H, preserving matrix order and symbol weights"
    },
    {
      "id": "GE:D15",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "The compact-time extension",
      "dependencies": [
        "GE:D14",
        "P2:OP3",
        "P2:OP5"
      ],
      "scope": "Global uniform symbol bounds after fixed compact extension, support of full residuals and finite-time correction integrals"
    },
    {
      "id": "GE:D16",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Correct the entire lower-order remainder.**",
      "dependencies": [
        "GE:D15",
        "P2:OP3",
        "G2:NF2",
        "G2:NF3"
      ],
      "scope": "Exact differential commutator, full residual recursion, correct -i sign and every differentiated order gain"
    },
    {
      "id": "GE:D17",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "Apply support-preserving symbol summation",
      "dependencies": [
        "GE:D16",
        "K:K1",
        "P2:OP3"
      ],
      "scope": "Supported asymptotic sum retaining M, all time derivatives and full smoothing intertwining remainder"
    },
    {
      "id": "GE:D18",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "Since \\(a=M+S^{-1}\\)",
      "dependencies": [
        "GE:D17",
        "K:K2",
        "K:K3",
        "P2:OP6"
      ],
      "scope": "Uniform inverse bounds, proper two-sided matrix parametrix and all-order conjugation"
    },
    {
      "id": "GE:D19",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Return to the canonical graph.**",
      "dependencies": [
        "GE:D18",
        "GE:D9",
        "GE:D10",
        "RC:C9",
        "CH:T2"
      ],
      "scope": "Return to supplied canonical coordinates, both inverse identities, arbitrary real graph order and ordered smoothing conjugation"
    },
    {
      "id": "GE:D20",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "## 8.",
      "dependencies": [
        "GE:D0",
        "GE:D9",
        "GE:D10",
        "ST:S4",
        "OSC:O6",
        "U001:P14.2-powers",
        "U001:P15.1"
      ],
      "scope": "Exact half-density pullback, change of variables, inverse and exponential Egorov model"
    },
    {
      "id": "GE:E1",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.1 ",
      "dependencies": [
        "GE:D0"
      ],
      "scope": "Exercise 9.1 and its full unchanged solution"
    },
    {
      "id": "GE:E2",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.2 ",
      "dependencies": [
        "GE:D0",
        "GE:D20"
      ],
      "scope": "Exercise 9.2 and its full unchanged solution"
    },
    {
      "id": "GE:E3",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.3 ",
      "dependencies": [
        "GE:D20",
        "GE:G1",
        "CT:T1"
      ],
      "scope": "Exercise 9.3 and its full unchanged solution"
    },
    {
      "id": "GE:E4",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.4 ",
      "dependencies": [
        "GE:D6",
        "GE:G2"
      ],
      "scope": "Exercise 9.4 and its full unchanged solution"
    },
    {
      "id": "GE:E5",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.5 ",
      "dependencies": [
        "GE:D3",
        "GE:G1",
        "OSC:O6"
      ],
      "scope": "Exercise 9.5 and its full unchanged solution"
    },
    {
      "id": "GE:E6",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.6 ",
      "dependencies": [
        "GE:D20",
        "ST:S4"
      ],
      "scope": "Exercise 9.6 and its full unchanged solution"
    },
    {
      "id": "GE:E7",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.7 ",
      "dependencies": [
        "GE:D10",
        "GE:D20"
      ],
      "scope": "Exercise 9.7 and its full unchanged solution"
    },
    {
      "id": "GE:E8",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.8 ",
      "dependencies": [
        "GE:D10"
      ],
      "scope": "Exercise 9.8 and its full unchanged solution"
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    {
      "id": "GE:E9",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.9 ",
      "dependencies": [
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        "U001:P15.1"
      ],
      "scope": "Exercise 9.9 and its full unchanged solution"
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      "id": "GE:E10",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.10 ",
      "dependencies": [
        "GE:D13",
        "G2:NF2"
      ],
      "scope": "Exercise 9.10 and its full unchanged solution"
    },
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      "id": "GE:E11",
      "source": "graph-operators-continuity-and-egorov.md",
      "source_sha256": "9ba81d9db2e7b1663c9f8c3a05585515d7e2eb131327fc3298d576426fe72902",
      "proof_locator": "**Exercise 9.11 ",
      "dependencies": [
        "GE:D16",
        "GE:D17",
        "P2:OP3"
      ],
      "scope": "Exercise 9.11 and its full unchanged solution"
    }
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