{
  "schema": "AN04-metric-positivity-proof-map/v1",
  "proofs": [
    {
      "id": "MP:B0",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "**The coefficient spaces and density.**",
      "dependencies": [
        "CE:H1",
        "CE:H2",
        "CE:H3",
        "CE:H4",
        "CE:H5",
        "CT:T1",
        "P3:M3",
        "P3:M4",
        "P3:M7",
        "P3:L1",
        "U001:Q5",
        "WY:A11"
      ],
      "scope": "Complete operator spaces and countable Hilbert direct sums, simple tensor density, Hilbert Fourier and symplectic unitaries with nonseparable ambient spaces"
    },
    {
      "id": "MP:B1",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 1.",
      "dependencies": [
        "MP:B0",
        "WY:W1",
        "WY:W2",
        "WY:W3",
        "WY:W9",
        "WY:W10",
        "P2:QG2",
        "P2:QG3",
        "P2:QG4",
        "P2:QG7",
        "P2:QG8",
        "CE:H1",
        "CE:H4",
        "P2:ML7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Norm-valued coefficient calculus with existence, all ordered remainders and finite dimension-independent constants, bounded-set continuity and no Banach Plancherel"
    },
    {
      "id": "MP:B2",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "Under (B2)–(B3), these symbols also act continuously",
      "dependencies": [
        "MP:B1",
        "WY:A1",
        "WY:A2",
        "WY:A3",
        "WY:A4",
        "WY:A5",
        "WY:A6",
        "WY:A7",
        "WY:A9",
        "MP:B0",
        "P3:L1",
        "P3:L3",
        "U001:Q3"
      ],
      "scope": "Full coefficient Schwartz action and norm topology, ordered product on actual domains, tested adjoints and compact-approximation limit"
    },
    {
      "id": "MP:B3",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 2.",
      "dependencies": [
        "MP:B0",
        "MP:B1",
        "MP:B2",
        "WY:W8",
        "WY:A2",
        "WY:A3",
        "P3:M4",
        "CE:H4",
        "U001:P21.4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Uniform Fourier-L1 bound and weighted frozen derivative norm in every ellipsoid, exact Jacobian cancellation and derivative threshold"
    },
    {
      "id": "MP:B4",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 3.",
      "dependencies": [
        "MP:B0",
        "CT:T1",
        "P3:M3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete two-matrix Cotlar summation, absolute quadratic series, signed coefficients, powers-of-two norm identity and enumeration-independent convergence"
    },
    {
      "id": "MP:B5",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 5.",
      "dependencies": [
        "WY:W1",
        "WY:W2",
        "P2:ML1",
        "P2:ML2",
        "P2:ML3",
        "P2:F2",
        "P3:M2",
        "P3:M3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Both neighborhood distance bases, reversal and polynomial counts, volume and overlap argument, both summable tails and frozen metric comparison"
    },
    {
      "id": "MP:B6",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 6.",
      "dependencies": [
        "MP:B1",
        "MP:B2",
        "MP:B3",
        "MP:B5",
        "WY:W3",
        "WY:W9",
        "P2:QG2",
        "P2:QG4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full mixed frozen Gaussian phase dual16, crossed-distance reflection, all diagonal derivative losses and independent two-family off-center operator decay"
    },
    {
      "id": "MP:B7",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 7.",
      "dependencies": [
        "MP:B0",
        "MP:B2",
        "MP:B3",
        "MP:B4",
        "MP:B5",
        "MP:B6",
        "P2:ML4"
      ],
      "scope": "Uniform metric Hilbert L2 with finitely many symbol derivatives, nonseparable coefficient spaces, degeneration of constant forms and infinite columns"
    },
    {
      "id": "MP:B8",
      "source": "metric-operator-bounds.md",
      "source_sha256": "599e4faf261aebb0ad88d7b96270bf45ca9e74c71e1df128a185c17e97bbf8ec",
      "proof_locator": "## 8.",
      "dependencies": [
        "WY:W1",
        "U001:Q5",
        "U001:P2",
        "QF:A1"
      ],
      "scope": "Complete positive-form symplectic axes, real skew blocks, intrinsic eigenvalue uniqueness, exact Planck parameter and alternate dilation"
    },
    {
      "id": "MP:F1",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "## 5.",
      "dependencies": [
        "FP0:S5",
        "MP:B0",
        "MP:B1",
        "MP:B2",
        "MP:B7",
        "WY:W11",
        "WY:A7",
        "WY:A9",
        "P3:M3"
      ],
      "scope": "Full squared operator localization, ordered first correction cancellation, weight mh2=1, infinite-column weak limit and lower-bound transfer"
    },
    {
      "id": "MP:F2",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "**Symplectic normalization.**",
      "dependencies": [
        "MP:B8",
        "WY:A13",
        "WY:A14",
        "U001:Q5"
      ],
      "scope": "Full symplectic normalization to lambda times Euclidean metric preserving all constant-form derivative hypotheses"
    },
    {
      "id": "MP:F3",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "**Induction.**",
      "dependencies": [
        "MP:F2",
        "MP:B0",
        "WY:W4",
        "WY:A13",
        "P3:L1",
        "P3:L3",
        "P3:M4",
        "U001:Q5"
      ],
      "scope": "Zero-dimensional base, one-direction independent symbols, delta-kernel fiber identity and orthogonal symplectic rotation"
    },
    {
      "id": "MP:F4",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "Apply the adaptive construction",
      "dependencies": [
        "MP:F3",
        "FP0:S1",
        "FP0:S2",
        "FP0:S3",
        "FP0:S4",
        "FP0:S6",
        "FP0:S7",
        "MP:B1",
        "MP:B2",
        "MP:B7",
        "MP:F1",
        "WY:W11",
        "U001:U001-A4"
      ],
      "scope": "Adaptive floor and normalized splitting branches, full transverse extension, uniform cutoff/product errors, real square positivity and global induction"
    },
    {
      "id": "MP:F5",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "**Finite derivative dependence and approximation.**",
      "dependencies": [
        "MP:F4",
        "MP:F1",
        "MP:B7",
        "MP:B1",
        "FP0:S4",
        "FP0:S6",
        "FP0:S7",
        "WY:W4",
        "U001:U001-A4",
        "P3:M3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Finite derivative recursion, normalization and compact approximation with R>=lambda^-1/2, all uniform constants and original form-domain limit"
    },
    {
      "id": "MP:F6",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "## 8.",
      "dependencies": [
        "MP:F1",
        "MP:F2",
        "MP:F3",
        "MP:F4",
        "MP:F5",
        "P2:ML2",
        "P2:ML3",
        "P2:ML4",
        "WY:W1",
        "WY:W2"
      ],
      "scope": "Complete scalar variable-metric Fefferman-Phong with exact h^-2 weight, frozen cutoffs and finite-seminorm uniformity"
    },
    {
      "id": "MP:F7",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "**The metric hypotheses in the original coordinates.**",
      "dependencies": [
        "WY:W1",
        "P2:ML1",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete classical metric slow variation, both weight ratios with actual dual distance, h and derivative class equivalence for0<=delta<rho<=1"
    },
    {
      "id": "MP:F8",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "The same conclusion, with another constant",
      "dependencies": [
        "MP:F6",
        "MP:F7",
        "MP:B7",
        "WY:Q1",
        "WY:Q2",
        "WY:Q3",
        "WY:Q4",
        "WY:W4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full classical endpoint, every remainder in quantization conversion, imaginary first correction and bounded error in symmetric left operator"
    },
    {
      "id": "MP:F9",
      "source": "scalar-fefferman-phong.md",
      "source_sha256": "133f02467366d991def07a02c2f80e56ad225d20483c93a10ed7cebc17fe9e1c",
      "proof_locator": "## F10.",
      "dependencies": [
        "WY:W4",
        "WY:W5",
        "WY:W11",
        "WY:W6",
        "P3:L3",
        "P3:M3",
        "P3:M4",
        "U001:Q2",
        "U001:P21.4",
        "U001:P14.2-powers",
        "U001:P15.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact Weyl square defect, log-coordinate isometry, smooth flat normalized Schwartz family and proved negative expectation and limit"
    }
  ],
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