{
  "schema": "AN04-restored-quadratic-forms-proof-map/v1",
  "proofs": [
    {
      "id": "QF:A0",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A0.",
      "dependencies": [
        "GEO:C0",
        "U001:F0-ALG",
        "U001:F0-COMP",
        "U001:P13.1"
      ],
      "scope": "Exact finite inputs and complex elimination, basis, rank and inverse constructions"
    },
    {
      "id": "QF:A1",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A1.",
      "dependencies": [
        "QF:A0",
        "U001:P16.3",
        "U001:F0-COMP"
      ],
      "scope": "Full complex polynomial root proof and finite factorization"
    },
    {
      "id": "QF:A2",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A2.",
      "dependencies": [
        "QF:A0"
      ],
      "scope": "Cofactor coefficient proof of the characteristic identity"
    },
    {
      "id": "QF:A3",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A3.",
      "dependencies": [
        "QF:A1",
        "QF:A2"
      ],
      "scope": "Polynomial Bezout projections, generalized direct sum, nilpotent restrictions and inverse formula"
    },
    {
      "id": "QF:A4",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A4.",
      "dependencies": [
        "QF:A0",
        "QF:A3"
      ],
      "scope": "Explicit commuting nilpotent chain projection, complete Jordan basis and all chain multiplicities"
    },
    {
      "id": "QF:A5",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A5.",
      "dependencies": [
        "QF:A3",
        "U001:P15.1",
        "U001:P16.2",
        "U001:P13.2-series",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Direct oriented circle moments and full generalized resolvent projection"
    },
    {
      "id": "QF:A6",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A6.",
      "dependencies": [
        "QF:A5",
        "QF:A0",
        "U001:F0-COMP",
        "U001:L7.1.4"
      ],
      "scope": "Uniform spectral gap by compactness, exact enclosing circle, continuous projections and range bases"
    },
    {
      "id": "QF:A7",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A7.",
      "dependencies": [
        "QF:A6",
        "QF:A0",
        "U001:F0-COMP"
      ],
      "scope": "Full semidefinite Hermitian Cauchy–Schwarz, open positivity and interval argument"
    },
    {
      "id": "QF:A8",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A8.",
      "dependencies": [
        "U001:Q5",
        "U001:P8-root",
        "QF:A0"
      ],
      "scope": "Actual positive symmetric square root and arbitrary-metric orthonormalization"
    },
    {
      "id": "QF:A9",
      "source": "spectral-algebra-and-contour-projections.md",
      "source_sha256": "92412459ab0edf8365d459df736d92b0fc3b9057122ef26f3e85ff138e112f23",
      "proof_locator": "## A9.",
      "dependencies": [
        "QF:A0",
        "U001:P15.1",
        "U001:P13.2-mertens",
        "U001:P16.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Convergent matrix exponential, all derivatives, group inverse and full symplectic flow"
    },
    {
      "id": "QF:H0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "## 1.",
      "dependencies": [
        "QF:A0",
        "G2:G0",
        "U001:F0-DIFF"
      ],
      "scope": "Critical-point intrinsic derivative, exact factor and Hessian block"
    },
    {
      "id": "QF:H1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Symmetry of \\(B\\) gives",
      "dependencies": [
        "QF:H0"
      ],
      "scope": "Hamilton skew identity, converse and complete symplectic covariance"
    },
    {
      "id": "QF:S0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Lemma 2.1",
      "dependencies": [
        "QF:H1",
        "QF:A3"
      ],
      "scope": "All generalized spaces, power transfer, nondegenerate opposite pairing and full orthogonal splitting"
    },
    {
      "id": "QF:R0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "U001:Q5",
        "ML:M0a",
        "QF:A0"
      ],
      "scope": "Signature obstruction with radical and exact negative-index bound"
    },
    {
      "id": "QF:R1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Real eigenvalues.**",
      "dependencies": [
        "QF:R0",
        "QF:S0"
      ],
      "scope": "Complex-quartet exclusion, real generalized dimension one and normalized hyperbolic block"
    },
    {
      "id": "QF:R2",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Imaginary eigenvalues.**",
      "dependencies": [
        "QF:R0",
        "QF:S0",
        "G2:G21"
      ],
      "scope": "Correct real eigenplane sign, oscillator, invariant-complement induction and repeated-frequency semisimplicity"
    },
    {
      "id": "QF:R3",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Rank-one and zero blocks.**",
      "dependencies": [
        "QF:R0",
        "QF:S0",
        "G2:G21"
      ],
      "scope": "Signed rank-one plane, zero symplectic plane and invariant orthogonal complements"
    },
    {
      "id": "QF:R4",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**The residual chain.**",
      "dependencies": [
        "QF:R3",
        "QF:A4",
        "G2:G21"
      ],
      "scope": "Residual isotropic kernel, exact sequence and single even nilpotent chain"
    },
    {
      "id": "QF:R5",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Moving powers of",
      "dependencies": [
        "QF:R4",
        "QF:R0"
      ],
      "scope": "Exact chain pairing, length-four bound and negative cubic pairing"
    },
    {
      "id": "QF:R6",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Put \\(b=\\omega(X,F^3X)\\)",
      "dependencies": [
        "QF:R5"
      ],
      "scope": "Signed scaling, corrected generator, whole canonical chain basis and exact exceptional quadratic form"
    },
    {
      "id": "QF:R7",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Theorem 4.1",
      "dependencies": [
        "QF:R1",
        "QF:R2",
        "QF:R6",
        "QF:A4",
        "ML:M0a"
      ],
      "scope": "Complete nonnegative and index-one classifications, all dimension constraints, invariants and sufficiency"
    },
    {
      "id": "QF:M0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Example 4.2",
      "dependencies": [
        "QF:R7",
        "QF:H1",
        "U001:P14.2-powers"
      ],
      "scope": "Full oscillator shear/scaling and exact primitive"
    },
    {
      "id": "QF:M1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Example 4.3",
      "dependencies": [
        "QF:R6",
        "QF:A9"
      ],
      "scope": "Actual four-coordinate nilpotent orbit, projection and time factor"
    },
    {
      "id": "QF:C0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "## 5.",
      "dependencies": [
        "QF:H1",
        "QF:A0"
      ],
      "scope": "Sectorial complex form with explicit nonnegative real part even for zero sector constant"
    },
    {
      "id": "QF:C1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Proof.** We first identify",
      "dependencies": [
        "QF:C0"
      ],
      "scope": "Polarized common real radical, including the small complex rotation argument"
    },
    {
      "id": "QF:C2",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "For \\(X=U+iV\\), symmetry",
      "dependencies": [
        "QF:C1"
      ],
      "scope": "All complex kernel equivalences and real spanning"
    },
    {
      "id": "QF:C3",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "If \\(F^3X=0\\)",
      "dependencies": [
        "QF:C2",
        "QF:A4"
      ],
      "scope": "Full zero nilpotency bound using the conjugate Hamilton map"
    },
    {
      "id": "QF:C4",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Identity (3.7) remains",
      "dependencies": [
        "QF:C3",
        "G2:G21"
      ],
      "scope": "Intrinsic generalized zero space via exact orthogonal image"
    },
    {
      "id": "QF:C5",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Finally let",
      "dependencies": [
        "QF:C2",
        "QF:S0"
      ],
      "scope": "Spectral sector, nonzero real pairing and absence of nonzero real eigenvalues"
    },
    {
      "id": "QF:C6",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Proposition 5.2",
      "dependencies": [
        "QF:C2",
        "QF:C3",
        "QF:C4",
        "QF:S0",
        "G2:G21"
      ],
      "scope": "Correct real symplectic radical quotient, strict descended real part and actual nonzero spectral map"
    },
    {
      "id": "QF:P0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "## 6.",
      "dependencies": [
        "QF:H1",
        "QF:A7"
      ],
      "scope": "Hermitian convention and strict/weak positive plane definitions"
    },
    {
      "id": "QF:P1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Theorem 6.1",
      "dependencies": [
        "QF:P0",
        "QF:C5",
        "QF:C2",
        "QF:S0",
        "U001:F0-COMP"
      ],
      "scope": "Half-dimensional invariant upper plane, uniform sector and real oscillator starting value"
    },
    {
      "id": "QF:P2",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "For the general form use \\(Q_t",
      "dependencies": [
        "QF:P1",
        "QF:R7",
        "QF:A6"
      ],
      "scope": "Actual uniform common circle, correctly oriented generalized projection and continuous range"
    },
    {
      "id": "QF:P3",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Strict positivity is open",
      "dependencies": [
        "QF:P2",
        "QF:A7",
        "QF:C2"
      ],
      "scope": "Open and closed strict positivity through Jordan degeneracies"
    },
    {
      "id": "QF:P4",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "For the numerical range",
      "dependencies": [
        "QF:P3",
        "QF:C2"
      ],
      "scope": "Full numerical range in the sector with the exact scalar normalization"
    },
    {
      "id": "QF:I0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Theorem 7.1",
      "dependencies": [
        "QF:A0",
        "QF:P0",
        "G2:G21"
      ],
      "scope": "Real-part bijection, induced complex structure and correct imaginary sign"
    },
    {
      "id": "QF:I1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Expand \\(\\omega(Z(U),Z(V))=0\\).",
      "dependencies": [
        "QF:I0"
      ],
      "scope": "Both compatibility identities, positive metric, Hermitian construction and factor one-half"
    },
    {
      "id": "QF:I2",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "For uniqueness,",
      "dependencies": [
        "QF:I1"
      ],
      "scope": "Unique full Hermitian structure with fixed imaginary part"
    },
    {
      "id": "QF:I3",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Corollary 7.2",
      "dependencies": [
        "QF:I2",
        "QF:A8",
        "G2:G21"
      ],
      "scope": "Whole basis adapted to prescribed real and strict complex Lagrangians"
    },
    {
      "id": "QF:I4",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Apply the corollary",
      "dependencies": [
        "QF:I3",
        "QF:P3"
      ],
      "scope": "Invariant-plane quadratic vanishing and full complex polynomial ideal membership"
    },
    {
      "id": "QF:G0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Proposition 8.1",
      "dependencies": [
        "QF:P0",
        "QF:A0"
      ],
      "scope": "Strict transversality in arbitrary real canonical coordinates and full graph equivalence"
    },
    {
      "id": "QF:G1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "The whole real symplectic coordinate change",
      "dependencies": [
        "QF:G0",
        "QF:A8",
        "U001:F0-DIFF"
      ],
      "scope": "Complete real shear/scaling, exact primitive and graph normalization"
    },
    {
      "id": "QF:G3",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "The compatible structure and metric can also",
      "dependencies": [
        "QF:G1",
        "QF:I2"
      ],
      "scope": "Full complex-structure matrix, compatible metric and explicit positive expression"
    },
    {
      "id": "QF:N0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "**Theorem 9.1",
      "dependencies": [
        "QF:P0",
        "QF:A7",
        "G2:G21"
      ],
      "scope": "Nullspace equals complexification of real isotropic part"
    },
    {
      "id": "QF:N1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "Real vectors in \\(L_{\\mathbb R}\\)",
      "dependencies": [
        "QF:N0",
        "G2:G21"
      ],
      "scope": "Exact symplectic quotient dimension, Lagrangian image and strict positivity"
    },
    {
      "id": "QF:N2",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "For the direct-sum version,",
      "dependencies": [
        "QF:N1",
        "G2:G21"
      ],
      "scope": "Full real symplectic completion and direct sum including both endpoints"
    },
    {
      "id": "QF:J0",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "## 10.",
      "dependencies": [
        "QF:H0",
        "QF:P3",
        "QF:A4"
      ],
      "scope": "Actual strictly sectorial nonsemisimple Hamilton form and its entire upper generalized plane"
    },
    {
      "id": "QF:J1",
      "source": "quadratic-hamilton-maps-and-positive-complex-planes.md",
      "source_sha256": "67ed2931738c6b79a412e7f161a2cdca8084cc4695af60ebdc90f9c9976f8189",
      "proof_locator": "With \\(h((x,ix),(x,ix))",
      "dependencies": [
        "QF:J0",
        "QF:A7",
        "U001:P16.3",
        "U001:P14.1"
      ],
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