{
  "title": "Quadratic Hamilton maps and positive complex planes — restored proof edition",
  "original_source_sha256": "19a7c37f2b5fcb1d7d34c90089cbfcaa6ad9bfa3c056c41ba6b0386986be0615",
  "restored_source_sha256": "11bee27367f59f85f306f7eb7c02e056edb0cbed4ae024f768f30c34139c7db1",
  "original_authorship": "GPT-6.1 Sol (OpenAI), Ultra, October 2026",
  "restoration_review": "GPT-6 Astra (OpenAI), Ultra, 5 October 2026",
  "source_policy": "Independent exposition with complete programme proofs and exact scholarly references.",
  "source_references": [
    {
      "edition_id": "ISBN-978-3-540-49938-1-pdf",
      "pdf_pages_read": [
        336,
        337,
        338,
        339,
        340,
        341,
        342,
        343
      ],
      "relevant_printed_pages": [
        321,
        322,
        323,
        324,
        325,
        326,
        327,
        328
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "3e4e537bb528733d58bbfd6cd76581fa203668dae7f982be2f09cee680bed19f",
    "results": "model-check.json",
    "original_script_sha256": "58d16671e0415c78a67d0592d7241c97d7311b0676637ca3b7d73d025ee00c13",
    "finite_checks": 7,
    "algebra_families": 8,
    "algebra_script": "check_algebra.py",
    "algebra_script_sha256": "af2b27dcb15fff31d8f5aafcb4679accf5776d86c78600ed9d0a2ae0cde3c466",
    "original_algebra_script_sha256": "fb3e65ecb7de62bc451459b518bbc79e72fd3526a47439bee3469119b4e398ee",
    "portable_algebra_changes": "Removed unused historical source-access and state-writing branches from the algebra companion; every finite mathematical check is retained. The main verifier changes input/output paths only.",
    "passed": true,
    "mathematical_checks_unchanged": true,
    "historical_archive_checks_removed": false,
    "removed_historical_check_flags": [],
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "figure": {
    "path": "figures/quadratic-block-flows.svg",
    "sha256": "eb567b8a8c01c252293289d1f924b93637e493820bdaf1dde305b9d23b7d4e65",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_quadratic_block_flows.py",
    "original_script_sha256": "dbd75dcdbb904b695fcf5e02ba74d2c69087ea19a3291aa2cc38911cf6494f90",
    "portable_script_sha256": "361e3749e47396e4529195d6d4bcd1ede239b3dc4e704965a8888c3644707de3",
    "license": "CC0-1.0",
    "coordinates": "Seven exact half-Hamiltonian orbit curves: two oscillator circles with radii 1 and 0.55 over 0<=t<=2pi; four hyperbolic branches with x0,xi0=+-0.65 over -0.9<=t<=0.9; the labelled (x2,xi2) projection (-t,t^2/2) of the full orbit (-t^3/6,-t,1,t^2/2) for -2<=t<=2.",
    "font_outline_notices": [
      "figures/notices/LICENSE_DEJAVU.txt"
    ]
  },
  "new_companion": {
    "source": "spectral-algebra-and-contour-projections.md",
    "sha256": "e4e9adb5841062e20fe11469f471662c8e20c5d556f3f741a6c4cad959ae9b61",
    "complete_proofs": "A0–A9",
    "finite_checks": {
      "script": "check_spectral_algebra.py",
      "results": "spectral-algebra-checks.json",
      "passed": true,
      "count": 2
    }
  },
  "new_figure": {
    "path": "figures/common-upper-contour.svg",
    "sha256": "86462fa95b13bbea56b2085c602d00884c5c5fceb85091394520ef4f531b5243",
    "source": "figures/draw_common_upper_contour.py",
    "source_sha256": "12b1c18729df240d90648ee45f597263805055bc2b258df7a6202e369eb93d69",
    "constants": {
      "M": 2,
      "delta": 1,
      "center_imaginary": 5,
      "radius": "9/2"
    },
    "scope": "Exact admissible regions and common contour, with a magnified panel; no particular eigenvalues are asserted.",
    "font_notice": "figures/notices/LICENSE_DEJAVU.txt"
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_fourteen_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the full original U027 lesson, all fourteen solutions, all 71 target entries, historical preparation, original figure source, seven-check verifier and all eight preparation model families. Retained every original display and complete solution.",
    "Verified the exact admitted H III register entry and PDF hash; read current PDF 336–343 from Section 21.5 to its concluding non-strict decomposition. The old preparation page offset was not reused.",
    "Consulted the shared AN-04 source map, existing proof library and bridge leads before writing. The free positive-canonical-relation and analytic minicourse routes remained leads, not unread substitute proofs. Reused the actual current diagonalization, inertia, complex arithmetic and symplectic algebra proofs.",
    "Checked the critical-point derivative under coordinate changes, exact Hf/2 factor, polarization, all Hessian blocks, skew identity and full congruence/conjugacy relation.",
    "Checked the generalized eigenspace power argument, opposite nondegenerate pairing, dimension equality and both symplectic and quadratic orthogonality. Added the complete elementary spectral foundation rather than assuming diagonalizability.",
    "Checked the negative-index obstruction away from the radical, exclusion of non-real quartets, real generalized dimension one and normalization of the hyperbolic block.",
    "Checked every imaginary eigenplane pairing and sign, exact oscillator form, invariant orthogonal-complement induction and semisimplicity even at repeated real-form frequencies.",
    "Checked both signed rank-one nilpotent planes, zero symplectic planes and exhaustion. Checked the residual exact sequence, one-dimensional kernel, even chain length, N=4 bound, cubic sign, corrected generator and all pairings of the full canonical chain basis.",
    "Checked the complete nonnegative and index-one classification, exact dimensional constraints and sufficiency as well as necessity of all spectral, Jordan and inertia invariants.",
    "Checked the complete oscillator shear/scaling and one-form identity, full nilpotent flow and conserved form, exact projection labels and the distinction between F and the full Hamilton field 2F.",
    "Checked the sector hypothesis retains P>=0 when C=0, real radical from polarized small complex rotations, all conjugated kernel equivalences, F3-to-F2 implication, intrinsic zero space and exact spectral sector.",
    "Checked the actual real symplectic quotient Wperp/(W intersect Wperp), descended form and positive real part, complete nonzero spectral isomorphism and induced Hamilton map.",
    "Added a full elementary complex polynomial root proof using a compact minimum and the first nonconstant term, finite factorization, characteristic identity by cofactor coefficients and explicit polynomial Bezout projections.",
    "Added an invariant nilpotent chain projection and complete complementary-chain induction, including the number of chains and all length invariants. Checked the formula on a nonorthogonal six-dimensional mixed-chain model.",
    "Added direct circle moments using uniform geometric series and exact primitives, the finite nilpotent resolvent formula and the full oriented generalized spectral projection.",
    "Replaced the unbound continuity-of-roots assertion by compactness and determinant continuity. Proved a uniform imaginary gap, constructed an explicit circle with center i(M^2/delta+delta) and radius M^2/delta+delta/2, and proved all inclusions and continuous projection ranges.",
    "Checked half-dimensional upper spectral isotropy, the real oscillator starting plane, open and closed strict positivity through Jordan degeneracies and the full Hermitian numerical range in the sector.",
    "Checked the real-part bijection, complex-structure sign, symplectic compatibility, positive metric and unique Hermitian form with its necessary factor one-half. Supplied full Hermitian Cauchy–Schwarz and strict-positivity openness.",
    "Checked the full adapted real canonical basis for a given real Lagrangian and a positive complex plane, actual Gram–Schmidt construction and exact complex polynomial ideal argument.",
    "Checked arbitrary-coordinate strict transversality, symmetric graph equivalence, full real canonical shear/scaling, the positive square root and its uniqueness, and the entire complex-structure and metric matrices.",
    "Checked the non-strict nullspace equals the complexification of the real isotropic part, all quotient dimensions and forms, strict quotient positivity and full orthogonal direct sum, including real and strict endpoints.",
    "Checked the actual nonsemisimple positive Hamilton example, all attained points of its numerical-range disk, the complete Jordan homotopy and full flow. Rechecked the counterexamples, sector failures, complete graph metric and all fourteen solutions.",
    "Retained seven original model checks and all eight preparation families. Two independent added checks verify the mixed-chain projection and exact Bezout splitting, with an independently integrated circle resolvent. These finite checks supplement the complete written proofs."
  ],
  "changes": [
    "Retained the full real classification, all sectorial kernel and quotient assertions, strict positive spectral geometry, compatible metric, arbitrary-coordinate graph theorem and non-strict decomposition.",
    "Reconnected actual finite algebra, compactness, real diagonalization, inertia, symplectic and complex arithmetic proofs.",
    "Added complete polynomial-root, characteristic identity, Bezout decomposition and invariant Jordan chain proofs.",
    "Proved the exact contour integral by geometric series and finite nilpotent resolvents, and proved the common uniform gap and explicit enclosing circle by compactness.",
    "Wrote the full Hermitian null-vector and Cauchy–Schwarz proof, positive openness, root and Gram–Schmidt constructions, and convergent matrix flow.",
    "Replaced the unbound appeal to continuity of polynomial roots with the fully proved uniform compactness argument without restricting repeated eigenvalues or Jordan blocks.",
    "Retained all original displays and fourteen solutions, the original seven-curve flow figure, seven verifier checks and all eight preparation families; added the exact common-contour illustration."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}