{
  "title": "Reducing Gaussian symbols and composing linear relations — restored proof edition",
  "original_source_sha256": "2e612bb7c788105b47372304896327b97e3eb377922552f9b8d06d44346622d2",
  "restored_source_sha256": "156a026a16e385ff975552529254ca6a3b0571feb0281d257416e90cf1e07895",
  "original_authorship": "GPT-6.1 Sol (OpenAI), Ultra, September 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": [
        351,
        352,
        353,
        354,
        355,
        356,
        357,
        358,
        359,
        360
      ],
      "relevant_printed_pages": [
        337,
        338,
        339,
        340,
        341,
        342,
        343
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_six_exercise_solutions_unchanged": true,
  "proof_review": [
    "Checked the transpose rank calculation, isotropy and quotient nondegeneracy for every real quadratic phase, including zero-dimensional phase or base blocks.",
    "Checked compact radical testing, projection translations, complementary sections and the density exact sequence. The fixed original phase-variable count gives exactly the excess factor (2pi)^(-e/2).",
    "Checked both positive density sequences and the perfect symplectic pairing, including lift independence and the full radical density.",
    "Completed the adapted symplectic basis with explicit vertical corrections that keep every prescribed horizontal constraint vector fixed.",
    "Checked the restricted critical equations, the exact isomorphism between the phase radical and lambda intersect Delta, and the reduced Lagrangian image.",
    "Checked phase elimination, unequal variable counts, translations by base variables, adapted symmetric shears, triangular base changes and inverse constraint half-density cancellation.",
    "Replaced the insufficient eigenvalue sentence by a smoothness proof valid through phase-Hessian rank changes. Fixed-size input phases, constant-rank kernel minors, radical quotients and transverse full-Hessian frames supply all steps. Read the existing Z2 Fourier-test proof of smooth Gaussian frames.",
    "Checked contained-constraint normalization and the zero-intersection common-horizontal construction, including the eta-map rank, transverse lifted plane and invertible B11 block.",
    "Checked the Schur complement, negative Fresnel signature, determinant half-density, all powers of 2pi and eighth-root coefficients between distinct-dimensional Gaussian frames.",
    "Completed staged reduction for every isotropic flag: simultaneous adapted coordinates, nested radicals, identical final quotient phase, telescoping normalization ratios, exact radical densities and constraint half-densities. No divergent-integral interchange is used.",
    "Checked linear canonical composition with the reflected middle symplectic form, diagonal constraint, symplectic volume contraction and exact (2pi)^(-e/2) coefficient.",
    "Checked graph transport, unique matching-space decomposition and both density and Maslov identity units. All six original solutions and every original display equation are unchanged.",
    "Read the admitted HIII radical/reduction/composition treatment and compared the actual mathematics and exposition. This lesson retains independent explanations, its density-first order, explicit calculations and six original exercises. It distributes no book text, figure or exercise.",
    "Read and retained the existing C1/C4 matching-density and G1-G5 quadratic/Maslov proofs; stated their exact normalization and phase-frame relationship rather than duplicating them."
  ],
  "changes": [
    "Exact current programme proof links and approved-edition bibliography.",
    "Explicit symplectic-basis completion preserving all prescribed constraint vectors.",
    "Smoothness proof through changing phase-Hessian rank using full transverse test Hessians and smooth radical quotients.",
    "Full staged-reduction proof for an arbitrary isotropic flag using nested radicals and equality of quotient phases.",
    "Explicit normalization and phase-frame comparison with the existing composition alternatives.",
    "Original authorship, every original displayed equation and all six exercise solutions retained."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}