{
  "title": "Gaussian lines, densities and invariant symbols — restored proof edition",
  "original_source_sha256": "b3a4345894097600e659f566a449b18eea9c2e13f78a0d7582ecf7df48bf1dc6",
  "restored_source_sha256": "6c5e711f9226611203204986041252582e08d2f37d5e6514be04376aa5485576",
  "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-978-3-642-00136-9-pdf",
      "pdf_pages_read": [
        24,
        25,
        26,
        27
      ]
    },
    {
      "edition_id": "ISBN-978-3-540-49938-1-pdf",
      "pdf_pages_read": [
        344,
        345,
        346,
        347,
        348,
        349,
        350,
        351
      ]
    }
  ],
  "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_five_exercise_solutions_unchanged": true,
  "proof_review": [
    "Checked affine Lagrangian charts, the power-series inverse and the Hausdorff/second-countable topology through orthogonal projections.",
    "Checked Schur-complement intersection strata, all endpoint ranks, compact graph covers of vanishing volume and the countable-subcover argument for simultaneous complements.",
    "Checked Gaussian-line uniqueness before invoking Fourier transformation: compact test division gives a simple layer, compact marginal primitives prove the zero-gradient distribution is constant, and the resulting Gaussian layer is tempered.",
    "Checked chirp signs, maximality of the annihilating system, dual-basis half-density factors, the full Gaussian transition matrix, determinant cancellation and even signature.",
    "Checked the exact integer cocycle, perturbation of singular B with all transverse intersections retained, pair reversal and cross-pair antisymmetry.",
    "Checked geometric Maslov gluing in adapted symplectic bundles and its identification with the existing phase-chart convention, including its dimension-dependent frame factor.",
    "Checked the direct quadratic critical-space bijection and isotropy, including no phase variables; Gaussian/delta evaluation, critical-density independence and the signature-difference proof with singular full Hessians.",
    "Checked conic half-density degree, ordinary symbol classes and the precise n/4 shift.",
    "Repaired symbol localization to a cutoff with full symbol one modulo smoothing on the smaller cone. Checked compact Fourier comparison with all derivatives, nonlinear cotangent transport and its extra base Hessian, and arbitrary bundle frames.",
    "Read the actual PS3–PS6 proofs of the local lower-order criterion, supported amplitude lift, base-locally-finite partition and global surjectivity/kernel. Their closedness and manifold hypotheses are stated explicitly in the restored lesson.",
    "Checked clean fiber reduction with the excess-dependent prefactor, all determinant/fiber/density degrees, and the corrected normalization cross-reference.",
    "Checked covector reflection and conjugate Maslov transitions for arbitrary smooth antilinear maps, including noninjective and unequal-rank maps. All five exercises and solutions and every original displayed equation are unchanged.",
    "Compared the admitted Hörmander III and IV treatments. Preserved independent explanation and exercises, accurate source credit and existing component rights. No book text, figure or exercise is distributed."
  ],
  "changes": [
    "Exact current programme proofs and explicit global-manifold hypotheses.",
    "Power-series justification of analytic charts and the Hausdorff second-countable projection topology.",
    "Complete countable null-volume argument for common complements.",
    "Explicit compact test division and marginal primitives proving Gaussian-line uniqueness without assumed temperedness.",
    "Repaired principal-symbol localization: full symbol one modulo smoothing on a smaller cone.",
    "Bound the global construction to the actual PS5 base-locally-finite partition.",
    "Corrected clean-formula cross-reference and stated its precise normalization.",
    "Approved-edition bibliography; original authorship and every original displayed equation and exercise retained.",
    "Proved the quadratic critical-space bijection directly, including an empty phase block."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}