{
  "title": "Recognizing a Lagrangian distribution intrinsically — restored proof edition",
  "original_source_sha256": "fbc79bd559a47881b260b342e094ca862dd5fc71afcd6969fad4434988c7711e",
  "restored_source_sha256": "08a0386104e6d1d866e33b76a287825d881e3d2d90036ae65eb31876a03abf29",
  "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": [
        15,
        16,
        17,
        18,
        19,
        20,
        21
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "all_original_displayed_equations_unchanged": true,
  "first_four_exercise_solutions_unchanged": true,
  "fifth_exercise_retained_and_microlocal_obstruction_completed": true,
  "proof_review": [
    "Checked the L2 extension, exact B(2,infinity) endpoint and ordinary differentiated restriction ideal. CH T3 covers both base coordinates and arbitrary smooth bundle frames, including half-densities.",
    "Checked matrix commutators without assuming scalar order drop. K3–K7 supply the fixed conic inverse, finite cosphere reconstruction and actual Besov membership of each finite word.",
    "Checked proper real regularization of the graph phase, compact kernels, all balanced words, their exact Fourier gauge and annular rescaling. P1 B2 supplies every pointwise derivative bound.",
    "Checked the converse for full ordinary matrix symbols, compact input/output cutoffs, Taylor division on the graph and repeated Fourier reduction with every differentiated remainder.",
    "Checked general base coordinates and smooth remainders. Existing GEO C3–C4, CH T3 and SB S3 prove the exact coordinate and Sobolev inputs.",
    "Checked proper homogeneous pullback including the low-frequency support condition and zero extension, using HT H2/H5.",
    "Checked both prescribed-phase right inverses, normal determinant degrees, normalized compact fiber densities, support-preserving summation and full clean remainders. The preserved excess-dependent prefactor differs by the explicitly stated constant from the earlier companion.",
    "Checked the strict Sobolev gain for every derivative, including t<=0, and the finite Fourier split at K>|beta|+n/2.",
    "Checked the point mass and all five solutions: matrix commutator, balanced-word polynomial, clean normalization, and the original R_j=2^(j^2) endpoint bump sequence. Completed the precise microlocal endpoint obstruction at (0,e1).",
    "Read the exact approved Hörmander IV pages 4–10 and compared the definition, localization, graph criterion, Sobolev gain, pullback and both phase converses. The lesson provides independent explanations and full programme proofs; no book text, figure or exercise is distributed."
  ],
  "changes": [
    "Replaced broad AN-03 citations by exact current programme providers.",
    "Specified differentiated S0 bounds on the restricted ordinary symbol.",
    "Explained the constant conversion between the two clean-phase conventions.",
    "Completed the endpoint counterexample at the exact covector (0,e1).",
    "Cited the exact edition, retaining original authorship."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}