{
  "title": "Positive Lagrangian ideals and distributions — restored proof edition",
  "original_source_sha256": "090502674cf4b3ce37806f5b27ecee15a9eaba5459d3747114c2562a88a2e96f",
  "restored_source_sha256": "5477f8105fa2a8d3ce17b928ac5555e86b4a7283d73a8a5ed794f59784766af8",
  "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.",
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  "original_lesson_preserved": true,
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  "original_lesson_licence": "CC0-1.0",
  "earlier_complete_revision": {
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    "rights_record_sha256": "70343ee8c9f8daed62d14a145acb381e534b761abfba0fb40df28cf53dc7ed2c",
    "scope": "Preserved U028 revision and full original nine-section companion; current proofs and prerequisite bindings reviewed afresh."
  },
  "historical_comparisons": [
    {
      "title": "Melin–Sjöstrand, Fourier intégraux à phases complexes, 1974 seminar exposition",
      "url": "https://www.numdam.org/item/SEDP_1973-1974____A12_0/",
      "scope": "Comparison credited in the earlier companion; the complete smooth theorem is proved here, not imported from its sketch."
    },
    {
      "title": "Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982)",
      "url": "https://www.numdam.org/item/AST_1982__95__R3_0/",
      "scope": "Holomorphic comparison credited in the earlier companion; the almost-analytic defect estimates are proved here."
    }
  ],
  "reproducible_model_checks": {
    "script": "check_models.py",
    "results": "model-check.json",
    "finite_groups": 7,
    "algebra_families": 8,
    "algebra_script": "check_algebra.py",
    "uniform_division_script": "check_uniform_division.py",
    "uniform_division_results": "uniform-division-checks.json",
    "uniform_division_groups": 3,
    "portable_changes": "Input/output paths only in the main and division verifiers; unused historical source and state branches omitted from algebra preparation.",
    "all_original_mathematical_controls_retained": true,
    "passed": true
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  "added_foundation_checks": {
    "script": "check_contour_foundations.py",
    "results": "contour-foundation-checks.json",
    "groups": 4,
    "passed": true,
    "scope": "Exact nonholomorphic determinant identity, noncommuting matrix-root derivative, independent full Gaussian quadrature and branch control, and all M1–M7 model identities."
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  "analytic_companion": {
    "source": "complex-stationary-contract.md",
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  "added_foundations": {
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    "proofs": "A0–A4"
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  "linked_component_rights": "All earlier proof components retain their individual notices and licences, including the complete inherited AN-03 GFDL notices.",
  "all_seventeen_original_exercise_solutions_unchanged": true,
  "original_display_typography_correction": "Only (5.12): replaced an accidental comma in the conjugate-transpose exponent by a thin space; all mathematical values and other original displayed equations unchanged.",
  "proof_review": [
    "Retained the complete preserved U028 revision, its full analytic companion, seventeen unchanged solutions and both original figures.",
    "Bounded all complex division, graph, flat-ideal, geometry, spectral and symbol inputs to actual current programme proofs.",
    "Supplied exact real base transversals, tensor interpolation, the complex critical-image Lagrangian proof and the fully homogeneous partial Legendre block.",
    "Proved complex symmetric square factorization and every noncommuting matrix-series derivative.",
    "Proved oriented contour deformation from an exact determinant identity and the fundamental theorem, with all lateral-boundary and parameter estimates.",
    "Proved the full matrix Gaussian by integration-by-parts moments and a path ODE, including Abel endpoints, orientation and the normal-phase block.",
    "Supplied full inverse-jet and residue algebra, retained the correct absolute parameter losses, and made representative-dependent constants explicit.",
    "Corrected an overbroad local smoothness sentence to the actual microlocal conclusion with the exact Fourier-cutoff proof.",
    "Retained the complete uniform symbol-ideal qualification and counterexample, all symbol orders, absolute remainders and the same-support converse.",
    "Read the complete original main lesson and complete nine-section analytic companion, every exercise and model, all 93 target entries, both figure sources and every finite mathematical control.",
    "Read all of exact admitted H IV Section 25.4 and the listed current H I division, flatness, complex stationary/nonstationary, oscillatory and wavefront pages; the current H I carrier offset replaces its historical preparation offset.",
    "Checked multivariable Schur induction, graph generation, common bounded patches, almost-analytic jets, positive path-error sign and smooth zero extension across singular real sets.",
    "Checked the ideal-power interpolation at i, all mixed derivative flatness, exact real transversals, Euler degree and full parameter elimination.",
    "Checked the quantitative critical-value inequality with semipositive imaginary Hessian, matrix perturbation bounds, actual isotropic critical image and radial nullspace.",
    "Checked the two complete contour homotopies, invertibility neighborhoods, uniform positivity, both lateral boundaries and every fixed parameter derivative.",
    "Checked all Gaussian Taylor moments, inverse-coordinate jets, residue representatives, leading branch and absolute differentiated remainder orders.",
    "Checked every Mellin rescaling coefficient, output/input comparison cone, all symbol derivatives and the uniform ideal qualification.",
    "Checked the whole dyadic oscillatory extension and separated-cone wavefront proof, determinant degree, two-pi factors, absolute Fourier comparison and full same-support converse.",
    "Checked every original model and all seventeen complete solutions, including the genuinely nonuniform smooth-ideal counterexample.",
    "Seven original verifier groups, eight algebra families, three uniform-division controls and four added foundation/model controls pass. Finite checks supplement rather than prove the general arguments."
  ],
  "changes": [
    "Retained the complete preserved U028 revision, its full analytic companion, seventeen unchanged solutions and both original figures.",
    "Bounded all complex division, graph, flat-ideal, geometry, spectral and symbol inputs to actual current programme proofs.",
    "Supplied exact real base transversals, tensor interpolation, the complex critical-image Lagrangian proof and the fully homogeneous partial Legendre block.",
    "Proved complex symmetric square factorization and every noncommuting matrix-series derivative.",
    "Proved oriented contour deformation from an exact determinant identity and the fundamental theorem, with all lateral-boundary and parameter estimates.",
    "Proved the full matrix Gaussian by integration-by-parts moments and a path ODE, including Abel endpoints, orientation and the normal-phase block.",
    "Supplied full inverse-jet and residue algebra, retained the correct absolute parameter losses, and made representative-dependent constants explicit.",
    "Corrected an overbroad local smoothness sentence to the actual microlocal conclusion with the exact Fourier-cutoff proof.",
    "Retained the complete uniform symbol-ideal qualification and counterexample, all symbol orders, absolute remainders and the same-support converse."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}