{
  "title": "Folds, reflections and uniform smooth descent — restored proof edition",
  "original_source_sha256": "84a8ea1464d38071deaad014845503685abf75ad22d600c936c03ddf2ba28a3c",
  "restored_source_sha256": "237e28393aadc0687ff2dda4c0fac7446d544da5a3ca1f54aa7bba7ff9bd8309",
  "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": [
        507,
        508,
        509
      ],
      "relevant_printed_pages": [
        492,
        493,
        494
      ]
    },
    {
      "edition_id": "ISBN-978-3-642-00136-9-pdf",
      "pdf_pages_read": [
        44,
        45
      ],
      "relevant_printed_pages": [
        33,
        34
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "02f906c73f580f756703c23d62681224a34f21360512f3e109469e5f99fa40f6",
    "results": "model-check.json",
    "original_script_sha256": "332216b5a8d5ab9eae67cbe1b539295ed77db984b1e0d2d870efeb9484e16126",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "figure": {
    "path": "figures/fold-and-smooth-descent.svg",
    "sha256": "c4dbd3653580da7d63b5ed305d4a4d2d97b5fcc90a88bdfc550c63f25af35983",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_fold_descent.py",
    "original_script_sha256": "891d5544b455404496cff92c1ff8cda5bafc8a49c25d22bd55b10cf7562d4ee2",
    "portable_script_sha256": "bc08b0e4bfe58f4dab38b6393176b043efdf8385918f84a507b219205df853cb",
    "license": "CC0-1.0",
    "coordinates": "Exact normal slice r=s squared, two sheet points at s=-1 and s=1, reflection identification and unattained r<0",
    "font_files_redistributed": false
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_nine_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the full preserved U014 lesson, nine complete solutions, exact normal-slice caption and six finite model calculations. All original displays, exercises and SVG bytes are retained.",
    "Checked the intrinsic kernel-to-cokernel Hessian, coordinate-change terms, equal dimensions, ordinary fold normal form, exact Taylor factor and coordinate-invariant determinant test, including dimension one.",
    "Checked uniqueness of the smooth sheet exchange across the fold, its transverse reflection line, and exact parity coordinates for any smooth involution with hypersurface fixed set.",
    "Checked odd division, the iterated integral square-derivative operator, all mixed parameter estimates, actual one-sided derivatives and exact boundary Taylor coefficients.",
    "Supplied the complete finite interpolation proof from elementary polynomial factorization. Checked both infinite products, uniform rapid coefficient bounds and an explicit finite-head/uniform-tail proof of every infinite moment identity.",
    "Checked the locally finite extension formula off the boundary, every mixed derivative, summable uniform bounds up to zero, compact-uniform boundary limits, FTC gluing, finite seminorm control and unchanged parameter support.",
    "Checked invariant descent in both directions, full-neighborhood even/odd decomposition and nonuniqueness only on the unattained side. No convergent Taylor-series assumption is used.",
    "Checked the ordinary-symbol argument for every real order, fixed extension kernels commuting with external derivatives and the total derivative loss of homogeneous ratio restrictions.",
    "Read the existing U018 explicit homogeneous fold phase and U019 radial order proof before supplying the companion. Checked all n+1 independent phase-critical differentials, the exact relation through the fold, both directions of every radial symbol estimate, the actual distributional cutoff-limit identity and the normalized kernel order.",
    "Checked both fractional coefficient orders, exact negative-i representation and positive-i solving sign, converse construction and compact base support. Full Airy representation, its noncompact tails and its continuity theorem are separate scopes.",
    "Read the exact approved H III printed 492–494 at PDF 507–509 and H IV printed 33–34 at PDF 44–45. Corrected the old private H III PDF locators, which refer to earlier appendix material in this reprint.",
    "Compared the source mathematics with the retained independent exposition. The controlled extension, explicit finite seminorms, model order companion, examples, nine exercises and original coordinate figure are retained course exposition. No book text, figure, exercise collection or book file is redistributed.",
    "Retained the six finite checks without changing their mathematics and made script destinations portable. Exact symbolic models and finite numerical comparisons support the written proof; they do not replace its general arguments."
  ],
  "changes": [
    "Restored all original displays, nine complete solutions and the exact fold diagram with current prerequisite bindings.",
    "Proved the finite polynomial interpolation identity used by the extension coefficients and made the finite-head/uniform-tail moment limit explicit.",
    "Reused the preserved model to prove phase nondegeneracy, all radial symbol derivatives, actual cutoff-limit kernel equality and normalized order before the critical amplitude theorem.",
    "Retained the full controlled extension, every mixed parameter estimate, both Airy coefficient orders and the exact odd sign; did not claim the later Airy integral or continuity theorem.",
    "Accurate approved-edition citations, exact source-page mapping and independent exposition; all alternatives retained.",
    "Retained six finite model calculations and the exact SVG, changing only reproducible-script destinations."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}