{
  "title": "Fold amplitudes and critical densities — restored proof edition",
  "original_source_sha256": "3ac921ff57f216f2927850c610077ad958b36767d91f3869627d2cb0a3efad3f",
  "restored_source_sha256": "6639aaa65967c3b76b3636d839b554b06e970021b488aa95e429872f2e7816da",
  "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": [
        43,
        44
      ],
      "relevant_printed_pages": [
        32,
        33
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "cdb08f87fedd9d47f40e63cbb7a0714100a121dcd2b9f25f3f6b11e9c08304d0",
    "results": "model-check.json",
    "original_script_sha256": "6fd25e77388dfae7eb89c9f590a83620d6398ed44dd0e7ae2c850be5a73b0531",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "historical_archive_checks_removed": true,
    "removed_historical_check_flags": [
      "prior_edition_local_history_checked",
      "obsolete portability statement"
    ],
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "figure": {
    "path": "figures/fold-critical-density.svg",
    "sha256": "b8d93cdf10eeb37eb2599490bf52de57d6823cb548071c9676d4173a0a509d91",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_fold_critical_density.py",
    "original_script_sha256": "36a8da4fc334d7493b3da329d7125e8c6cb1d30b7c3ef7812e76039fd37ab412",
    "portable_script_sha256": "4617dbe8ca57151ee529cac813e8ea0737b864b90621edbd84095e38797ae8d1",
    "license": "CC0-1.0",
    "coordinates": "Exact diagram: ambient radial Jacobian rho, constraint determinant rho^-1, critical density rho^2 and cancellation of amplitude rho^-1 with its positive half density",
    "font_outline_notices": [
      "figures/notices/LICENSE_DEJAVU.txt"
    ]
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_ten_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the full original U019 lesson, all ten complete exercise solutions, its exact density diagram and all six mathematical model checks. Original displays, solutions and SVG bytes are unchanged.",
    "Read the exact H IV, corrected second printing (1994), printed pages 32–33 / PDF 43–44, using this edition’s pagination.",
    "Consulted the existing fold-density target inventory and shared AN-04 source map, including H FIO I critical densities and Melrose–Taylor fold decomposition leads. Reused exact current programme proofs and the earlier phase/order companion; did not adopt an unchecked external lead.",
    "Checked both directions of the ordinary-symbol equivalence under tau=rho s with all mixed derivatives, compact degree-zero support, comparable homogeneous frequencies and exact 2n-base/n+1-phase normalization.",
    "Checked all independent raw critical constraints, their full normal determinant and positive quotient density. Supplied the normal-coordinate proof for variable-dependent constraint matrices.",
    "Checked the complete homogeneous constraint transformation, fixed-tau radial correction, ambient and inverse-normal radial factors and exact amplitude/half-density cancellation.",
    "Checked every mixed derivative under restriction to the critical set, including rho factors from s derivatives. Verified ordinary scalar order m+1/2, positive frame degree (n-1)/2 and total symbol order m+n/2.",
    "Checked reduced-phase critical independence, nonzero full differential, both covector signs, critical dimension, whole relation embedding and exact density through the fold.",
    "Checked the actual degree-one auxiliary diffeomorphism and inverse. Specified determinant +1 for ordered pairs (xi1,tau) and (v,u), and absolute determinant one for either ordering; verified exact hyperbolic phase identity, zero signature and Gaussian radial factor.",
    "Proved exact distributional first-frequency Fourier elimination by pairing with a compact test, bounding remaining frequencies, applying Fourier inversion and then using arbitrarily rapid partial Fourier decay to remove the remaining cutoff.",
    "Checked the omitted first-frequency tail with a fixed compact s support, uniform nonstationary denominator and comparability of full frequency to xi1. Every differentiated integral has arbitrary negative order after sufficiently many transposes; the full tail kernel is smooth.",
    "Checked the conically truncated amplitude has the claimed full ordinary-symbol order and equals the reduced coefficient on the entire critical support.",
    "Checked the full Gaussian principal-symbol isomorphism and exact lower-order kernel, including the shared Maslov frame and every local symbol class. No citation replaces the symbol theorem.",
    "Made the all-derivative rapid decrease of localized graph coefficients off wavefront explicit using compact polynomial multipliers, the actual W1 convolution argument and K7 cutoffs. Checked finite graph partition, homogeneous coordinate/density changes and fixed support at each residual step.",
    "Checked the complete lower-order residual induction, actual support-preserving K1 summation in (x,s;xi-prime), every remainder order and intrinsic all-order smoothness. Linked the exact complete earlier E6 provider instead of relying on its old remote label.",
    "Checked the operator formula on compact smooth inputs is absolutely convergent, and a compact base cutoff equal to one near the entire critical support changes only a smooth kernel.",
    "Checked the explicit elliptic coefficient and both numerical order examples without claiming any Airy or continuity estimate from those order counts.",
    "Retained six mathematical checks unchanged, removing only obsolete history/archive assertions. Preserved original evidence, exact density diagram, portable drawing source and DejaVu notice. Finite models support the written argument and do not certify arbitrary-symbol representation."
  ],
  "changes": [
    "Restored the full normalized amplitude and density proof, reduced homogeneous phase, all-order first-frequency tail and complete reduced-amplitude representation.",
    "Reconnected exact current canonical-model, symbol, microlocal localization, Fourier and summation providers; retained earlier alternative phase-order proof.",
    "Explained the critical-density constraint transformation for variable-dependent matrices directly in normal coordinates.",
    "Made nonzero reduced phase differential, critical-set dimension and full relation embedding explicit.",
    "Specified the ordered auxiliary Jacobian: determinant one for (xi1,tau) to (v,u), absolute determinant one for either ordering.",
    "Proved the distributional first-frequency elimination with compact test functions and uniform rapid Fourier decay before removing the remaining cutoff.",
    "Made all-derivative rapid decrease of microlocal symbol representatives, finite graph partition and fixed support under recursion explicit.",
    "Preserved all original displayed equations, ten complete solutions, exact figure and six mathematical checks; removed only obsolete historical archive assertions from the portable verifier."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}