{
  "title": "Simultaneous reflections and flat corrections — restored proof edition",
  "original_source_sha256": "a9c6cc236c6a01db3a445e9943506f9e9f2646d5c5ada859bf529a17ca5dcd96",
  "restored_source_sha256": "f2f8251c913b455633a6f88882b5964257f66b1fc97e55ae42c14a1fbfc35aaa",
  "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": [
        293,
        509,
        510,
        511,
        512,
        513,
        514,
        515
      ],
      "relevant_printed_pages": [
        278,
        494,
        495,
        496,
        497,
        498,
        499,
        500
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "de857d3b2825b0b3c9d39680e25bab8ceb8d5cae271856b29505ea22abc3f3ca",
    "results": "model-check.json",
    "original_script_sha256": "de7bf14fcbc90954e9ac1241a0e525c577c514711fe935e1dc5a4f94bbc46975",
    "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/two-reflections-and-shear.svg",
    "sha256": "2b8ec1765cae275c65b2700035a92cde633aaa5fdb0c3b7f2219b8392211d1a0",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_two_reflections.py",
    "original_script_sha256": "efb157765e13ade49bea8455f2111ca41c01e27ee44e05b059e9d57e1aa37d84",
    "portable_script_sha256": "d53639b1c8f66a00a42c39b0ec69819657a7ca4aacd23e7a3f1b54606ca130b7",
    "license": "CC0-1.0",
    "coordinates": "Reflection vectors (0,1) and (-1/2,1); P=(0.15,0.9), gP=(1.05,-0.9), fgP=(1.05,0.9); arrows depict maps, not flows",
    "font_outline_notices": [
      "figures/notices/LICENSE_DEJAVU.txt",
      "figures/notices/LICENSE_STIX.txt"
    ]
  },
  "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 entire preserved U015 lesson, all nine complete solutions, both panels and exact coordinate caption of its figure, and all six reproducible finite checks. Original displays, exercises and SVG bytes remain unchanged.",
    "Checked the common fixed hypersurface, distinct transverse reflection lines, exact two-dimensional eigenvectors and opposite ordered shear products. The ordinary theorem is a smooth germ theorem with no symplectic conclusion.",
    "Checked the nonzero tangential vector field, its transverse flow chart, explicit exponential and integral solutions of all first-normal equations, invertible coordinate differential and unequal error orders.",
    "Supplied the weighted-remainder composition proof using parity and integral division. Checked even-order cancellation, odd inverse remainder, both conjugation signs and the two homological integrals.",
    "Made stabilization of every finite normal jet and its inverse explicit on a common tangential box; no convergence of coordinate diffeomorphisms is assumed. Checked the local parameter Borel construction, cutoff derivative powers, summable tails, exact jets and parity. Supplied the FTC proof of termwise smooth differentiation.",
    "Checked the actual realized coordinate map and every mixed flat jet after conjugation. Supplied the integral Taylor remainder giving uniform arbitrary-power derivative estimates after compact extension.",
    "Checked the exact summed recurrence for the nilpotent shear, its linear matrix-power growth, all repeated-chain-rule losses, the normal-coordinate and derivative bootstrap and its strict improvement for each fixed derivative order.",
    "Checked finite escape for both signs of the normal variable, monotone steps, the O(1/abs(s)) count and exact no-return translations. Added the locally uniform escape index that justifies differentiating the series without differentiating a stopping time.",
    "Checked every mixed derivative of the correction, smooth flat extension by difference quotients and exact telescoping. Made the common initial domain for y and delta(y), and the restriction back to the original germ, explicit.",
    "Checked both parity averages, spectator and normal coordinate corrections, preservation of flatness under composition and inverse, the final tangential defect sign and exact simultaneous coordinate identities.",
    "Checked the invariant transverse slice and its retained reflection lines. Read and restored the source definition of conic manifold, then proved the unique smooth product along a full positive ray using an equivariant cone chart, a positive linear radius and the inverse theorem. Checked every homogeneous degree and the independent radial hypothesis.",
    "Read the exact approved H III treatment at printed 494–500, PDF 509–515, and the conic definition at printed 278, PDF 293. The old PDF 505–511 locators belong to a different pagination and are retained only in historical evidence.",
    "Compared the source argument with this independently written course lesson. The complete parameter realization, summed recurrence, all derivative/domain details, exact examples, nine exercises and original figure supply the lesson exposition. No book text, figure, exercise collection or book file is redistributed.",
    "Retained all six finite exact/numerical checks with destination-only changes, and preserved the exact figure with its reproducible source and both embedded-font notices. Finite calculations support the written proof and do not certify a general smooth germ."
  ],
  "changes": [
    "Restored all original displays, nine complete solutions and exact two-reflection diagram with current prerequisite bindings.",
    "Added explicit first-coordinate flow formulas, weighted smooth-remainder composition and finite-jet stabilization arguments.",
    "Justified smooth Borel-series differentiation by compact FTC, arbitrary-power flat bounds by Taylor remainder, and local finite stopping times without differentiating the stopping index.",
    "Made the common telescoping domain and flatness under the final coordinate changes explicit.",
    "Restored the source meaning of conic manifold and proved a unique smooth product on the full positive saturation of a transverse slice.",
    "Retained all six finite calculations and the exact SVG; made script destinations portable and retained both embedded-font notices."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}