{
  "title": "Two reflections in folded symplectic coordinates — restored proof edition",
  "original_source_sha256": "70f5e7b742725a6d1cc3fdc29dd4f82ea693d1c0672cc87cfff73d1bb4506c05",
  "restored_source_sha256": "1c619a26fe49895aabc17e3d14ce9ed74748364bd4111159adf71eab572c0e9d",
  "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": [
        322,
        323,
        324,
        331,
        332,
        333
      ],
      "relevant_printed_pages": [
        307,
        308,
        309,
        316,
        317,
        318
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "0dfe29c62e5e5c3702e2b46bd0031894e8ea79a10554fb62c2dd9ee8b51202ae",
    "results": "model-check.json",
    "original_script_sha256": "068df9f239b5066163b18d4bf515910ce02dde56379532306181944d5f5ad557",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "historical_archive_checks_removed": true,
    "removed_historical_check_flags": [
      "prior_lessons_verified_unchanged",
      "prior_lessons"
    ],
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "figure": {
    "path": "figures/homogeneous-fold-shifts.svg",
    "sha256": "fbac45f1bfc22ce8c6e00d44058124ea5a8411bcb4842e3fe391fba1d80e4221",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_homogeneous_fold_shifts.py",
    "original_script_sha256": "bcfb377135a89b29abf0640337c637a9672425661bf3c71372793da19e8e0859",
    "portable_script_sha256": "069ea3d661a2ac5a5925f9ed56d96b558e8ddb0605e174dc6e654c1d442a0431",
    "license": "CC0-1.0",
    "coordinates": "Ratio t in [-1,1]; exact fixed-frequency shifts V=2t and W=-2t^3/3, coefficient cancellation 2t^2-2t^2=0; marked t=1/2 gives shifts 1 and -1/12",
    "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 preserved U017 lesson, all ten complete solutions, both panels and the exact caption of its shift/cancellation figure, and all six reproducible mathematical checks. Original displayed equations, exercises and SVG bytes are unchanged.",
    "Checked the simple top-wedge zero, restricted-rank condition, distinct reflection lines, ambient radical and characteristic line. Both common-invariant differentials and the simultaneous tangential direction are identified intrinsically.",
    "Checked the exact Hamiltonian signs and smooth extension criterion. Supplied integral division with every parameter derivative, proving tangency and the full limiting field on the fold.",
    "Checked actual commuting-flow charts and their inverse differential, unique solutions of the constant equations, and all symmetry conditions. Corrected the general symmetry statement to include the equations and right-hand sides, and proved the exact weighted pullback law.",
    "Checked the full canonical-plane splitting and its sign on the fold, smooth contraction identities, form preservation and preservation of the critical hypersurface by actual flows.",
    "Checked the entire spectator induction: full boundary differentials, invariant seeds, zero cross brackets, transverse sections, retained fold values and nonzero equation transported by commuting flows.",
    "Checked the odd normal coordinate and its full smooth characteristic field, first-position equation and parity, complete coordinate differential and bracket inversion on both sides, including the empty-spectator case n=1.",
    "Checked the one-variable residual shift, fixed-set/oddness conditions and nonzero derivative. Checked the exact cubic integral, smooth even coefficient, signed real root for either sign, actual inverse, smooth quotient and full nonzero Jacobian.",
    "Checked homogeneous radial independence, marked single-involution chart, common invariant seeds and every field/function degree. The positive last momentum is retained without imposing a nonconic zero level.",
    "Checked the homogeneous normal coordinate and first position through independent commuting fields. Supplied the exact smooth transverse field and its normal value, all singular-commutator cancellations, parity and weight.",
    "Checked the three-field construction of the last position and every cross bracket. Proved the last-momentum differential on the one-dimensional conic section is nonzero using Rp_n=p_n. Checked the full half-to-full degree coordinate change and mixed form terms.",
    "Proved the full conic coordinate domain by restricting the local construction to p_n=1 and extending its coordinate functions with their exact degrees along the unique positive-ray product. The recovered radius and normalized point prove a full coordinate inverse.",
    "Checked both residual homogeneous shifts, their one-ratio dependence and the exact differential constraint. Proved necessity and sufficiency of the full form-preservation equations, not only a sufficient primitive identity.",
    "Checked the actual signed cubic inverse in the homogeneous correction, all smooth even quotients and its full Jacobian. The first shift is exactly 2t and the form equation forces the last shift -2t^3/3.",
    "Checked every original solution, including the genuinely nonlinear ordinary and homogeneous maps, negative-sign normalization, exact full conjugacy equations, true radial-exclusion counterexample and ordered products.",
    "Read the approved H III pages printed 307–309 and 316–318, PDF 322–324 and 331–333,. The historical plus-11 pagination is not used for this approved edition.",
    "Compared the source argument with the independently written lesson. The intrinsic reflection-line proof, explicit full-flow construction of both final homogeneous positions, parameter arguments, ten solutions and shift/cancellation diagram supply its teaching exposition. No book text, figure, exercise collection or book file is redistributed.",
    "Consulted the existing private target inventory, source-use map and shared AN-04 source map, including the Melrose–Taylor alternative route. Those source leads do not certify their proof coverage. Reused the exact existing flow, form, division, simultaneous-involution and folded-form providers instead of duplicating prerequisite lessons.",
    "Retained all six mathematical model checks unchanged; only obsolete historical edition/archive assertions were removed from the portable verifier with their exact preamble recorded. Preserved the original evidence, exact SVG, portable drawing source and embedded-font notice. Finite models do not establish the general smooth theorem."
  ],
  "changes": [
    "Restored the full ordinary and homogeneous simultaneous folded-form arguments, all original displays, ten complete solutions and exact shift/cancellation figure with current programme providers.",
    "Made smooth parameter division and actual Hamiltonian preservation through the fold explicit.",
    "Corrected the general symmetry statement to require preservation of the differential equations and right-hand sides; proved the exact weighted transformation law.",
    "Made compatibility of the spectator induction explicit through commuting propagation.",
    "Supplied complete smooth coefficients for the characteristic field and exact transverse normal value of the last auxiliary field.",
    "Proved the nonzero last-momentum coordinate on the conic initial section and the full coordinate extension along a positive ray.",
    "Proved necessity as well as sufficiency of the full homogeneous form-preservation equations.",
    "Retained all six mathematical model checks, removing only obsolete historical archive/edition assertions from the portable verifier. Original evidence, SVG bytes, drawing source and font notice are preserved."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}