{
  "title": "Canonical relations with two folding projections — restored proof edition",
  "original_source_sha256": "6145addee7a6f33c2fb6796f27aea75921a26289adfed589646e2b7bd91b94ec",
  "restored_source_sha256": "69518d9ee9afdcf7a965293c40bf3ae045ee2cc65eba1b7c1380e4b4afcd7e2b",
  "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": [
        325,
        333,
        334
      ],
      "relevant_printed_pages": [
        310,
        318,
        319
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "7a8c3338ea00ba55ab3288c503e02f328fcff1e3d8f97f583562f9761fa9760f",
    "results": "model-check.json",
    "original_script_sha256": "6ec1c3469deecfbd88c7240eab46f1058eeec2d1cc88843dc287b21743741c7c",
    "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/canonical-fold-sheets.svg",
    "sha256": "b9a780909eb19d111cb6c98e88182d8807c277714579b2d4f7fd5fb172278162",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_canonical_fold_sheets.py",
    "original_script_sha256": "16b734bd74a4160a43ddc3199095afee733b6656815c9fce028fcba410181b0b",
    "portable_script_sha256": "ee1f35c6f4129df2f95dd2cdd8dad72ca1775cc299216d26d0c12892dbb64858",
    "license": "CC0-1.0",
    "coordinates": "Fixed output x=0, rho=1: p=s^2 and input positions (s,-s^3/3); s=+/-3/4 share p=9/16 and give positions (+/-3/4,-/+9/64)",
    "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 entire preserved U018 lesson, all ten complete solutions, the exact two-sheet figure and caption, and all six mathematical model checks. Original displays, exercises and SVG bytes are unchanged.",
    "Verified the actual local embedded-branch reduction by choosing an invertible ambient coordinate minor and exhibiting a smooth graph inverse.",
    "Checked equal source/target dimensions, the signed Lagrangian identity, common closed pullback and equality of the two simple critical hypersurface germs through their top forms.",
    "Checked the fold restriction to each critical image and supplied a direct proof of the symplectic hyperplane radical using its annihilator and nondegenerate pairing. Verified the full restricted-rank and ambient-rank conditions.",
    "Checked that the two genuine projection kernels are transverse, independent by product immersion, and span the ambient radical. Checked uniqueness, form preservation and reflection lines of both actual fold exchanges.",
    "Checked the complete ordinary simultaneous source theorem, both full target converses, exact half-shift and square, whole-relation equality in both directions and dimension-two case. Checked the ordinary generating family and exact signed one-form, without claiming homogeneous phase properties for that ordinary family.",
    "Checked Cartan and projection-equivariant contraction give exactly equal pulled-back canonical one-forms. Proved the relation inherits an equivariant conic chart using a positive normalized pulled-back radius, transverse slice and its unique positive saturation.",
    "Checked nonzero common one-form is exactly radial independence from the two-kernel plane and forces n>=2. Dilation conjugacy and uniqueness prove both sheet exchanges homogeneous.",
    "Checked every hypothesis of the full homogeneous simultaneous form theorem, the degree-zero ratio, full half-degree normal coordinate and both homogeneous target converses with their marked values and parities.",
    "Checked the input first and last position changes are full coordinates and cancel all mixed frequency/normal form terms. Both target charts retain symplecticity on the unattained side through the exact earlier Hamiltonian extension proof.",
    "Checked the entire homogeneous relation, eliminated equations and smooth inverse recovering all parameters and sheets. No formal or tangent-only equivalence is substituted for the full target maps.",
    "Checked the ordinary-to-conic auxiliary change, every phase derivative at fixed tau, degree one, all n+1 independent critical gradients, nonzero full phase differential and exact critical dimension. The critical map is the actual product embedding with correct covector signs and zero critical value.",
    "Checked both full model one-forms, exact signed projection determinants, ranks and distinct kernel lines. Bound the determinant kernel derivatives to the proved intrinsic fold Hessian criterion; checked top-power constant, radial field and nonzero marked primitive.",
    "Checked both signed sheets and attained-side ranges, coalescence and exact fixed-output diagram values. The figure represents relation points and projection fibers, not trajectories.",
    "Checked the full excluded homogeneous relation, exact Jacobians equal to u, rank-three rows, transverse kernels, nonzero target covectors and common zero pulled-back primitive. Supplied its smooth product left inverse and the invariant obstruction to the marked model.",
    "Read the approved H III pages printed 310 and 318–319, PDF 325 and 333–334, after verifying the current exact registered copy and reading/citation permissions. Historical plus-11 page locators are not used for this approved edition.",
    "Compared source and independently written lesson: intrinsic projection geometry, full target-converse conditions, genuine degree-one phase, original excluded relation, ten solved exercises and exact two-sheet figure are all retained. No book text, figure, exercise collection or book file is redistributed.",
    "Consulted the existing private canonical-fold target inventory and shared AN-04 source map, including Melrose–Taylor and Greenleaf–Seeger leads. Reused the already proved flow, fold, form, target and phase providers; no new external lead is represented as an adopted proof.",
    "Retained all six mathematical checks unchanged; removed only obsolete historical edition/archive assertions from the portable verifier, recording the exact preamble. Preserved original evidence, exact SVG, portable drawing source and embedded-font notice. These finite models support but do not certify the arbitrary-relation theorem."
  ],
  "changes": [
    "Restored both complete two-sided fold normal forms, explicit homogeneous phase, every original display, all ten solutions and exact two-sheet figure.",
    "Reconnected exact simultaneous folded-form, full target-converse, fold, form and elementary calculus providers; linked the existing earlier homogeneous-phase companion.",
    "Proved the local embedded-branch reduction from an actual invertible coordinate minor.",
    "Made the hyperplane radical and restricted-rank proof explicit through the nondegenerate pairing.",
    "Constructed the relation conic chart from a positive pulled-back target radius and a unique normalized slice.",
    "Verified the nonzero full phase differential and exact critical-manifold dimension, with explicit link to the determinant-to-intrinsic-fold criterion.",
    "Supplied exact counterexample Jacobians, rank-three rows and smooth left inverse for its product embedding.",
    "Retained six mathematical checks unchanged, removing only obsolete historical edition/archive assertions from the portable verifier. Preserved all original evidence, exact SVG, portable figure source and font notice."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}