{
  "title": "Homogeneous submanifold normal forms — restored proof edition",
  "original_source_sha256": "547b0dd5a5c840337a8446c607fe4858ea1038d79273a1911d245a8b19fc8c5c",
  "restored_source_sha256": "3a182f28488d8023ad6ddf5129fd46fd43de65c58f70a37a47c140e2192ac4a6",
  "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": [
        298,
        299,
        300,
        301
      ],
      "relevant_printed_pages": [
        283,
        284,
        285,
        286
      ]
    },
    {
      "edition_id": "ISBN-978-3-642-00136-9-pdf",
      "pdf_pages_read": [
        42
      ],
      "relevant_printed_pages": [
        31
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "a4d8f92adbd1b2347e2c86dd99364b4a31729ba4a77eb56a8361c292562e5a8e",
    "results": "model-check.json",
    "original_script_sha256": "86128f69594048fa9f9d80875d846d53750309597886c5a124a79a44fbd11f6b",
    "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/characteristic-leaves-and-dilation.svg",
    "sha256": "0351c02eaf4efa1ea8fece358bd70544fcb80b65c7b29bfe42fb6f484045dbff",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_characteristic_leaves.py",
    "original_script_sha256": "0ea923e3cccb58985e96232dae3c77162ffa99eacb5e5a5ce245c276c2ceab02",
    "portable_script_sha256": "b2aa405670f6be2b191ee318977b8e70ca8c00566a56d454589981f5dfb757c0",
    "license": "CC0-1.0",
    "coordinates": "Exact slice x1=0 of x2=0, xi2>0; vertical characteristic leaves and arrows 0.24 times the radial vector",
    "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 entire original U012 lesson, all nine complete solutions, its exact coordinate caption and all six finite-model calculations. Retained all original displayed equations, the complete exercise section and the original SVG bytes.",
    "Checked the three-block symplectic splitting, quotient radical and dimensions, defining-function brackets and both Lagrangian-containment implications.",
    "Checked the actual homogeneous canonical pair, joint smooth-flow coordinates, independence from the radial field, exact momentum and parameter signs, symplectic slice, product pullback and pushforward dilation weights. Supplied the complete commuting-flow and variational proofs.",
    "Checked both induction branches of the homogeneous normal form: remove a symplectic normal pair or a characteristic direction. Verified dimension, restricted rank, smoothness, homogeneity and persistence of the nonzero restricted canonical one-form at each step.",
    "Checked the ordinary counterpart without a radial or nonzero-one-form hypothesis, including isotropic submanifolds and the zero-dimensional final slice.",
    "Supplied explicit smooth kernel/image frames using an invertible minor, quotient/complement constructions and smooth descended symplectic forms. Checked both quotient summands and their identification along a characteristic leaf.",
    "Supplied full plaque-chain construction of maximal connected immersed leaves, chart compatibility, Hausdorffness, countable atlases, tangent-curve containment and the action of distribution-preserving diffeomorphisms. Checked the leafwise radial dichotomy and the local symplectic quotient, without asserting a global Hausdorff leaf space.",
    "Checked the exact coisotropic model, all coordinate brackets and canonical-one-form equality, actual characteristic curves and the reduced symplectic form.",
    "Checked constant-rank projection dimensions, local embedded conic image germs and coisotropy, separate flattening, characteristic cylinders, bijective reduced graph and the full homogeneous canonical product change.",
    "Checked sharpness for the universal operator-class quantifier: retained the positive-order flat model, made compact-cutoff preservation of its nonzero modulation limit explicit, and checked the actual proper graph inverse transfer and smoothing errors. Kept the positive-bundle-rank qualification.",
    "Read the admitted H III printed pages 283–286 and H IV printed page 31, with exact file hashes checked. Compared their mathematical coverage to the whole restored scope. Our explicit pair construction, supporting flow/bundle/leaf proofs, explanations, operator transfer, examples and nine exercises are independent exposition with accurate credit; no book text, figure or exercise collection is redistributed.",
    "Reused the current full geometry, corank, graph, analytic-composition and constant-rank coordinate proofs before writing supporting details. All alternative proofs remain available.",
    "Retained all six reproducible finite checks with destination-only changes, and retained the exact coordinate SVG and its reproducible drawing script with portable destinations. These computations support the proof review and do not replace the general proofs."
  ],
  "changes": [
    "Restored all original displayed equations, nine complete solutions and exact coordinate figure with current prerequisite bindings.",
    "Complete companion proofs of tangency, commuting flows, homogeneous pushforward weights, smooth constant-rank bundles and maximal connected immersed leaves.",
    "Explicit constant-rank image branch and conic saturation, actual compact-cutoff modulation limit and proper-support inverse transfer.",
    "Accurate approved-edition citations and source comparison; independent exposition and all alternatives retained.",
    "Retained all six finite model checks and the reproducible figure source, changing only file destinations."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false,
  "formatting_correction": {
    "lesson_id": "AN04-U012",
    "previous_source_sha256": "2bfa0cac7b0ed1f7673dd89708e807128897f7e3d5e919a2fd434a7f6bc38c0f",
    "corrected_source_sha256": "3a182f28488d8023ad6ddf5129fd46fd43de65c58f70a37a47c140e2192ac4a6",
    "scope": "Restore TeX escapes in the newly added compact-modulation paragraph and split that addition into a readable paragraph. Original mathematics, displayed equations, all nine solutions and the figure are unchanged.",
    "cause": "The file-writing string consumed TeX backslashes before Markdown rendering. Corrected the literal writer and rechecked actual rendered formulae.",
    "mathematical_argument_changed": false,
    "original_exercises_and_displays_unchanged": true
  }
}