{
  "title": "Corank geometry and sufficient continuity — restored proof edition",
  "original_source_sha256": "56df76ccd50c2ab2d2ed7312b834933d5da7ce187f4afbde9402ca4876ad9a29",
  "restored_source_sha256": "5f4f7b9863fd0d677fbec271a57c36c44be65bab876c0be05aa1e00f10f270bf",
  "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": [
        39,
        40,
        41,
        42
      ],
      "relevant_printed_pages": [
        28,
        29,
        30,
        31
      ]
    },
    {
      "edition_id": "ISBN-978-3-540-49938-1-pdf",
      "pdf_pages_read": [
        294,
        295,
        296,
        297,
        298,
        308,
        309,
        310,
        311
      ],
      "relevant_printed_pages": [
        279,
        280,
        281,
        293,
        294
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "48dee30627fc060eb25761ff688bd643749a5ea8005550b710f137f3c52268aa",
    "results": "model-check.json",
    "original_script_sha256": "6b4811793d084c44ce91dad52d819e1e83b984ec92cb07f172fec629e613aa67",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "linked_component_rights": "All earlier proof components retain their individual notices and licences.",
  "all_original_displayed_equations_and_eight_exercise_solutions_unchanged": true,
  "proof_review": [
    "Read the entire original U011 lesson and all eight solutions. Preserved every displayed equation and the full exercise section.",
    "Checked the full signed linear relation splitting: projection images equal the kernel orthogonals, quotient sections are symplectic, the quotient relation is a bijective symplectic graph and the radical is exactly the two kernel summands.",
    "Checked initial homogeneous coordinates with every prescribed radial-compatible first derivative, using the exact transverse-slice and smooth flow proofs.",
    "Checked the symmetric first coefficient jet of the radial primitive, its radial null vector, and the exact degree-one quadratic function whose Hessian cancels that jet.",
    "Bound the homogeneous Moser argument to the complete current flow and pullback proofs. Made the common time-one domain, identity variational derivative, dilation commutation and homogeneous inverse explicit.",
    "Checked the equivalence of individual radial exclusions with nonzero common one-form, actual radial representatives in both quotient sections, and separate homogeneous tangent normalization. No nonlinear flattening is claimed by the tangent result.",
    "Checked the partial generating function and input sign, invertible mixed matrix, nondegeneracy, exact ambient and frequency dimensions, critical density and quarter-corank order shift.",
    "Justified the actual partial kernel action by rapid partial Fourier decay and convergence with every base/parameter derivative. Checked the supported amplitude reduction against the exact existing graph proof.",
    "Checked the zero two-jet and cubic homogeneous estimate, including the radial/angular scaling and the case of a one-dimensional shared graph.",
    "Checked the uniform graph norm on a single compact parameter family, actual parameter integration, exact volume factors and zero-dimensional cases.",
    "Checked return through properly supported elliptic graph quantizations and their inverses, wavefront exclusion for errors, finite compact normalized covers and the allowance for changing rank.",
    "Checked every real Sobolev shift using exact reducers and local inverses, smooth kernel bounds, common compact approximation and equality with the original distributional action.",
    "Supplied the vector integration proof required by the measurable Hilbert-family exercise. It retains arbitrary finite parameter measures: constructed their product measure via the generating class, proved Tonelli/Fubini and completion, then proved strong vector approximation, actual section measurability and the integral norm bound.",
    "Checked flat-family sharpness using fixed-norm modulated compact inputs, exact multiplier shift, an integrable Fourier majorant, Plancherel and a nonzero limiting output.",
    "Compared all eight exercises and their signs, normalizations, geometric hypotheses and norm calculations. Reused all six existing finite model checks with only input/output paths changed; the exact tangent, phase and flow calculations and independent finite parameter operator checks all pass.",
    "Read the admitted H IV printed pages 28-31 and H III 279-281 and 293-294. The full primitive-jet/Moser proof, explanatory organization, flat model and eight original exercises are independent exposition. Accurate source credit is retained, and no book text, figure or exercise collection is redistributed.",
    "Compared the already available WM slicing argument and current source/coverage records before adding prerequisites. WM keeps its base-submersion assumptions; the restored proof handles the stated individual radial exclusions by homogeneous coordinate changes. All alternative proofs remain available."
  ],
  "changes": [
    "Exact current geometry, smooth-flow, graph, composition, measure and Fourier providers, preserving all original displayed equations and eight complete solutions.",
    "Explicit common time-one domain, variational identity and dilation-compatible inverse for the prescribed tangent Moser construction.",
    "Actual cutoff convergence, compact parameter integration and uniform graph estimates.",
    "Full strong vector integration and measurability proof for the Hilbert-valued exercise, with its measurable-family convention explicit.",
    "Preserved flat sharpness model, exact unequal-dimensional normalization, cubic estimate and changing-rank examples; retained all six finite model checks.",
    "Accurate approved-edition citations and independent provenance; existing alternative slicing proof retained with its own hypotheses."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}