{
  "title": "Prescribed canonical coordinates and isotropic fibers — restored proof edition",
  "original_source_sha256": "143a18118f4b05cf4504494c5c70b8b92d0ad9053b2df9d664fc89af014b3b32",
  "restored_source_sha256": "0822251b59c9279212f165831dce586f34d6a32844659fa76da0ecb490e2b14b",
  "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": [
        288,
        289,
        294,
        295,
        296,
        297,
        300,
        302,
        303
      ],
      "relevant_printed_pages": [
        273,
        274,
        279,
        280,
        281,
        282,
        285,
        287,
        288
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "b15c407933bef55e22539ea3e6a2c0d9955fcc88caac38001f204566c0e8f5ae",
    "results": "model-check.json",
    "original_script_sha256": "8ea08b8a6cbc128709587e5a55ae5e32788d632a43fcfad8adda37a0dbd3efd6",
    "finite_checks": 6,
    "passed": true,
    "mathematical_checks_unchanged": true,
    "historical_archive_checks_removed": false,
    "removed_historical_check_flags": [],
    "scope": "Finite models support the written proof; they do not establish the general theorem."
  },
  "figure": {
    "path": "figures/isotropic-cusp-and-covector-direction.svg",
    "sha256": "61d895226c488ae28a454e0c88ac1033bac382289eecec9d650e378c4392f820",
    "original_bytes_unchanged": true,
    "reproducible_source": "figures/draw_isotropic_cusp.py",
    "original_script_sha256": "cceea9739794b8b35a4c66d0c95ae8367bfc8b67315e8edc2c874180554f9e6e",
    "portable_script_sha256": "9f816ea05c2ce750b1d302246e55ca35fefb41cad2c5350efe7021fbedcd6b4e",
    "license": "CC0-1.0",
    "coordinates": "Numerical samples of the exact rho=1 slice for -0.24<=r<=0.24: upper (r^2,-2r^3/3-r^4/2); lower (r+r^2,r^2). The lower tangent at zero is (1,0). The proof, not the samples, establishes the full conic embedding and primitive identity.",
    "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 complete original U022 lesson, every exercise solution, the reproducible cusp figure, all six finite checks and the 45-item target inventory. Preserved every original displayed equation and all ten complete solutions.",
    "Verified the exact approved H III source register and PDF bytes; read PDF 288–289, 294–297, 300 and 302–303, including the full ordinary and marked homogeneous coordinate theorems, the normal-pair reduction, conic isotropic normal form and constant-rank conormal theorem.",
    "Consulted the existing shared AN-04 source map and current submanifold flow material. Its Duistermaat–Hörmander and Bates–Weinstein free leads were not treated as proof replacements without reading. Reused exact current programme prerequisites.",
    "Checked the alternating-subspace isometry extension: nondegenerate radical complement, independent dual functionals, signed skew correction, paired blocks and equal-dimensional symplectic complements, including empty blocks.",
    "Checked the ordinary marked tangent completion, Jacobi commutator, actual independent commuting-flow chart and constant field equations. Added the explicit affine section function realizing every compatible value and covector.",
    "Checked the finite extension retains old functions identically, preserves independence after each shrinking and yields the complete standard Poisson matrix; verified the nonlinear shear and its exact primitive difference.",
    "Checked the radial primitive, Cartan identity and weights [R,Hf]=(degree-1)Hf, the marked radial linear completion and necessity of a nonzero momentum outside the prescribed positions.",
    "Checked the entire semidirect flow chart with positive tau contribution to R. Bound the commuting and dilation-conjugation argument to the full variational proof SN:F1; checked all position/momentum Euler equations and every marked differential.",
    "Checked the exceptional last position: the ordinary completion retains the radial-compatible tangent map, d lambda forces h to depend only on pJ, and h(bJ)=0. Its integral correction preserves every bracket and the marked first jet on either sign interval avoiding zero.",
    "Checked the transverse pJ=bJ section, unique positive-ray saturation, degree of each bracket, full conic inverse and identity of all originally prescribed functions. The resulting map preserves the canonical primitive.",
    "Checked isotropy is equivalent to an identically zero pulled-back primitive for conic submanifolds, not just a zero value at a point. Constructed the full degree-one conormal functions on a positive-ray section.",
    "Checked the k<n normal-pair induction: positive normal-form rank, nonradial Hamilton field, zero marked momentum, nonzero distinct marked momentum, implicit graph and second exact prescribed-pair completion. Made the full product map, inverse and homogeneity explicit.",
    "Checked the Lagrangian cylinder induction using tangency and actual flow uniqueness, dimension n-1 of the transverse reduced germ and retained nonzero radial field. Checked the n=k=1 base and the full auxiliary model.",
    "Checked the full homogeneous rotation, exact primitive, inverse, all spectator coordinates, both model inclusions and empty sums. A plain position-momentum exchange would have the wrong degrees.",
    "Supplied the complete map constant-rank proof from an invertible minor and the inverse theorem: pi=(u,h(u,v)), constant rank forces all v derivatives to vanish, and line-segment calculus gives an embedded image. The rank-zero case is proved as well.",
    "Checked zero primitive implies conormal inclusion, the equal-dimensional inclusion gives equality of germs, and the nonzero radial kernel gives r<n. Checked the maximal-minor open-dense argument and the distinction from a global conormal bundle.",
    "Checked the deformed cusp primitive, smooth inverse via rho and r+r^2, full immersion at zero, changing base ranks, exact cusp equation, lower projected tangent and obstruction to conormal-germ recognition at the caustic.",
    "Retained all six finite algebraic models unchanged apart from portable paths, including the dual correction, ordinary shear, semidirect equations, exceptional coordinate, five rotations and cusp. They support the complete written proofs rather than certify arbitrary smooth germs."
  ],
  "changes": [
    "Restored the complete ordinary and homogeneous prescribed-coordinate theorems with exact functions, arbitrary compatible marked values, tangent jets and full conic maps.",
    "Reconnected exact programme proofs of smooth flows, commuting fields, dilation conjugation, exterior calculus and inverse/implicit function theorems.",
    "Wrote the affine section-jet construction and made the full homogeneous symplectic product extension explicit.",
    "Proved the map constant-rank argument from the inverse function theorem and line-segment calculus, including rank zero; preserved the actual conormal germ conclusion.",
    "Retained both isotropic induction cases, the exact primitive-preserving rotation and inverse, the lifted cusp and all projection qualifications.",
    "Retained every original display, all ten solutions, the numerical cusp illustration, six finite model checks 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
}