{
  "title": "Clean Lagrangian pairs and common transversals — restored proof edition",
  "original_source_sha256": "7a4536ceae5e9c88b485cddfc81e85c4f0ca1f07755a42281a308e4ee283eced",
  "restored_source_sha256": "29da3b27f73df43bf3a6f0f145d3f817f93265d86e2770e3b722e0f71bc7338a",
  "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": [
        303,
        304,
        305
      ],
      "relevant_printed_pages": [
        288,
        289
      ]
    }
  ],
  "original_lesson_preserved": true,
  "book_files_redistributed": false,
  "original_lesson_licence": "CC0-1.0",
  "reproducible_model_checks": {
    "script": "check_models.py",
    "script_sha256": "c31e9798763640f6a7da77385aa06e2957dd9174d2bd4a2738a4559fba06f87f",
    "results": "model-check.json",
    "original_script_sha256": "2fd309f7620fffe2c99df70cce835caa399365614489af49af5babcb62ea1226",
    "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."
  },
  "original_lesson_has_figure": false,
  "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 U023 lesson, all ten solutions, all six finite model checks, the 36-target inventory and historical proof preparation. Retained every original display and solution.",
    "Verified the admitted H III source-register entry and exact PDF bytes, then read PDF 303–305, including the full Theorem 21.2.10 and Corollary 21.2.11 at. The historical preparation page offset belonged to a different carrier and was not used as the current edition locator.",
    "Consulted the shared AN-04 source map and existing preparation. Reused actual restored U022 and smooth-flow proofs; the free FIO I and Guillemin–Sternberg leads were not treated as read replacements.",
    "Checked cleanliness as an embedded actual intersection with nearby tangent equality, not merely a pointwise tangent count. Proved the common parameter extension in each submanifold chart via a fixed tangent complement and inverse theorem.",
    "Checked the full simultaneous chart F1+F2-i+Zz, invertibility of its differential, exact restrictions and both germ equalities. Its common normal coordinates give the full annihilator of the tangent sum.",
    "Checked the nonzero radial tangent forces 1<=k<=n, the positive ray section gives actual products, and unique tangent decompositions preserve cleanliness and give dimensions 2n-1,n-1,n-1,k-1.",
    "Checked the slice still has k common normal coordinates, their degree-one rho extensions have independent differentials, and the common Hamilton span equals the intersection tangent. Every chosen field is tangent to both actual Lagrangians; when k>1 one is independent of the radial field.",
    "Added the full cotangent-lift formula P=(D psi)^(-T)p, its smooth inverse, exact primitive, homogeneous degrees and fiber images. Proved marked nonzero covector normalization by completing that covector to a row basis.",
    "Checked the k=1 base straightens the first entire fiber, bounds the second projection rank below by a minor and above by the nonzero radial kernel on a whole neighborhood, and therefore applies the proved conormal-germ theorem. The n=1 rank-zero case is included.",
    "Checked the hypersurface defining function is scaled by its actual signed marked coefficient; added an explicit base-coordinate completion. The full lift keeps the first fiber, straightens the second germ and fixes (0,e1) without changing the positive dilation parameter.",
    "Checked the k>1 common degree-one momentum is zero on both submanifolds, while an independent positive first momentum remains. Actual flow uniqueness gives both cylinders and their embedded reduced sections.",
    "Checked the actual intersection is also a cylinder, its transverse section has dimension k-1, all tangent equalities hold nearby, and both reduced submanifolds are conic Lagrangian with nonzero radial field.",
    "Made the full product extension, inverse, primitive summands, degrees and conic domain explicit; checked both auxiliary models and the correct induction in k and ambient dimension.",
    "Checked the entire homogeneous rotation, exact primitive cancellation and inverse, every spectator, marked point, both image equalities and reverse containments. Verified the paired index permutation and k=n endpoint.",
    "Checked the linear quotient pairing is well defined and perfect, its exact kernels are the common intersection, and alternating-subspace extension gives the full linear pair form including k=0 and k=n.",
    "Checked the explicit common transverse plane has dimension n, vanishing restricted symplectic form and zero intersection with each model plane, including both empty-block endpoints. No global field is asserted.",
    "Checked the full ordinary fiber induction from prescribed coordinates, actual cylinders, signed symplectic exchanges and dimension-zero base. In the transverse ordinary case, proved every component of the closed one-form radial integral and exact momentum translation, retaining the marked point.",
    "Checked the full ordinary clean-pair induction for k>0, actual reduced cleanliness, both auxiliary images and paired permutation. Its primitive changes by an exact differential; no homogeneous claim is made for an ordinary exchange.",
    "Checked the nonlinear hypersurface conormal and full six-coordinate lift, inverse momenta, exact primitive and radial clean intersection; checked the smooth but nonclean tangency example and its invariant dimension obstruction.",
    "Retained six finite checks unchanged except portable paths: clean/radial dimensions, seven full homogeneous model changes, seven ordinary changes and an exact graph, twenty common transversals, curved conormal lift with signed normalization, and nonclean tangency. These checks supplement the general proofs."
  ],
  "changes": [
    "Restored the complete clean conic, linear and ordinary pair theorems and the common transverse Lagrangian plane with all exact ranges and germ qualifications.",
    "Reconnected exact current coordinate-extension, isotropic-fiber, conormal, flow, exterior-calculus and inverse/implicit proofs.",
    "Proved extension of the common intersection parametrization directly in the two submanifold charts using the inverse theorem.",
    "Added the full cotangent-lift map, inverse and primitive identity, including signed marked-covector normalization.",
    "Made the full homogeneous product extension and conic inverse domain explicit.",
    "Expanded every component of the closed-form primitive integral used in the ordinary transverse base case.",
    "Retained every original display, all ten solutions and six finite model checks; the original lesson contains no figure."
  ],
  "all_alternative_proofs_preserved": true,
  "bounded_lesson_P514_closed": true,
  "human_review_claimed": false,
  "full_course_complete": false,
  "public_release_authorized": false
}