{ "title": "Real and complex symplectic function normal forms — restored proof edition", "original_source_sha256": "878af42ad671c028da044b54e999281e5031a9a3433fc7926f78c6a92a783724", "restored_source_sha256": "e62d1972dea872f5db4f9da248bb3dec413e799411f4470e12b0d899ada58af8", "original_authorship": "GPT-6.1 Sol (OpenAI), Ultra, October 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": [ 311, 312, 313, 314, 315, 316, 317, 318 ], "relevant_printed_pages": [ 296, 297, 298, 299, 300, 301, 302, 303 ] }, { "edition_id": "ISBN-978-3-642-61497-2-pdf", "pdf_pages_read": [ 213, 214, 215, 216 ], "relevant_printed_pages": [ 198, 199, 200, 201 ] }, { "edition_id": "ISBN-978-3-540-26964-9-pdf", "pdf_pages_read": [ 200, 201, 202 ], "relevant_printed_pages": [ 187, 188, 189 ] } ], "original_lesson_preserved": true, "book_files_redistributed": false, "original_lesson_licence": "CC0-1.0", "reproducible_model_checks": { "script": "check_models.py", "script_sha256": "cba67fb53d9ec98ff5ab594731651657555c2680617ea2dd528951faaa96de12", "results": "model-check.json", "original_script_sha256": "c48680a8d0f37ec12b0893c4a122d3fe37fa9baf90f48695befb085b93de3c68", "finite_checks": 7, "algebra_families": 8, "algebra_script": "check_algebra.py", "algebra_script_sha256": "6406a7d8acf7f5a2e43dbcffdee9b9800e03d103dcc6a4f42dadb0eba5d0761f", "original_algebra_script_sha256": "f7bd04268a68d489eb052c26d0974f9c26457643824a0bf28c2dd99ff4708582", "portable_algebra_changes": "Removed the unused historical source-access/state-writing branch from the eight-family algebra companion; retained every finite mathematical check. Main verifier uses portable paths and its fundamental-solution caption now identifies the polar proof.", "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/weighted-contraction-and-contact-signs.svg", "sha256": "f5580158e209220a0d9a3e2be5d810d5698f1640af0fa4b8b94ea28d0bd5e459", "original_bytes_unchanged": true, "reproducible_source": "figures/draw_weighted_contraction.py", "original_script_sha256": "b417bd2544872ca81f18cbbbc6e82b3dfc37918172437c704509ef4998d4e470", "portable_script_sha256": "9ac1971929078cb1a73fa9a1ecbace2efaf3cff8e5c191de921cc23a75df1b09", "license": "CC0-1.0", "coordinates": "Numerical projections (p,q)=(p0 exp(-2s/3),q0 exp(-s/3)) for four signed initial pairs and 0<=s<=6; tangential flow is omitted explicitly. The other two panels sample signed x1^2 and x1^3 at xi1=0,h=1 on -1.1<=x1<=1.1. They show even/odd contact slices; the proof establishes the full coordinate maps and marked signs.", "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_fourteen_exercise_solutions_unchanged": true, "proof_review": [ "Read the full preserved U026 lesson and all fourteen solutions, the 65-target inventory, the seven-check verifier and eight-family algebra companion, and the original illustration source. Retained every displayed equation and every original solution.", "Verified the current approved register entries and exact source bytes for H III, I and II. Read H III PDF 311–318, H I PDF 213–216, and H II PDF 200–202. The full fixed-contact sign cases and same-domain smooth solver were checked against those exact editions.", "Consulted the existing shared AN-04 source map and historical preparation. Its free FIO II and analytic minicourse leads were not adopted as unread replacements. Reused the actual prescribed-coordinate, submanifold, flow, measure and Fourier proofs.", "Checked Hamilton and dilation signs, multiplier effects and degree reduction, including the zero real principal-type mark and nonradial condition.", "Checked the complete weighted normalizer: exact scalar equation, smooth tangential Taylor division, compact common negative-time flow domain, integrable drift, all initial derivatives, positive infinite integral, its equation and uniqueness, and homogeneous extension. Wrote the integrable-coefficient scalar inequality explicitly.", "Checked the positive commuting momentum construction on the symplectic zero section, unique commuting-flow transport, radial scaling, independence and complete marked coordinate completion. Checked both signs of the simple complex model.", "Checked the parameter almost-analytic extension, complex root via a real contraction, exact segment division and the all-order flat quotient across a possibly singular zero graph. Verified the common division domain depends on the divisor and that no uniqueness of smooth division is claimed.", "Checked homogeneous one-momentum elimination and its full prescribed-pair map, nonzero multiplier and independence of the eliminated momentum.", "Checked the full fixed-contact hypothesis includes a zero on every nearby curve. Verified the implicit zero graph, kth-order Taylor factor and ideal filtration for all bracket trees. Added the actual transverse section argument for transformed curve zeros.", "Checked every differentiated term in the multiplier law, the factor F=A-bB, all lower-order cancellations, the kth leading coefficient and the odd-sign invariant. No statement is made for a transformed real Hamilton field that vanishes at the point.", "Checked finite order excludes a radial real field, the actual division multiplier preserves the contact sign, the homogeneous zero graph and kth root, exact fractional weights and full model multiplier. Both parity operations and the negative odd marked-point obstruction are retained.", "Checked the elliptic two-real-variable frozen operator and supplied a direct polar proof of its distributional fundamental solution, including the 1/(2 pi) factor and sign. No Cauchy integral or Green theorem remains as an unproved substitute.", "Checked the compact fundamental kernel, annular error support, Fourier decay and all-real derivative gain. Proved the actual convolution equals the multiplier using weighted Cauchy–Schwarz, Tonelli, Fubini and compact smooth density.", "Checked the uniform small-coefficient bound, complete-space Neumann series and uniqueness on one fixed L2 space, support of the unknown and the exact 3 epsilon support margin for the right inverse.", "Checked all parameter derivatives of the operator and inverse on that same space, the exact spatial difference-quotient equation and all mixed derivative inductions. Replaced the unbound weak-compactness invocation with Fourier Fatou, dominated convergence and strong difference-quotient convergence.", "Proved the joint smooth representative explicitly from compactly localized mixed weak derivatives, Fourier differentiation, the weighted annular integral, Cauchy–Schwarz and absolutely integrable inverse Fourier derivatives. The parameter domain is not shrunk with the derivative order.", "Checked the actual involutive zero set is coisotropic with nonzero restricted primitive, the full conic mark and n>=3, and the elliptic restricted Hamilton field with homogeneous common-domain solver.", "Checked normal jet ideals, all four terms of the exponential multiplier bracket, the special first correction and every higher paired coefficient equation, including the even middle coefficient and omitted order 2k-1>=k+1.", "Checked the restricted operator stays fixed after the first correction, the parameter Borel construction realizes all corrections on one domain, and the resulting bracket is flat. Checked homogeneous extension from a positive-ray section.", "Checked flat real bracket division, both sequential real transport corrections, flatness and homogeneous uniqueness, the all-order smooth complex flat quotient and the final exact commuting-momenta canonical map.", "Checked all three worked examples and all fourteen exercise solutions, including the full real exchange, tangential-drift normalizer, contact parity and homogeneous flat corrections. Retained seven finite checks and all eight algebra families; these support the written proofs, not arbitrary smooth-germ certification." ], "changes": [ "Restored all real, simple complex, one-momentum, finite-contact and involutive normal forms without narrowing their hypotheses or conclusions.", "Reconnected exact finite-flow, prescribed-coordinate, coisotropic, measure, Fourier, Sobolev and finite-calculus proofs.", "Wrote the integrable-coefficient derivative bound used for the entire backwards-flow normalizer.", "Made the transformed fixed-contact flow section explicit, retaining existence of the zero on every nearby curve.", "Proved the fundamental-solution constant by real polar integration and proved equality of actual convolution and the Fourier multiplier.", "Wrote the uniform complete-space Neumann argument and replaced the unbound weak-compactness step by a full Fourier Fatou/dominated-convergence difference-quotient proof.", "Wrote the joint Fourier smooth-representative argument from all mixed weak derivatives on the same parameter domain.", "Retained every original display, all fourteen solutions, the exact sampled illustration, seven finite checks with eight algebra families, 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 }