{ "schema": "lie-tresse-proof-providers/v1", "license_expression": "CC0-1.0", "lesson": "LIE-TG-09", "edition": { "authors": [ "Boris Kruglikov", "Valentin Lychagin" ], "title": "Global Lie–Tresse theorem", "url": "https://arxiv.org/abs/1111.5480v3", "revision_date": "2015-06-13" }, "providers": [ { "path": "NOE-SYM/src/NOE-SYM-03.md", "href": "../NOE-SYM/src/NOE-SYM-03.md" }, { "path": "NOE-SYM/NOE-SYM-03.html", "href": "../NOE-SYM/NOE-SYM-03.html" }, { "path": "NOE-RAT/src/NOE-RAT-01.md", "href": "../NOE-RAT/src/NOE-RAT-01.md" }, { "path": "NT-LOC/src/NT-LOC-04.md", "href": "../NT-LOC/src/NT-LOC-04.md" }, { "path": "NT-LOC/NT-LOC-04.html", "href": "../NT-LOC/NT-LOC-04.html" }, { "path": "NOE-RAT/NOE-RAT-01.html", "href": "../NOE-RAT/NOE-RAT-01.html" }, { "path": "AG-CA/src/noetherian-and-artinian-rings.md", "href": "../AG-CA/src/noetherian-and-artinian-rings.md" }, { "path": "AG-CA/noetherian-and-artinian-rings.html", "href": "../AG-CA/noetherian-and-artinian-rings.html" }, { "path": "AG-MO/src/quasi-finite-morphisms-and-chevalley.md", "href": "../AG-MO/src/quasi-finite-morphisms-and-chevalley.md" }, { "path": "AG-MO/quasi-finite-morphisms-and-chevalley.html", "href": "../AG-MO/quasi-finite-morphisms-and-chevalley.html" }, { "path": "AG-CA/src/kahler-differentials.md", "href": "../AG-CA/src/kahler-differentials.md" }, { "path": "AG-CA/kahler-differentials.html", "href": "../AG-CA/kahler-differentials.html" }, { "path": "AG-CA/src/krull-dimension-and-noether-normalization.md", "href": "../AG-CA/src/krull-dimension-and-noether-normalization.md" }, { "path": "AG-CA/krull-dimension-and-noether-normalization.html", "href": "../AG-CA/krull-dimension-and-noether-normalization.html" }, { "path": "AG-CA/src/integral-extensions-lying-over-going-up-and-going-down.md", "href": "../AG-CA/src/integral-extensions-lying-over-going-up-and-going-down.md" }, { "path": "AG-CA/AG-CA-05.html", "href": "../AG-CA/AG-CA-05.html" } ], "uses": [ { "result": "Proposition 4.2", "inputs": [ "AG-CA-09 Lemma 1.3, including the finite exchange bound and persistence of algebraic independence under an algebraic extension", "NOE-RAT-01 Lemma 1.1 and Theorem 1.2", "NT-LOC-04 Proposition 0.1, infinite-base branch" ], "scope": "Integral finite-type fibre over a characteristic-zero field; field generation only, not differential generation or orbit separation." }, { "result": "Lemma 4.3", "inputs": [ "Algebraic independence and a full group action preserving the coefficient field and fixing the polynomial coordinates" ], "scope": "Conditional coefficient descent as stated in Lemma4.3; actual equation and invariant affine coordinates are provided by Theorems4.32,4.38 and4.45." }, { "result": "Section 4.5", "inputs": [ "NOE-SYM-03 Section 6, equations 6.9–6.11 and Figure 6.1" ], "scope": "Counterexample to individual real-orbit separation, not to finite differential generation." }, { "result": "Example 4.4", "inputs": [ "AG-CA-03 Theorem 2.1 and Proposition 1.2" ], "scope": "Noetherian is not free; generator degrees do not bound the first Koszul homology. This refutes those implications, not the Lie–Tresse theorem." }, { "result": "Lemma 4.5", "inputs": [ "AG-CA-03 Theorem 1.1, Proposition 1.2 and Theorem 2.1" ], "scope": "One fixed contraction-closed symbolic system on finite-dimensional vector spaces in characteristic zero; uses finite generation of its annihilator relations, not a free basis of its graded dual." }, { "result": "Section4.7 constructibility comparison", "inputs": [ "AG-MO-06 Theorem3.1 and Theorem4.2" ], "scope": "Constructible total image and dense-open containment only; no fibrewise image equality inferred." }, { "result": "Example4.6", "inputs": [ "Explicit polynomial equations, coordinate projections and infinite-field polynomial nonvanishing, all proved locally." ], "scope": "Counterexample to the algebraic total-image inference; not asserted to be a pseudogroup stabilizer realization." }, { "result": "Lemma4.7", "inputs": [ "AG-CA-03 Theorem1.1,Proposition1.2,Theorem2.1", "LIE-TG-09 Lemma4.5 evaluation pairing" ], "scope": "Uniform eventual prolongation for images of one graded relation submodule under all field specializations. The actual orbit-symbol application is Theorem4.28, with its regular Cartan action identification in Theorem4.45." }, { "result": "Example4.8 and Proposition4.9", "inputs": [ "Explicit affine action, graph-jet chain rule, scalar stabilizer equations and rational functions on irreducible affine space, all proved locally." ], "scope": "Actual finite-order transitive algebraic formally integrable pseudogroup; rank-jumping whole-space stabilizer family and explicit invariant regular domain." }, { "result": "Example4.10 and Lemmata4.11-4.12", "inputs": [ "Local transformation equations, Leibniz differentiation, finite-dimensional kernel/image exactness and finite Taylor expansions, proved explicitly in Section4.10." ], "scope": "Actual regular counterexample to the displayed normalization inference; exact general linear obstruction and conditional contraction lemma, not a complete uniform theorem." }, { "result": "Lemma4.13,Corollary4.14,Example4.15", "inputs": [ "Local ordinary determining equations and all their differential consequences; constant-rank kernel extension; Section2 point prolongation; linear insertion of symmetric tensors. All steps of the new construction are proved in Section4.11." ], "scope": "Spencer difference gives admissible orbit-symbol contractions on the stated finite-order regular chart, provided the representative derivative on the tangent plane vanishes. Equality of adjacent first-derivative stabilizer images supplies that normalization. No uniform common-domain result asserted." }, { "result": "Lemma4.16,Proposition4.17,Example4.18", "inputs": [ "AG-CA Theorems1.1/2.1 and Proposition1.2; LIE-TG-09 evaluation pairing and uniform relation-module Lemma4.7; finite-order ordinary symbols; all elimination, specialization and pure-action comparisons proved in Section4.12." ], "scope": "One first-jet parameter localization gives every pure transformation-action symbol annihilator and uniform eventual prolongation. Full orbit-symbol annihilators are not identified by this result." }, { "result": "Theorem 4.19, Example 4.20 and Exercise 8", "inputs": [ "AG-CA Theorem 2.1 and LIE-TG-09 Lemma 4.16 for finite monic elimination.", "AG-MO Theorem 4.2 and its dense-constructible criterion.", "Section 4.13 proves the point test, orbit closure comparison, specialization and coefficient rewriting." ], "scope": "Fixed finite algebraic group action; geometric orbits over algebraically closed extensions, including disconnected groups. No uniform higher-jet domain or individual real-orbit separation is asserted." }, { "result": "Theorem 4.21, Corollary 4.22 and Example 4.23", "inputs": [ "Commuting total derivatives and graph-transversality covariance from Section 2.", "Integral finite-level fibre function fields with injective pullbacks.", "Section 4.14 supplies the finite maximal basis, minor identities, explicit kernel basis, quotient duality, coefficient covariance, representative independence and filtration arguments." ], "scope": "Uniform horizontal rank and quotient invariant derivations under the stated tower/fibre hypotheses. Does not supply common-domain full orbit-symbol stabilization or differential generation." }, { "result": "Theorem 4.24 and Example 4.25", "inputs": [ "AG-CA-03 Theorems1.1/2.1 and Proposition1.2: polynomial/module Noetherianity.", "Section4.15 proves the ordered coefficient bases, full polynomial observation gauge, weighted leading shifts, finite monic localization, exact ordinary-field specialization, graph elimination, finite normal truncation and observed-space duality." ], "scope": "Uniform finite observations under the explicitly stated coefficient-algebra and derivation hypotheses. The actual geometric coefficient algebra and transport to one invariant finite-level domain are additional application requirements." }, { "result": "Lemmas4.26–4.27, Theorem4.28 and Example4.29", "inputs": [ "Section2 graph/characteristic chain rule; Lemma4.5 symmetric insertion pairing; Theorem4.24 full graph elimination and ordinary-field specialization.", "Section4.16 proves raw regular determining ideals, full formal germ transport, bounded equation coefficient algebra, actual tangential kernel strictness, point-row highest-symbol duality, finite Taylor coefficient module, invariant saturation and overlap gluing.", "AG-CA-03 Theorems1.1/2.1 and Proposition1.2 for one finite coefficient module." ], "scope": "Induced transitive finite-order formally integrable point actions with the stated algebraic equation fibres. One finite invariant domain works at every higher geometric jet; the general finite differential-generation theorem and action-only extension are not asserted." }, { "result": "Theorem4.30 and Example4.31", "inputs": [ "Lemma4.26 regular equation ideals and formal integrability; Theorem4.28 actual finite-level full-symbol annihilator presentation; Theorem4.21 and Corollary4.22 horizontal domain and quotient derivations.", "AG-CA-03 Theorems1.1/2.1 and Proposition1.2: Noetherian polynomial modules.", "Section4.17 proves monic coefficient freeness, exact annihilator-quotient specialization, the Koszul homotopy and absolute degree bound, split homology specialization, Spencer duality and the common invariant domain." ], "scope": "Uniform eventual quotient-symbol Spencer vanishing for the precise induced point-action setting of Section4.16. Affine quotients, polynomial generation and the compatible action-only extension require further arguments." }, { "result": "Theorem4.32 and Example4.33", "inputs": [ "Lemma4.26 actual regular equation ideals, finite point-arrow transport and raw FI affine prolongations.", "Theorem4.24 monic leading-module division; Theorem4.30 actual equation-symbol images and coefficient specialization.", "AG-CA-03 Theorems1.1/2.1 for finite polynomial ideals and modules.", "Section4.18 proves the fixed denominator derivative rule, actual monic pivot realization, triangular inverse polynomial maps and whole affine-fibre/scheme equality." ], "scope": "One nonempty affine chart in a fixed finite equation fibre has compatible polynomial coordinates on every complete higher inverse image. Coordinates and chart are not asserted invariant; quotient and generation assertions remain separate." }, { "result": "Lemma4.34, Theorems4.35–4.36 and Example4.37", "inputs": [ "Lemma4.26 actual finite arrow sections; Theorem4.19 full finite orbit quotient, geometric fibres, parameter elimination and local closedness; Proposition4.2 primitive-element step.", "AG-CA-16 Section5 Theorem5.1/equation13 andSection6 Lemma6.1/Theorem6.2; AG-CA-09 Lemma1.3,Corollary3.2,Lemma4.1/Theorem4.2; AG-CA-05 Theorem5.1; AG-MO-06 Theorem4.2. The geometric generic fibre dimension is proved directly inTheorem4.35 from finite normalization andTheorem4.19 full orbit fibres.", "Section4.19 proves field differential rank, componentwise orbit tangent dimension, triangular separable smooth lifting, actual stabilizer kernels, finite generic coefficient flatness, reduced fibre pairs, faithful chart-cover equalizers, invariant sheaf, openness and categorical descent." ], "scope": "Finite algebraic fibre quotient over an algebraically closed characteristic-zero field, with transitive point-groupoid invariant extension. No all-order invariant coordinate or generation theorem is inferred without its further construction." }, { "result": "Theorem4.38 and Example4.39", "inputs": [ "Theorems4.19/4.21/4.24/4.28/4.30/4.32/4.35/4.36, Corollary4.22, Lemmas4.3/4.34 and their complete arguments in this lesson.", "Theorem4.36 proves the invariant field as a consequence of flat full-orbit descent; no new-level field-generation premise is assumed.", "Section4.20 proves the fixed order before field generators, full quotient-derivative symbol basis, homogeneous free kernel splittings, invariant saturation at every higher extension, full affine orbit levels, geometric quotient induction, low-polynomial coefficient invariance and compatible formal transporter images." ], "scope": "Complete finite differential generation for the induced point-action setting of Section4.16. Theorem4.45 derives the actual action identification for a regular Cartan jet action." }, { "result": "Theorem4.1 local infinitesimal count", "inputs": [ "LIE-TG-04 Section2 Theorem2.1 complete analytic first-integral proof; Section1 prolongation bracket identity." ], "scope": "Analytic first integrals and identity-local invariance on a constant-rank infinitesimal distribution. Full disconnected-group differential spanning is Theorem4.35 on its stated open." }, { "result": "Theorem4.40, Example4.41, Lemma4.42 and Proposition4.43", "inputs": [ "Section4.21 proves the exact Cartan pullback recurrence, all-order point-jet bound, holonomic plane injectivity and infinitesimal characteristic dictionary.", "The intrinsic example has a full group law/inverse, first-order FI determining presentation, trivial fixed-point jet groups, explicit orbit normalization and polynomial invariant induction.", "The rational constant lemma and full original-class invariant field/nonseparation argument are proved in Section4.21; neither source citation substitutes for those proofs." ], "scope": "Point-covering Cartan reconstruction and an intrinsic example with a proved coefficient obstruction. The regular finite-jet theorem and specified intrinsic-presentation transport are Theorem4.45 and Proposition4.46." }, { "result": "Lemma4.44, Theorem4.45, Proposition4.46 and Example4.47", "inputs": [ "Lemma4.44 proves propagation of algebraicity using local transitive sections, finite polynomial equations, finite-jet inversion and ordinary prolongation.", "The finite Cartan-plane identity4.117 and the transformed-graph chain rule force equality of actual action maps at every order. The reduced point test identifies full stabilizer schemes and vertical orbit symbols.", "Theorems4.19/4.21/4.24/4.28/4.30/4.32/4.35/4.36/4.38 and Lemmas4.26/4.27/4.34 have their complete proofs in this lesson and bound earlier programme inputs.", "Proposition4.46 proves every holonomic jet identification and the exact coefficient comparison relative to the same intrinsic base. Example4.47 computes the higher-jet domain of the contact map." ], "scope": "Full Lie–Tresse conclusion for the defined regular Cartan finite-jet action on its specified base, and transport for specified regular intrinsic finite presentations. It does not assert that an arbitrary differential substitution has such a presentation." } ], "complete_general_proof": true, "authorship": { "lesson": "GPT-6.1 Sol (OpenAI)", "sections_4_2_through_4_10": "GPT-6 Astra (OpenAI)", "sections_4_11_and_4_12": "GPT-6.1 Sol (OpenAI)", "primitive_element_prerequisite": "NT-LOC-04, Proposition 0.1: Claude Opus 5.5 (Anthropic)." }, "source_corrections": [ { "title": "Erratum to Global Lie-Tresse theorem", "url": "https://serre.mat-stat.uit.no/ansatte/boris/corr/Erratum-SM2016.pdf", "pages": 1, "scope": "Corrects the invariant derivation and revisited example in Sections 3.5 and 4.5; does not amend the Section 2.2 module argument." } ], "symbolic_stabilization": { "source_version": "arXiv:1111.5480v3", "source_section": "2.2", "fixed_symbolic_system_proved": true, "uniform_finite_level_open_proved": true, "total_image_inference": { "status": "refuted_as_general_algebraic_inference", "counterexample": "LIE-TG-09 Example4.6", "stable_total_images": true, "affine_bundle_tower": true, "constant_rank_subbundles": true, "actual_pseudogroup_realization_established": false, "general_proposition_refuted": false }, "uniform_relation_module_lemma_proved": true, "actual_symbol_specialization_comparison_proved": true, "actual_homothety_diagnostic": { "proof_locator": "LIE-TG-09 Section4.9, Example4.8 and Proposition4.9", "field": "R or C", "action": "(x,u) -> (lambda*x+b,lambda*u+c),lambda nonzero", "prolonged_action": "u_j -> lambda^(1-j)*u_j", "stabilizer": "Scalar a with (j-1)*a*u_j=0 for2<=j<=k", "stable_whole_space_total_image": "End(tau)=F at every order", "whole_space_rank_constant": false, "rational_sections_at_order_at_least_two": "All zero on every nonempty open of the irreducible graph-jet chart.", "refuted_inference": "Whole-space stable total image implies rational stabilizer sections spanning that image and stable images at every higher lift.", "source_locator": "arXiv1111.5480v3 Section2.2 proof of Proposition4, native lines1218–1238", "correct_domain": "U^2={u_2!=0}", "orbit_rank_on_higher_pullback": 3, "orbit_symbols_zero_from_order": 3, "uniform_inclusion_from_order": 2, "arbitrarily_delayed_full_space_failure": "At jet of u=x^r/r!,varpi_(r-1)=0 and varpi_r!=0,for anyr>=2.", "preceding_constant_rank_example_realization_claimed": false, "general_proposition_refuted": false, "general_theorem_proved": false }, "bracket_representative_diagnostic": { "proof_locator": "LIE-TG-09 Section4.10,Example4.10,Lemmata4.11-4.12", "source_locator": "arXiv1111.5480v3 Section2.2 proof ofProposition4,native1240-1278", "actual_pseudogroup": "(x,u)->(f(x),u+g(x)),f prime nonzero", "stabilizer_rank": "k at every graph k-jet,k>=1", "full_stabilizer_projections_surjective_in_example": true, "stable_derivative_images_in_example": true, "example": "For k>=4,Xstar=x^2 partial_x+x^k partial_u and Xzero=x^k partial_u represent the same nonzero orbit symbol and have zero first jet; their brackets with partial_x+x partial_u differ by a nonzero order2 component.", "failure": "Zero first derivative does not justify the claimed representative-independent bracket calculation or its asserted lower-order vanishing.", "good_representative_exists_in_example": true, "general_proposition_refuted": false, "exact_sequence": "0->A_k(g_k)->varpi_k->S_(k-1)/p(S_k)->0", "sufficient_condition": "The entire ordinary(k-1)-jet of the representative vanishes; full stabilizer lifting supplies this normalization.", "stronger_condition_proved_in_general": false, "general_theorem_proved": false }, "spencer_section_construction": { "proof_locator": "LIE-TG-09 Section4.11,Lemma4.13 and Corollary4.14", "determining_equations": "Finite-order homogeneous linear system with all ordinary prolongation equations through the indicated orders.", "regularity": "Locally constant rank of A_(k-1) composed with transformation-jet truncation, on the equation (k-1)-jet base.", "normalization": "The representing transformation jet has value zero and first derivative zero on the tangent plane; no zero entire lower ordinary jet required.", "construction": "Extend the representing jet as a kernel section; take minus the horizontal Spencer difference. Its admissibility follows by subtracting a differentiated determining equation from the derivative of the section equation.", "representative_independence": "The admissible transformation jet can depend on its extension; its prolonged action is the same intrinsic insertion, so differences lie in the lower stabilizer.", "image_lifting_corollary": "Equality of adjacent first-derivative stabilizer images supplies the normalization. It is not full stabilizer lifting and does not validate the raw bracket.", "full_common_domain_proved": true, "actual_symbol_specialization_comparison_proved": true, "complete_general_theorem_proved": false, "common_domain_scope": "Theorem4.28 induced point-action hypotheses; whole stabilizer surjectivity is not asserted." }, "pure_symbol_presentation": { "proof_locator": "Section4.12,Lemma4.16 and Proposition4.17", "ordinary_symbol_hypothesis": "Finite defining order with higher ordinary symbols given by algebraic prolongation.", "base": "Integral affine chart of allowed tangent planes at one base point; nonzero first-jet parameter f inverted.", "construction": "Finite homogeneous annihilator module; monic module elimination over the parameter fraction field; clear finitely many denominators from generators, representations and S-vector reduction identities.", "specialization": "After this localization, the eliminated relation module specializes to the actual pure-action symbol annihilator in every degree and at every field-valued parameter.", "all_degree_specialization_proved": true, "uniform_pure_symbol_prolongation_proved": true, "full_orbit_symbol_comparison_proved": false, "common_invariant_general_domain_proved": false, "separate_full_symbol_comparison": "Lemma4.27 and Theorem4.28 use the full action and retain the point-value rows." }, "scope_limits": [ "Theorem4.45 gives the complete regular Cartan finite-jet theorem from the actual action identification and Theorems4.28–4.38.", "Proposition4.46 uses a specified regular intrinsic finite presentation and its own base. It does not infer that presentation from an arbitrary finite-order differential substitution.", "Geometric separation does not separate individual real orbits inside a geometric orbit, and formal arrows do not assert analytic convergence." ], "proved_scope": "Induced point actions under the precise hypotheses of Section4.16, for each preassigned finite observation order.", "action_only_extension_proved": true, "uniform_quotient_symbol_cohomology_proved": true, "point_action_finite_generation_proved": true }, "figures": [ { "path": "docs/courses/LIE-TG/figures/lie-tresse-fibre-tower.svg", "license_expression": "CC0-1.0", "authorship": "Original diagram by GPT-6 Astra (OpenAI), Ultra.", "proof_locator": "Example4.6", "scope": "Exact algebraic fibre maps, not a pseudogroup realization.", "source_format": "Editable SVG with complete geometry and labels." }, { "path": "docs/courses/LIE-TG/figures/lie-tresse-homothety-stabilizers.svg", "license_expression": "CC0-1.0", "authorship": "Original GPT-6 Astra (OpenAI), Ultra.", "proof_locator": "Example4.8 and Proposition4.9", "scope": "Exact two curves and prolonged common scaling; actual rank-jumping family, not the preceding constant-rank algebraic tower." }, { "path": "docs/courses/LIE-TG/figures/lie-tresse-bracket-obstruction.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6 Astra (OpenAI), Ultra", "proof_locator": "Example4.10 and Lemma4.11" }, { "path": "docs/courses/LIE-TG/figures/lie-tresse-spencer-contraction.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI), Ultra", "proof_locator": "Lemma4.13,Corollary4.14 and Example4.15" }, { "path": "docs/courses/LIE-TG/figures/lie-tresse-pure-symbol-specialization.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI), Ultra", "proof_locator": "Lemma4.16 and Proposition4.17" }, { "path": "docs/courses/LIE-TG/figures/finite-level-orbit-quotient.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Exercise 8 and Solution 8", "scope": "Exact additive action, its nonzero-x orbit quotient, and the different orbit ideals at x=0." }, { "path": "docs/courses/LIE-TG/figures/horizontal-quotient-derivations.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Example 4.23, Exercise 9 and Solution 9", "scope": "Exact values at (1,0) of two horizontal representatives and their annihilating difference." }, { "path": "docs/courses/LIE-TG/figures/normal-curvature-observation.svg", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Example4.25 and Exercise10/Solution10", "scope": "Exact tangential translation on a line and its full gauged normal-commutator graph relation." }, { "path": "docs/courses/LIE-TG/figures/formal-transport-common-domain.svg", "reproducible_source": "docs/courses/LIE-TG/figures/formal-transport-common-domain.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.28, Example4.29 and Exercise11/Solution11", "scope": "Finite coefficient selection and faithful formal-field descent; analytic jet formulas sampled at481 points." }, { "path": "docs/courses/LIE-TG/figures/quotient-symbol-cohomology.svg", "reproducible_source": "docs/courses/LIE-TG/figures/quotient-symbol-cohomology.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.30, Example4.31 and Exercise12/Solution12", "scope": "Exact complete Koszul homology support of F[t1,t2]/(t1^2,t2^3), absolute Spencer degree conversion and the variable-killing homotopy." }, { "path": "docs/courses/LIE-TG/figures/equation-polynomial-pivots.svg", "reproducible_source": "docs/courses/LIE-TG/figures/equation-polynomial-pivots.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.32 and Example4.33 equation4.75, Exercise13/Solution13", "scope": "Exact order-three affine equation plane in four graph coordinates and monic symbol pivots for (1+p^2)u_xx=u_xy on1+p^2!=0." }, { "path": "docs/courses/LIE-TG/figures/cusp-geometric-quotient.svg", "reproducible_source": "docs/courses/LIE-TG/figures/cusp-geometric-quotient.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorems4.35–4.36 and Example4.37, Exercise14/Solution14", "scope": "Cusp parametrization sampled in601 points for-1.15<=t<=1.15; exact cusp ring, coordinate differential, nonzero fibre-pair nilpotent and generic inverse identities." }, { "path": "docs/courses/LIE-TG/figures/point-affine-quotient-tower.svg", "reproducible_source": "docs/courses/LIE-TG/figures/point-affine-quotient-tower.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.38", "scope": "Exact commutative quotient square, polynomial affine split and eventual formal transporter system." }, { "path": "docs/courses/LIE-TG/figures/point-invariant-generation.svg", "reproducible_source": "docs/courses/LIE-TG/figures/point-invariant-generation.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Example4.39 equations4.91–4.99, Exercise15/Solution15", "scope": "Exact low ring, order-three affine inverse, all-order triangular symbol determinant and polynomial affine-plane quotient tower." }, { "path": "docs/courses/LIE-TG/figures/intrinsic-cartan-reconstruction.svg", "reproducible_source": "docs/courses/LIE-TG/figures/intrinsic-cartan-reconstruction.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.40 and scalar projection of Example4.41", "scope": "Exact coordinate reconstruction and coefficient-class comparison; all plotted points and maps are specified in the caption." }, { "path": "docs/courses/LIE-TG/figures/intrinsic-coefficient-classes.svg", "reproducible_source": "docs/courses/LIE-TG/figures/intrinsic-coefficient-classes.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Example4.41, Lemma4.42 and Proposition4.43", "scope": "Exact coordinate reconstruction and coefficient-class comparison; all plotted points and maps are specified in the caption." }, { "path": "docs/courses/LIE-TG/figures/cartan-finite-jet-action.svg", "reproducible_source": "docs/courses/LIE-TG/figures/cartan-finite-jet-action.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.45, equations4.116–4.120", "scope": "Exact finite Cartan identity and one-domain consequence." }, { "path": "docs/courses/LIE-TG/figures/holonomic-intrinsic-presentation.svg", "reproducible_source": "docs/courses/LIE-TG/figures/holonomic-intrinsic-presentation.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Proposition4.46 and Example4.47, equations4.121–4.126", "scope": "Exact holonomic order/coefficient shift and explicit contact-map domain." }, { "path": "docs/courses/LIE-TG/figures/generic-fibre-orbit-rank.svg", "reproducible_source": "docs/courses/LIE-TG/figures/generic-fibre-orbit-rank.py", "license_expression": "CC0-1.0", "authorship": "GPT-6.1 Sol (OpenAI)", "proof_locator": "Theorem4.35, geometric generic fibre and equation4.77", "scope": "Finite normalization, scalar extension, full geometric orbit fibre and invariant differential rank." } ], "classical_symbolic_source": { "title": "Invariants différentiels d'un pseudogroupe de Lie. II", "author": "A. Kumpera", "journal": "Journal of Differential Geometry", "volume": 10, "issue": 3, "year": 1975, "pages": "347–416", "doi": "10.4310/jdg/1214432952", "url": "https://doi.org/10.4310/jdg/1214432952", "exact_results": [ { "locator": "H1,p.363; equivalent finite-jet H1-prime,p.365", "scope": "There is an order k1 such that the vertically reduced jet bundles have locally constant rank at every order k>=k1. The finite-jet version is over J_(k+1), not one common finite-level base." }, { "locator": "Lemma22.3,pp.361–362; Lemma22.5,p.364; Lemma23.1,p.368", "scope": "For every fixed infinite jet X, there is an order l(X) beyond which the symbol contractions/prolongation inclusion hold, under the stated regularity hypothesis." }, { "locator": "Theorem22.1,p.364; Theorem23.1,p.369", "scope": "The eventual equality with algebraic prolongation likewise has an order k1(X) depending on X." }, { "locator": "H1-prime-X and Theorem23.6,pp.378–379", "scope": "The localized regularity hypotheses and conclusion allow an open neighborhood of X_(k+1) that depends on k." } ], "quantifier_comparison": { "cited_pointwise": "For every X there exists l(X) such that for every k>=l(X), P(X,k).", "cited_local": "For a fixed X and each sufficiently large k there is a neighborhood W_(k+1) of X_(k+1) on which the stated order-k property holds.", "required_uniform": "There exist a finite l and a nonempty invariant Zariski open U in E^l such that P(a,k) holds for every k>=l and every compatible higher jet a above U.", "required_uniform_follows_from_cited_statements_alone": false }, "actual_geometric_comparison_closed": true, "general_theorem_refuted": false, "actual_comparison_proof": "Theorem4.28 gives the actual common finite-level full-symbol comparison; Theorem4.45 identifies the regular Cartan action with those action morphisms. The pointwise source statements alone do not supply the uniform quantifier." }, "finite_level_orbit_quotient": { "proof_locator": "Section 4.13, Theorem 4.19", "hypotheses": "Smooth affine algebraic group with regular action on an integral separated complex variety; real descent requires a geometrically integral variety and geometrically reduced affine algebraic group.", "construction": "Reduced monic basis of the full restricted orbit graph; coefficient functions separate orbit closures and normal-form coefficient ratios generate the invariant field.", "domain": "Nonempty invariant open in this fixed variety; target is an integral finite-type variety.", "fibres": "Single full-group orbits on all algebraically closed extension fields.", "field": "k(Y)=k(X)^G=k(c_1,...,c_N).", "scope": "This is the pointwise orbit quotient and field identification defined in Theorem 4.19; no additional scheme-theoretic quotient universal property is asserted.", "common_finite_level_domain_for_all_higher_orders": false, "source": { "authors": [ "Evelyne Hubert", "Irina A. Kogan" ], "title": "Rational invariants of a group action. Construction and rewriting", "url": "https://iakogan.math.ncsu.edu/papers/rinvHubertKogan.pdf", "version": "Author manuscript, 30 March 2006", "locators": [ "Proposition 2.7", "Theorem 2.14", "Lemma 2.15", "Theorem 2.16" ] } }, "horizontal_rank_and_quotient_derivations": { "proof_locator": "Section 4.14, Theorem 4.21 and Corollary 4.22", "hypotheses": "Integral algebraic jet tower with dominant projections, regular finite-level actions and covariant regular total differentiation; smooth-base version uses transitivity and integral jet fibres.", "domain": "Full regularity and independence locus of finitely many invariant horizontal differentials, at one finite level.", "uniformity": "All larger minors are zero rational identities; they vanish at every higher field-valued jet where their entries are regular.", "quotient": "The horizontal space modulo the common annihilator is canonically dual to the span of invariant horizontal differentials.", "derivatives": "Coefficient derivations dual to the selected differentials, represented by inverse-matrix coefficients at the finite level; invariant under the full action as operations on invariants.", "representative_independence": "Differences annihilate the selected forms and hence every invariant; annihilating fields form a Lie ideal among preserving fields.", "commutation": "The selected quotient derivations commute on the invariant field, even if their horizontal representatives have a nonzero annihilating bracket.", "function_class": "Rational through the chosen finite level and polynomial in higher graph jets; filtration shift at most +1 above that level.", "general_finite_generation_proved": false }, "finite_observations": { "proof_locator": "Section4.15 Theorem4.24; Example4.25 and Exercise10/Solution10", "hypotheses": "Integral Q-algebra, commuting tangential derivations with graph jet shift, arbitrary equation-coefficient normal derivations, finite determining and polynomial observation rows.", "conclusion": "One coefficient localization, one finite raw word/order bound and a split observed relation space valid for every ordinary field specialization and every higher truncation.", "normal_curvature": "Whole unipotently gauged observation copies retain curvature. No normally prolonged free scalar observation is substituted.", "finite_order": "Dummy components receive the maximum observation derivative degree; a bounded weighted normal slice is a finite tangential module.", "specialization": "Monic coefficient splittings and elimination, with rows evaluated after differentiation; field maps need not be differential.", "general_lie_tresse_proved": false, "geometric_point_application": "Lemmas4.26–4.27 and Theorem4.28." }, "common_domain_point_action": { "proof_locator": "Section4.16 Lemmas4.26–4.27 and Theorem4.28", "hypotheses": "Induced point action; transitive finite-order FI smooth jet groupoid with algebraic fixed-point groups; invariant nonsingular irreducible algebraic equation fibres and affine FI higher projections.", "quantifiers": "For every preassigned d>=1 there exist ell and an invariant nonempty fibrewise Zariski-open U in E^ell; all geometric higher jets above U have the stated properties at every k>=ell.", "coefficient_algebra": "Union R_j[[w]] with one finite fibre order per element; formal-unit graph inverses and equation-tangent normal lifts.", "real_convention": "Polynomial functions on the actual real algebraic set; irreducibility implies complex integrality by the explicit sum-of-squares proof. Arbitrary real schemes require geometric integrality.", "full_symbols": "Ann(varpi_k) equals the highest tangential characteristic relation image after inclusion of point-value rows; eventual equality with algebraic prolongation.", "finite_presentation": "Finite coefficients of finitely many transported fixed-degree rows generate one finite-level homogeneous module, with exact images at all higher field-valued jets.", "invariance": "Saturation by full isotropy, finite chart cover, intrinsic subbundle gluing and source-a smooth sections transport the domain over the base.", "stabilizers": "Stable low-d projections and adjacent first-derivative images; no surjectivity of whole stabilizers.", "general_finite_generation_proved": false }, "quotient_symbol_cohomology": { "proof_locator": "Section4.17 Theorem4.30, equations4.58–4.66; Example4.31 and Exercise12/Solution12", "scope": "The exact induced point-action and geometric/real-coordinate hypotheses of Section4.16.", "quantifiers": "There exist a finite L and a nonempty invariant fibrewise Zariski-open U in E^L such that H^(i,j)(g/varpi)=0 for every higher geometric jet above U, all i>=L and0<=j<=n.", "actual_module": "Truncated finite homogeneous equation annihilator J is included in the actual full-orbit annihilator I after one finite coefficient localization. Coefficient freeness of P/I and I/J splits both sequences, so specialized I/J is the actual quotient-symbol dual.", "homology": "Exterior multiplication contracts each polynomial variable. Finite homogeneous homology generators therefore give finite internal-degree support, including relation and higher-cycle degrees.", "specialization": "Coefficient-free differential cokernels split images and kernel sequences; the projective image splits the homology kernel sequence. Thus every field specialization preserves graded homology.", "indexing": "Internal degree a=i+j; choose L>=max(b+1,N+n) and all coefficient orders. Polynomial differentiation and insertion compare by degree-i rescaling1/i!.", "domain": "One finite product of coefficient denominators, saturated under full isotropy and base transport, intersected with the finite horizontal domain in an integral equation fibre.", "affine_bundles_proved": false, "polynomial_generation_proved": false, "action_only_extension_proved": false }, "equation_coordinate_pivots": { "proof_locator": "Section4.18 Theorem4.32, equations4.67–4.73; Example4.33, Figure4.12 and Exercise13/Solution13", "hypotheses": "Regular reduced finite-order equation ideal, ordinary FI prolongations, onto affine higher projections and the point-action finite fibre transport of Lemma4.26.", "quantifiers": "For a fixed basic point there exist finite L and a nonempty affine W in E_a^L such that every complete higher inverse image has coordinate algebra B[y_(L+1),...,y_k].", "actual_equations": "Finite expressions of monic symbol rows are realized by coefficients multiplying already ordinarily differentiated original ideal generators; coefficient order<=L