{ "course": "GL-GLC", "planned_lessons": 15, "written_drafts": 15, "whole_course_complete": false, "original_text_license": "CC0-1.0", "lessons": [ { "id": "GL-GLC-01", "source": "src/from-automorphic-functions-to-automorphic-sheaves.md", "proof_obligations": [ "full arithmetic Langlands and excursion theorems", "function–sheaf trace formalism and complete earlier dependency chains" ] }, { "id": "GL-GLC-02", "source": "src/the-moduli-stack-of-bundles.md", "proof_obligations": [ "remaining recursive Lie and reductive-group-scheme foundations", "remaining recursive local-algebra and projective-scheme foundations" ], "nilpotent_triple_field_transfer": { "theorem": "Every nonzero nilpotent element of a semisimple Lie algebra over an algebraically closed characteristic-zero field lies in an sl2 triple.", "root_data": "The finite rational Cartan/root table and semisimple-centralizer steps transfer to that field.", "filtration": "The kernel/image weight filtration is invariant under nonzero scaling and field extension; it is intrinsic over inner forms.", "boundary": "No rational triple over a smaller field or functorial arbitrary-ring choice is asserted; group/flag and recursive global nilpotent foundations remain required." }, "algebraic_triple_group_integration": { "theorem": "Every sl2 Lie algebra homomorphism into a smooth affine finite-type group over any characteristic-zero field integrates uniquely to an algebraic SL2 homomorphism.", "cocharacter": "The diagonal torus gives a cocharacter with exact differential h and the exact integer weight filtration.", "boundary": "A triple over the given field is an input. Parabolic classification/properness, inner-form subgroup descent and recursive global foundations remain required." }, "canonical_nilpotent_parabolic_and_curve_reduction": { "theorem": "The full intrinsic filtration scheme stabilizer is a canonical parabolic over any characteristic-zero field, including inner forms, with a projective flag quotient.", "curve": "Its generic reduction extends on every smooth projective curve; reduced finite-type families have finite reduction stratifications.", "boundary": "The extended special reduction need not be canonical; nonreduced-family canonical base-change, recursive global foundations and spectral regularity are not inferred." }, "parabolic_bundle_section_atlas_and_nilpotent_chart_isotropy": { "theorem": "Every parabolic bundle stack has an arbitrary-test projective section atlas, affine finitely presented diagonal and absolute smoothness.", "geometry": "The relative tangent complex and nilradical section cone supply dense-open chart isotropy and the nilpotent upper dimension bound.", "boundary": "The forgetting map need not be smooth; recursive global lower-dimension and spectral regularity foundations remain required." } }, { "id": "GL-GLC-03", "source": "src/sheaves-and-d-modules-on-bun-g.md", "proof_obligations": [ "Extension of the full independent nilpotent projector-image and ind-regularity theorem to other geometric ground fields", "derived tempered Satake", "microlocal support comparison and its ind-extension for nilpotent Riemann–Hilbert", "categorical descent/base-change foundations beyond the proved smooth quotient, closed/open transfer and projective correspondence class", "recursive curve, reductive-group and local-algebra foundations" ], "finite_point_detection_foundations": { "support": "closed conical N in T*V with dim N <= dim V; V smooth separated finite type in characteristic zero", "scope": "all unbounded D-module complexes in the cohomological characteristic-support convention", "theorem": "finite point !-fibres jointly detect the supported category", "generation_hypotheses": "P left adjoint to continuous conservative U, with U-image contained in the stated supported category", "global_nilpotent_dimension_and_projector_geometry": "General nilpotent chart upper bound proved and applied; lower dimension equality remains a separate Lesson2 obligation, and spectral projector geometry remains required.", "regularity": "not inferred from point detection or from the dimension bound" }, "nilpotent_chart_detection_and_ind_holonomic_cohomology": { "support": "actual global nilpotent cone on Bun_G; algebraically closed characteristic-zero field, smooth projective connected curve, arbitrary connected reductive group, all genera and components", "upper_bound": "proved in Lesson2 SG.8–SG.9; sufficient without lower dimension equality", "cohomology": "on each smooth affine chart, every cohomology module is a filtered union of coherent holonomic submodules", "detection": "finite normalized chart-point !-fibres per quasicompact open; all unbounded objects and morphisms detected across a cofinal exhaustion", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "formal_hecke_cotangent_and_levi_normalization": { "scope": "Algebraically closed characteristic-zero field; arbitrary connected reductive group with faithful closed representation and arbitrary invariant nondegenerate B", "jordan_and_cocharacter": "Compatible matrix Jordan components lie in Lie(G), the torus and nilpotent exponential lie in G scheme-theoretically, and C nonnilpotent gives C_G(lambda)=C_G(H) and B(C,dlambda) nonzero, including central directions", "formal_normalization": "Every coefficient k-algebra R, including nonreduced R: A0 in Lie(M), ad(A0) invertible on the M-stable complement; normalization h0=1 exists and its left M-arc coset is unique", "cotangent": "Actual framed arc-quotient tangent in degrees -1,0 and its continuous classical cotangent as regular compatible covector pairs", "curve_residue": "Crystal minus coordinate lift equals g^{-1}g-prime; at u^lambda the curve covector is the intrinsic pairing of A0 with dlambda", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "bounded_affine_springer_tangent_isolation": { "scope": "Algebraically closed characteristic-zero field, arbitrary connected reductive group, Levi M=C_G(lambda), A(u) in Lie(M)[[u]], complementary ad(A0) invertible", "bound": "Reduced Schubert closure of the L+G orbit; full affine Grassmannian nilpotents retained separately", "scheme_equations": "Actual finite principal-part coefficient equations on a sufficiently high smooth jet cover; closed ideals descend on all coefficient tests", "isolation": "The formal regular adjoint inverse kills the arc-orbit tangent; a complete Noetherian and determinant proof gives an open reduced Spec(k) point, also for the inverse modification in the opposite bound", "examples": "SL3 has six root-jet coefficients; arbitrary theta over arbitrary coefficient k-algebras has determinant729; torus nilpotent Laurent direction remains outside the bound", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "global_hecke_cotangent_and_compatible_higgs_fibres": { "scope": "Geometric-point classical cotangents over an algebraically closed characteristic-zero field, every connected reductive group and smooth connected projective curve; compatible-Higgs fibre and invariant transport on all ordinary coefficient algebras", "gluing_tangent": "Explicit degrees-minus-one-and-zero groupoid with complement-isomorphism tangent b and local image b-a1+a2; perfect globally normalized residue pairing", "compatible_fibre": "With an identified fixed input, all-test compatible-Higgs fibre equals the full Springer functor; both reduced bounded central-cocharacter points are open and reduced", "nilpotent_equations": "All homogeneous positive-degree invariant sections agree by Cartier injectivity, including nonreduced coefficient tests", "curve_relation": "Crystal minus coordinate transport gives positive Res dx; central cocharacter gives intrinsic nonzero B(C,d-lambda) dx", "examples": "Global SL3 bundle O(-x)^2 plus O(2x), full Jordan trace residue6q(0)dx and both adjoint blocks; P1 complement tangent residue1; dual-number matrix with nonzero invariant trace", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "bounded_hecke_IC_conormal_and_global_source": { "scope": "Algebraically closed characteristic-zero geometric-point classical cotangent fibres at the open reduced Schubert orbit; original finite-stage nilpotents retained; full unbounded external inputs and all ordinary coefficient tests for finite frames and bundle gluing", "kernel": "Actual middle extension restricts to smooth closed-orbit direct image; normal PBW gives all conormal symbols", "finite_stages": "Ambient quotient and tangent-space injection prove embedding independence and transition compatibility for this kernel", "unbounded": "Exact external/closed transfer and finite coefficient generation give the coherent-cohomological union product without closure", "global_source": "Finite congruence subgroup fixes the entire open orbit on all rings; compatible formal frames lift on affine tests and effective gluing gives the product; the compatible covector restricts to the input support times orbit zero section", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "isolated_cotangent_tests_and_cycle_summand": { "scope": "Algebraically closed characteristic-zero smooth cotangent geometric points; proper complex-sheaf formal deduction with every required premise explicit; full-unbounded categorical lemma only under exact Ind/left-completion hypotheses; complete analytic ramification and positive-dimensional cancellation examples", "geometry": "Symmetric Q graph transverse to any tangent subspace of dimension at most dimY; Hessian realizes a regular test; zero local tangent makes its intersection isolated; quasifinite cotangent fibre transfers isolation", "summand": "Disjoint closed support gives a unique functorial point inclusion/projector on complexes; extension by colimits and truncation limits retains the finite biproduct under explicit hypotheses", "proper_image": "Universal-cover nearby cycles, the actual specialization map and cone give proper comparison subject to exact base-change inputs; the point retract survives with the same coefficient complex", "examples": "z^m produces m-1 normalized cycles and cotangent fibre length; every nonzero target covector detected by m contractible sectors; compact torus monodromy(2,1) yields an explicitly contractible cochain complex and fails quasifiniteness", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "unbounded_residue_fields_and_field_extension_point_detectors": { "scope": "Full unbounded Noetherian residue-field detection and smooth affine D-module point Hom after field extension; global orthogonality only under explicit adjunction and category base-change premises", "residue": "Prime filtration, dimension induction through Frac(A)/A and stable tensor kernel prove conservativity without boundedness", "point": "A detecting prime gives a closed K-rational point after extension; right-PBW and a finite normal Koszul resolution compute delta Hom for all unbounded complexes", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "full_Dmodule_categorical_field_extension": { "scope": "All unbounded module categories; arbitrary fields; full strong-descent D-modules on smooth algebraic stacks with affine diagonal, including BunG and specified half twist", "category_duality": "Evaluation by relative tensor and coevaluation by the regular diagonal bimodule; augmented bar contractions prove both triangles", "descent": "Categorical tensor commutes with limits by duality; affine operator and smooth/regular transfer base change identify full nerve and open diagrams", "compact_tests": "Free/forgetful adjunction proves compact Hom and generator scalar extension", "fixed_support_boundary": "Extended zero-support connection contains an original-field coherent free D-module; continuous full presentable inclusions obey the internal-action image criterion", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "rigid_action_enhanced_Hecke_induction": { "scope": "Full unbounded balanced eigen-objects for a specified continuous compact rigid action, compactly generated automorphic module and affine DG coefficient algebra", "regular_algebra": "Explicit multi-object coend, continuous multiplication right adjoint and coherent commutative algebra/balancing", "enhancement": "Actual free enhanced induction and conservative continuous forgetful adjunction under the specified action data; full action-module equivalence", "field_extension": "Every field extension carries the regular algebra, adjunction and compact generators to those for the extended specified data", "examples": "Laurent nonidempotence, two-character idempotents and the derived quotient Ext/Tor obstruction", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "actual_proper_curve_Dmodule_Ran_diagram": { "scope": "Full unbounded D-module Ran category on a smooth projective curve over any characteristic-zero field, for specified compact rigid coefficient category with compact unit", "projective_compacts": "All bounded coherent operator complexes equal all compact D-modules; induced ample twists generate", "transitions": "Actual coordinate direct images are continuous and preserve compacts, with coherent composition and product compatibility", "colimit": "Explicit coherent Grothendieck module localization proves the actual full colimit and its compact generators, compact unit and symmetric tensor", "examples": "Sequential reflector, finite-point normal form and unit, nonproper Weyl compact failure, two-weight Laurent regular algebra", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "proper_Dmodule_duality_and_full_structure_map_adjunctions": { "scope": "All full unbounded D-modules on every smooth projective variety of pure dimension d over any characteristic-zero field", "constant": "O[-d] represents normalized de Rham direct image; p* left adjoint to p_*", "proper": "O[d] tensored with a k-complex is right adjoint to p_*; all-coherent duality and compact bar extension prove the full statement", "external": "Compact coherent Verdier duality respects external products with their density factors and Koszul signs", "examples": "Projective-line ordered residue and both shifts; nonholonomic regular module on P^n; exact affine-line failure of the proposed right adjunction", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "actual_Dmodule_Ran_kernel_contraction_and_rigidity": { "scope": "Actual full D-module Ran category for smooth projective curves over every characteristic-zero field, with a specified compact rigid coefficient category", "pairing": "Full unbounded operator pairing with compact first input; no finite-dimensional Hom assumption", "kernel": "Canonical diagonal mates and signed three-coordinate contraction before forgetting coefficient-labelled geometry", "rigidity": "Both actual Ran triangles, all compact objects dualizable and the actual regular-algebra coend", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "chartwise_ind_regular_RH_and_half_twist_Morita": { "field": "Complex ground and coefficient field", "charts": "Full bounded derived regular/finite algebraically constructible categories, ind-completed on each smooth finite-type affine chart", "stack": "Actual coherent inverse-image descent limit; no interchange of Ind with a global limit", "twist": "Actual normalized global root, operator-bimodule Morita triangles and intermediate correspondence-kernel cancellation", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "bounded_affine_springer_isolation_proved": true, "opposite_bound_inverse_modification_isolation_proved": true, "springer_scheme_local_equations_on_all_coefficient_tests_proved": true, "six_root_jet_family_calculation_all_coefficient_rings_proved": true, "global_hecke_cotangent_compatibility_proved": true, "full_satake_singular_support_calculation_proved": false, "proper_hecke_pushforward_covector_survival_proved": false, "full_suitable_open_nilpotent_support_preservation_proved": true, "general_nilpotent_regularity_proved": false, "spectral_projector_image_proved": false, "whole_lesson_complete": false, "whole_course_complete": false, "all_coefficient_compatible_higgs_springer_fibre_equivalence_proved": true, "all_coefficient_hitchin_invariant_transport_proved": true, "global_crystal_curve_covector_relation_proved": true, "bounded_global_compatible_higgs_fibre_isolation_both_directions_proved": true, "bounded_hecke_IC_open_orbit_characteristic_fibre_proved": true, "bounded_hecke_IC_nilpotent_stage_cotangent_compatibility_proved": true, "full_unbounded_external_IC_support_product_proved": true, "finite_jet_global_open_hecke_orbit_product_proved": true, "bounded_hecke_source_covector_membership_proved": true, "regular_holonomic_proper_pushforward_lower_bound_proved": false, "global_nilpotent_regularity_proved": false, "quadratic_cotangent_isolation_at_smooth_points_proved": true, "codifferential_quasifinite_isolation_transfer_proved": true, "isolated_cycle_functorial_support_summand_proved": true, "conditional_ind_left_completion_summand_lemma_proved": true, "conditional_proper_cycle_survival_deduction_proved": true, "finite_ramification_all_nonzero_covector_example_proved": true, "positive_dimensional_proper_fibre_cancellation_example_proved": true, "full_unbounded_noetherian_residue_field_conservativity_proved": true, "finite_type_noetherian_finite_dimension_proved": true, "field_extension_closed_rational_point_detection_proved": true, "full_unbounded_affine_delta_koszul_detection_proved": true, "operator_sheaf_field_extension_comparison_proved": true, "conditional_global_field_orthogonality_deduction_proved": true, "intrinsic_stack_point_adjunction_proved": false, "spectral_category_field_extension_equivalence_proved": false, "explicit_module_category_duality_proved": true, "arbitrary_field_category_scalar_extension_limit_comparison_proved": true, "full_strong_smooth_Dmodule_field_extension_proved": true, "half_twisted_strong_smooth_field_extension_proved": true, "compact_mapping_complex_scalar_extension_proved": true, "full_automorphic_compact_generators_field_extension_proved": true, "fixed_zero_support_transcendental_connection_obstruction_proved": true, "regular_algebra_multiobject_bar_proved": true, "rigid_action_enhanced_Hecke_adjunction_proved": true, "rigid_action_full_Hecke_module_category_proved": true, "rigid_action_affine_coefficient_compact_generators_proved": true, "rigid_action_Hecke_field_extension_proved": true, "Laurent_Hecke_monad_nonidempotence_proved": true, "derived_quotient_nonfullyfaithful_example_proved": true, "actual_geometric_Ran_Hecke_action_constructed": false, "actual_nilpotent_regular_image_proved": false, "full_projective_Dmodule_compact_classification_proved": true, "proper_coordinate_Dmodule_compact_preservation_proved": true, "actual_geometric_Dmodule_Ran_diagram_constructed": true, "actual_geometric_Dmodule_Ran_compact_generation_proved": true, "actual_geometric_Dmodule_Ran_symmetric_tensor_constructed": true, "finite_point_Ran_normal_form_proved": true, "nonproper_free_Dmodule_compact_failure_proved": true, "actual_geometric_Ran_rigidity_proved": true, "proper_coherent_Dmodule_duality_normalization_proved": true, "full_unbounded_projective_structure_map_adjunctions_proved": true, "compact_Dmodule_external_duality_proved": true, "nonproper_full_Dmodule_duality_obstruction_proved": true, "actual_Dmodule_Ran_kernel_contraction_proved": true, "actual_Dmodule_Ran_regular_algebra_constructed": true, "general_riemann_hilbert_proved": true, "chartwise_ind_regular_stack_RH_proved": true, "half_twisted_ind_regular_stack_RH_proved": true, "half_twist_kernel_Morita_cancellation_proved": true, "nilpotent_microlocal_RH_restriction_proved": false, "complex_classical_coefficient_fusion_Hecke": { "field": "Complex ground and coefficient field", "coefficients": "Ind of the actual bounded derived finite-dimensional dual-group representation category; full ambient spherical mapping complexes retained", "geometry": "Finite-support regular D-module fusion families and all finite-tuple Hecke functors on full unbounded Dmod(Bun_G), including the transported normalized global half twist", "unbounded": "Actual projective correspondence calculus on singular supports; arbitrary coefficient functors defined by colimits of bounded-support formulas", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_classical_coefficient_DG_Satake_functor_constructed": true, "complex_classical_coefficient_fusion_Hecke_action_constructed": true, "full_unbounded_projective_correspondence_calculus_proved": true, "full_D_module_valued_unital_Ran_Hecke_action_constructed": false, "complex_full_D_module_unital_Ran_Hecke_action": { "geometric_ground_field": "Complex numbers", "categories": "Full Ran coefficient category of all finite-set functions and full unbounded D-module parameter complexes; full unbounded Dmod(Bun_G) with the global half twist", "relations": "Actual diagram transitions, unital monoidal action, all coherent chains and compact dual adjunctions", "parameters": "Arbitrary commutative DG complex scalar algebra through full derived module bars", "enhanced": "Actual universal enhanced category with geometric first action and regular Ran second action", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_full_D_module_valued_unital_Ran_Hecke_action_constructed": true, "complex_full_Ran_Hecke_compact_dual_adjunction_proved": true, "complex_full_Ran_Hecke_derived_scalar_parameters_constructed": true, "complex_universal_enhanced_Ran_Hecke_category_constructed": true, "all_characteristic_zero_geometric_ground_field_Ran_action_constructed": false, "complex_derived_spectral_coefficient_and_enhanced_category": { "geometric_ground_field": "Complex numbers", "categories": "Full Ran representation category and every unbounded D-module parameter; quasicoherent complexes on the full derived de Rham tensor-functor prestack", "relations": "Every finite-set transition, coherent monoidality, nonflat derived affine-point base change and Cartesian gluing", "enhanced": "Actual regular-algebra spectral module category, affine compact generation, derived scalar category comparison and global enhanced adjunction", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_derived_de_Rham_spectral_coefficient_functor_constructed": true, "complex_spectral_coefficient_full_Ran_transitions_proved": true, "complex_spectral_coefficient_nonflat_derived_base_change_proved": true, "complex_enhanced_spectral_category_constructed": true, "complex_enhanced_affine_point_compact_generators_proved": true, "complex_global_enhanced_spectral_adjunction_constructed": true, "spectral_diagonal_regular_algebra_identification_proved": false, "global_enhanced_spectral_point_generation_proved": true, "geometric_Beilinson_projector_nilpotent_image_proved": true, "all_characteristic_zero_geometric_ground_field_spectral_action_constructed": false, "complex_full_Ran_tensor_universal_property_and_derived_spectral_paths": { "geometric_ground_field": "Complex numbers", "categories": "Full curve D-module category; actual full Ran category; every connective derived affine spectral point", "relations": "Entire tensor universal property; full compact-DG coend and all coherent derived path spaces; nonflat connective scalar extension", "example": "Actual projective-line torus diagonal algebra, full multiplicative Spencer–Cech model and degree-two paths", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_full_D_module_categorical_self_duality_proved": true, "complex_full_Ran_tensor_universal_property_proved": true, "complex_derived_spectral_point_pair_path_functor_identified": true, "complex_spectral_pair_algebra_nonflat_connective_base_change_proved": true, "projective_line_torus_derived_diagonal_algebra_proved": true, "general_spectral_pair_algebra_connectivity_proved": false, "complex_connective_affine_spectral_diagonal_and_unit_pushforward": { "geometric_ground_field": "Complex numbers", "categories": "All connective DG bases and full unbounded operator and quasicoherent module categories; all-DG target tests in algebra Yoneda", "coefficient": "Finite projective normalized coefficients from all tensor powers and the full derived residue criterion", "diagonal": "Actual connective affine spectral diagonal, nonflat derived fibres, regular-algebra unit pushforward and full affine module descent", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_derived_spectral_normalized_coefficients_finite_projective_proved": true, "complex_full_commutative_algebra_pushforward_left_adjoint_constructed": true, "complex_normalized_curve_algebra_left_adjoint_connectivity_proved": true, "complex_general_spectral_pair_algebra_connectivity_proved": true, "complex_spectral_diagonal_representable_affine_proved": true, "complex_spectral_diagonal_regular_algebra_unit_pushforward_proved": true, "complex_affine_diagonal_full_QCoh_module_descent_proved": true, "complex_global_QCoh_tensor_comparison_proved": true, "complex_global_spectral_localization_proved": true, "complex_full_derived_spectral_torsors_and_bounded_frames": { "spectral_geometric_ground_field": "Complex numbers", "categories": "Entire spectral tensor-functor infinity-groupoid, full operator torsors and every connective DG base", "bounds": "Degree zero and exact slope bound for flat bundles over every algebraically closed characteristic-zero field", "frames": "Actual compact Ran value gives full DG projective frames and evaluation kernel; one finite-type ordinary reduced bundle atlas", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_full_derived_spectral_torsor_connection_reconstruction_proved": true, "char0_flat_bundle_degree_zero_and_exact_HN_bound_proved": true, "complex_full_DG_spectral_projective_global_section_frames_proved": true, "complex_full_DG_spectral_evaluation_kernel_finite_projective_proved": true, "complex_bounded_classical_spectral_bundle_frame_atlas_proved": true, "complex_full_derived_spectral_connection_atlas_constructed": true, "complex_derived_connection_and_quotient_atlas": { "spectral_geometric_ground_field": "Complex numbers", "categories": "Entire derived connection fibre and every connective DG coefficient algebra, including all higher paths", "cohomology": "Full proper curve two-term projective cohomology and all nonflat derived base changes", "frames": "Full square-zero and effective Postnikov comparison for the bounded frame scheme", "atlas": "Finite derived affine connection scheme W over the smooth ordinary Q; actual spectral prestack [W/GL_b]", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_full_proper_curve_DG_cohomology_proved": true, "complex_full_DG_bounded_frame_functor_represented": true, "complex_actual_derived_spectral_quotient_atlas_proved": true, "complex_global_spectral_localization_and_projector": { "spectral_geometric_ground_field": "Complex numbers", "categories": "Entire unbounded DG quasicoherent, Ran and enhanced categories; all higher maps retained", "quotient_generators": "Inverse ample determinant powers tensored with every finite rational frame representation", "localization": "Actual global exterior tensor equivalence and continuous fully faithful localization right adjoint, with the genuine diagonal counit map", "projector": "Actual full de Rham categorical projector with its adjunction; the restricted-variation nilpotent Beilinson projector remains to be constructed", "point_generators": "All geometric residue-field points of the full bounded DG atlas, including generic coefficient fields over C; no identification with the restricted-variation functor is asserted", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_global_QCoh_compact_generation_proved": true, "complex_global_spectral_localization_right_adjoint_continuous_fully_faithful_proved": true, "complex_global_enhanced_spectral_compact_generation_proved": true, "complex_ordinary_spectral_projector_constructed": true, "complex_geometric_coefficient_field_point_generation_proved": true, "complex_full_quasilisse_curve_coefficient_construction": { "spectral_geometric_ground_field": "Complex numbers", "categories": "Entire completed lisse target on every smooth projective curve, with all unbounded objects and every higher mapping space", "positive_genus": "Full derived-heart model D(Ind finite connections), cofree acyclicity, injective dimension at most2 and effective arbitrary Postnikov towers", "projective_line": "All modules over the free associative primitive Hopf generator of degree-1; finite-lisse ind-category is its u²-torsion subcategory", "DG_coefficients": "Complete internal A-actions and scalar tensor bars for every connective DG A, including all negative coefficients", "P1_restricted_functor": "Actual full framed Spec Sym(h_dual[1]) and unframed quotient by H for every connected reductive complex group; all paths and nonflat changes", "rank_one_variation": "Ordinary original-field locally finite monodromy, finite coefficient algebra and its infinitesimal fixed-character description", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_full_quasilisse_curve_coefficient_target_constructed": true, "complex_positive_genus_derived_lisse_heart_comparison_proved": true, "complex_positive_genus_full_lisse_Postnikov_effectivity_proved": true, "complex_P1_full_quasilisse_Koszul_completion_model_proved": true, "complex_full_DG_quasilisse_scalar_action_bars_proved": true, "complex_P1_actual_restricted_spectral_functor_all_reductive_H_proved": true, "complex_classical_rank_one_original_field_restricted_variation_proved": true, "complex_general_curve_restricted_spectral_atlas_constructed": true, "complex_general_curve_restricted_spectral_localization_proved": false, "complex_restricted_spectral_enhanced_generation_proved": false, "complex_general_curve_restricted_prestack_foundations": { "spectral_geometric_ground_field": "Complex numbers", "categories": "Full continuous right-t-exact tensor functor infinity-space into the actual completed lisse target, with every DG coefficient and higher comparison", "convergence": "Entire connective internal-module action windows and scalar recovery; all coherent representation and tensor diagrams retained", "descent": "Full unbounded faithfully flat coefficient descent through the actual compact-DG module model", "frames": "Actual affine smooth H-frame torsor and full fpqc quotient of prestacks; not an ind-affine representability assertion", "bounded_coefficients": "Exact finite-lisse ind-target replacement when the coefficient algebra has finitely many nonzero cohomology degrees", "ordinary_condition": "Original-complex-field finite monodromy image algebra on one faithful representation and its dual", "remaining": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_general_curve_restricted_tensor_functor_constructed": true, "complex_general_curve_restricted_DG_convergence_proved": true, "complex_general_curve_restricted_faithfully_flat_descent_proved": true, "complex_general_curve_restricted_full_frame_quotient_proved": true, "complex_restricted_bounded_coefficient_ind_target_comparison_proved": true, "complex_classical_faithful_representation_original_field_finiteness_proved": true, "complex_general_curve_restricted_Hecke_action_constructed": false, "complex_ordinary_restricted_coefficient_charts": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Every ordinary commutative complex algebra; no reducedness, Noetherianity or connectedness assumption", "framed_points": "A-linear Hopf maps O(H)_A to the actual representative-function Hopf algebra C_X,A", "charts": "Finite antipode-stable coefficient subcoalgebras; finitely many polynomial relation/coproduct/counit equations; affine finite-type stages", "transitions": "Closed coordinate embeddings; all ordinary points factor through one finite stage", "quotient": "Conjugation-stable stages and actual fpqc frame descent; representable closed quotient transitions", "examples": "Triangular polynomial family, exact ordinary dual-number stage and full P1 derived point loop", "limitations": "Ordinary restriction only. Full DG framed ind-affine presentation, geometric charts, restricted localization/Hecke and nilpotent projector/image/regularity are additional assertions." }, "complex_ordinary_restricted_framed_ind_affine_presentation_proved": true, "complex_ordinary_restricted_finite_type_coefficient_stages_constructed": true, "complex_ordinary_restricted_closed_stage_transitions_proved": true, "complex_ordinary_restricted_finite_stage_fpqc_quotient_proved": true, "complex_derived_restricted_deformation": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Every connective commutative complex DG algebra, without a lower cohomological bound", "ind_target": "Every finite representation value of every restricted point lies in the actual finite-lisse ind-target", "embedding": "Fully faithful full DG operator map to the flat-torsor prestack", "nilpotent_lifts": "Full Cartesian lift comparison for every map with surjective H0 and nilpotent kernel", "gluing": "Finite projective frames glue on the derived scalar pullback when one H0 map is a nilpotent quotient; both full and restricted point functors are infinitesimally cohesive", "examples": "Sharp SL2 associative nilpotence, bi-infinite all-module obstruction, and infinitely many higher P1 coefficient point groups", "limitations": "Compact connective cotangent calculation, effective derived ind-affine charts, restricted localization and Hecke action, nilpotent projector with its support image and original-field ind-regularity remain additional assertions." }, "complex_restricted_all_connective_ind_target_comparison_proved": true, "complex_restricted_full_operator_embedding_proved": true, "complex_restricted_nilpotent_lifting_all_DG_coefficients_proved": true, "complex_restricted_formally_etale_embedding_proved": true, "complex_derived_nilpotent_finite_projective_gluing_proved": true, "complex_restricted_full_infinitesimal_cohesiveness_proved": true, "complex_restricted_framed_cotangent": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Every connective commutative complex DG algebra, with no lower cohomological bound", "actual_deformation": "The full split square-zero framed lift space is Omega-infinity of fib(adjoint deRham cohomology to its point fibre), tensored with the full ideal and shifted by1", "cotangent": "The fibre of dual evaluation from h-dual tensor A to the dual adjoint deRham complex", "finiteness": "A compact connective A-module of relative Tor amplitude[-1,0], natural under frame changes and arbitrary nonflat coefficient changes", "examples": "Constant points in every genus, a nonsemisimple elliptic point, a nonzero second-order SL2 obstruction, and every higher derivative group on unbounded coefficients", "limitations": "Effective derived ind-affine stages, their whole convergent point functor, restricted localization and Hecke action, nilpotent projector with support image and original-field ind-regularity remain additional assertions." }, "complex_restricted_full_split_derivative_computed": true, "complex_restricted_compact_connective_framed_cotangent_proved": true, "complex_restricted_framed_cotangent_Tor_minus1_0_proved": true, "complex_restricted_cotangent_nonflat_scalar_comparison_proved": true, "complex_effective_restricted_derived_charts": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Every connective commutative complex DG algebra, arbitrarily unbounded below", "ambient": "The actual affine based Betti mapping functor of a finite pointed homotopy model of the curve", "ordinary_stages": "Closed coordinate surjections from finite word evaluations of the actual SCI coefficient coalgebras, including nonreduced ordinary coefficients", "derived_stages": "Odd Koszul generators with differential f_i^N, closed transitions, finite derived presentation and quasi-smooth cotangent Tor amplitude[-1,0]", "effectivity": "A filtered infinity-category with full coherent cones represents the whole actual framed restricted point functor; all unbounded nullhomotopy fibres are identified", "quotient": "The original full faithfully flat H-frame torsor and intrinsic H-action, without equivariance of every chosen generator tuple", "examples": "Five complete solutions: projective-line/elliptic charts, surplus stage paths, original-field families, nilpotence threshold and a uniform unbounded DG homotopy", "limitations": "Actual restricted localization/Hecke, nilpotent projector with singular-support image and original-field regularity, and geometric ground-field extensions remain additional assertions." }, "complex_restricted_full_Betti_frame_comparison_constructed": true, "complex_restricted_full_Betti_nilpotent_comparison_proved": true, "complex_restricted_framed_derived_ind_affine_presentation_proved": true, "complex_restricted_effective_derived_Koszul_stages_constructed": true, "complex_restricted_stage_coherent_filtered_effectivity_proved": true, "complex_restricted_stage_finite_derived_presentation_proved": true, "complex_restricted_stage_quasismooth_cotangent_proved": true, "complex_restricted_actual_diagonal": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Every connective commutative complex DG algebra, arbitrarily unbounded below", "diagonal": "Affine on the full point functor, with actual Betti quotient fibre algebras and faithful flat unframed descent", "quasicoherent_algebra": "Diagonal unit pushforward on the actual product, full module adjunction, arbitrary derived base change and projection formula", "Ran_algebra": "Image of the full regular algebra after the canonical exterior functor to the actual product", "examples": "Three complete solutions for projective-line and elliptic whole automorphism fibres and exact nonflat Koszul reduction", "limitations": "Restricted global exterior-tensor/duality and relative tensor localization, compatible Hecke action, nilpotent projector/image/original-field regularity and geometric ground-field extensions are additional assertions" }, "complex_restricted_actual_affine_diagonal_proved": true, "complex_restricted_diagonal_all_DG_base_change_proved": true, "complex_restricted_diagonal_QCoh_module_adjunction_proved": true, "complex_restricted_Ran_regular_algebra_on_actual_product_proved": true, "complex_restricted_global_exterior_tensor_equivalence_proved": true, "complex_restricted_global_relative_tensor_localization_proved": true, "complex_koszul_torsion_completion": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Any connective commutative complex DG algebra, with modules unbounded in both directions", "power_stages": "Relative semifree polynomial model, finite strict nilpotent stage filtration and full coherent first-reduction cancellation", "equivalence": "Actual QCoh formal completion, compactly generated Koszul torsion modules and derived-complete modules, with explicit inverse functors", "monoidal_structure": "Ambient torsion tensor, completed tensor and colimits, actual units without a compact-unit assumption", "exterior_tensor": "Canonical equivalence for any two actual formal affine charts", "examples": "Four complete solutions for zero equations, doubly unbounded coefficients, noncompact formal-disc unit, two-variable principal parts and nontrivial full module actions", "limitations": "Geometric projector, compatible Hecke realization, nilpotent support image and original-field regularity require further geometric arguments; other ground fields require their comparisons" }, "complex_Koszul_module_tower_all_unbounded_counit_proved": true, "complex_formal_affine_Koszul_QCoh_comparison_proved": true, "complex_formal_affine_torsion_completion_equivalence_proved": true, "complex_formal_affine_external_tensor_equivalence_proved": true, "complex_supported_union_localization": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Actual restricted prestack on all connective complex DG algebras; modules unbounded in both directions", "supported_union": "Intrinsic continuous coreflections and explicit inverse functors between full supported modules and actual QCoh of the framed union", "quotient": "Finite equivariant Koszul generators constructed in derived rational H representations, with full torsor descent", "exterior_tensor": "Canonical quotient exterior equivalence and dualizability of the presentable DG category", "relative_tensor": "Actual multiplication A tensor over R with A to A is an equivalence, via the full Ran regular algebra and module bars", "examples": "Four complete solutions: all closed points of a line, equivariant weights, nonflat derived intersection and elliptic noncontinuous spectral right adjoint", "limitations": "Geometric projector, compatible Hecke realization, nilpotent support image and original-field regularity require further geometric arguments; other ground fields require their comparisons" }, "complex_restricted_framed_supported_union_QCoh_comparison_proved": true, "complex_restricted_framed_global_exterior_tensor_equivalence_proved": true, "complex_restricted_H_equivariant_Koszul_generators_constructed": true, "complex_restricted_QCoh_compact_generation_proved": true, "complex_restricted_QCoh_dualizability_proved": true, "complex_restricted_full_tensor_Ran_diagonal_algebra_identified": true, "complex_restricted_tensor_self_duality": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Actual restricted quotient and full unbounded modules", "compact_objects": "Finite perfect underlying modules, supported at one coefficient stage, with actual supported tensor duals", "global_sections": "Continuous rational-invariant functor agrees with ordinary global sections on all compact objects; compact-object extension uniquely determines it", "tensor_self_duality": "Canonical evaluation is continuous global sections after tensor; actual diagonal and full Ran regular algebra give its coevaluation", "examples": "Two complete solutions compare the formal-disc unit sections and prove failure of a left adjoint to the affine polynomial unit functor", "limitations": "Geometric projector, compatible Hecke realization, nilpotent support image, original-field regularity and other geometric fields require further arguments" }, "complex_restricted_compact_tensor_duals_proved": true, "complex_restricted_compact_global_sections_comparison_proved": true, "complex_restricted_continuous_renormalized_global_sections_constructed": true, "complex_restricted_canonical_tensor_self_duality_proved": true, "complex_restricted_actual_diagonal_coevaluation_proved": true, "complex_universal_restricted_idempotent": { "geometric_ground_field": "Complex numbers", "coefficient_domain": "Actual restricted quotient and full unbounded Ran and automorphic categories", "continuous_transpose": "Whole compact DG coend, exact R-module comparison and actual coevaluation triangle", "universal_idempotent": "Non-unital commutative object with explicit inverse to multiplication; no source-unit map or adjunction inferred", "geometric_realization": "Full restricted enhanced Hecke eigen-families; integration with continuous global sections is the actual idempotent geometric Ran action", "examples": "Three complete solutions for closed/open support, doubly unbounded coefficients and the completion tensor kernel invisible to all finite jets", "limitations": "Full spectral-subcategory comparison, nilpotent singular-support image, original-field regularity and other geometric fields require further arguments" }, "complex_restricted_continuous_transpose_R_linearity_proved": true, "complex_restricted_universal_spectral_idempotent_constructed": true, "complex_restricted_enhanced_geometric_Hecke_realization_proved": true, "complex_restricted_geometric_spectral_idempotent_constructed": true, "complex_bounded_charts_lisse_image": { "geometric_ground_field": "Complex numbers", "local_algebra": "Perfect cotangent amplitude [-1,0] and coherent finiteness give local finite Koszul models and bounded actual stages", "coefficient_comparison": "Full bounded-affine and frame descent lifts universal normalized coefficients to the ind-connection category for every connective parameter", "geometric_image": "Actual parameterized Hecke eigencomparison and continuous Gamma! integration give Hecke-lisse membership on full unbounded inputs", "coefficient_categories": "The whole ind-category need not be left-complete or equal to its completion", "examples": "Three complete solutions for local even-cell cancellation, signed odd actions over unbounded scalars and a non-effective completed tower", "limitations": "Nilpotent singular-support image, original-field regularity, full spectral-subcategory comparison and other geometric fields require further proofs" }, "complex_restricted_affine_stages_cohomologically_bounded_proved": true, "complex_quasi_smooth_local_Koszul_model_proved": true, "complex_restricted_universal_coefficients_ind_lisse_proved": true, "complex_restricted_normalized_ind_completed_functors_all_connective_coefficients_proved": true, "complex_restricted_parameterized_Hecke_eigencomparison_proved": true, "complex_restricted_geometric_idempotent_Hecke_lisse_image_proved": true, "complex_Hecke_lisse_coefficient_action": { "geometric_ground_field": "Complex numbers", "subcategory": "Full presentable simultaneous kernel, actual continuous ind-connection coreflector and coherent coordinate intersections", "geometric_action": "Every finite-coordinate Hecke operation takes values in the same full Hecke-lisse category with ind-connection coefficients; the entire full Ran action preserves it", "coefficient_category": "Canonical tensor pairing transposes the correct coalgebra, and a full large-cardinal DG presentation constructs its actual Ran category and continuous action", "spectral_comparison": "Actual universal-coefficient functor from the lisse-coefficient Ran category to the restricted spectral category, with composite equal to the original Ran coefficient functor", "examples": "Three complete solutions for unbounded coordinate torsion, the noncompact actual projective-line transposed diagonal and a full mapping-complex counterexample to an idempotence shortcut", "limitations": "Equivalence of the lisse Ran spectral comparison, full nilpotent-support image, original-field regularity and other geometric fields require separate proofs" }, "complex_ind_connection_continuous_coreflector_constructed": true, "complex_Hecke_lisse_subcategory_presentable_proved": true, "complex_Hecke_lisse_all_coordinates_and_Ran_stability_proved": true, "complex_ind_connection_canonical_tensor_self_duality_proved": true, "complex_lisse_transposed_coalgebra_constructed": true, "complex_lisse_coefficient_Ran_category_constructed": true, "complex_Hecke_lisse_coefficient_Ran_action_constructed": true, "complex_lisse_Ran_restricted_spectral_functor_constructed": true, "complex_lisse_Ran_restricted_spectral_equivalence_proved": true, "complex_full_restricted_spectral_equivalence": { "geometric_ground_field": "Complex numbers", "ambient": "Actual full framed Betti algebra is almost finite type, quasi-smooth and cohomologically bounded", "comparison": "Actual continuous symmetric monoidal lisse Ran functor to QCoh of restricted local systems is an equivalence, on full unbounded categories", "action": "Full Hecke-lisse category carries the actual unital restricted spectral action", "image": "Full essential image of the actual geometric idempotent equals the full Hecke-lisse category", "examples": "Three complete solutions for variable characters and generic residue fields, finite algebra filtrations of unbounded modules, and conservative change to nonzero field-linear categories", "limitations": "Nilpotent singular-support comparison, original-field ind-regularity and other geometric ground fields require separate geometric proofs" }, "complex_full_Betti_ambient_cohomologically_bounded_proved": true, "complex_full_Betti_spectral_equivalence_proved": true, "complex_restricted_supported_kernel_all_unbounded_residue_generation_proved": true, "complex_Hecke_lisse_restricted_spectral_action_proved": true, "complex_geometric_idempotent_full_Hecke_lisse_image_proved": true, "complex_curve_transpose_normalization": { "geometric_ground_field": "Complex numbers", "normalization": "Full proper-curve integral is represented by O[-1]; normalized ind-connection unit is O[1]; tensor transpose is t=g[2]", "membership": "Unshifted adjunction i adjoint g supplies membership and coordinate coreflectors", "coordinate_comparison": "The transpose t is a coalgebra map on every finite set; the equivalent g comparison has shift2(|J1|-|J2|)", "geometric_action": "Actual full unbounded Ran coefficient functor and full Hecke-lisse geometric action use t in each operator parameter coordinate", "examples": "Three complete solutions distinguish absolute projective-line counit degrees, mixed diagonal coevaluations and shifted versus unshifted membership projectors", "limitations": "Nilpotent singular-support identification, original-field ind-regularity and other geometric ground fields require separate geometric proofs" }, "complex_curve_ind_connection_transpose_normalization_proved": true, "complex_lisse_Ran_operator_parameter_shift_comparison_proved": true, "complex_P1_shifted_diagonal_coreflection_proved": true, "complex_ran_regular_kernel_colimit": { "geometric_ground_field": "Complex numbers", "kernel": "Actual full regular Ran kernel is the coherent counit colimit of finite-point insertion kernels", "parameters": "Actual spectral projector object has a colimit presentation with parameters in original ind-connection categories", "regularity": "Actual Hecke stages and spectral projector preserve the full chartwise ind-regular subcategory, including all its unbounded objects", "geometry": "Every curve genus, automorphic component, collision pattern and bounded finite Satake coefficient", "examples": "Proper projective-line trace, complementary insertions and spectral integration of the full torus regular kernel", "limitations": "Regularity of the entire projector image, nilpotent support identification and other geometric ground fields require separate arguments" }, "complex_actual_Ran_regular_kernel_counit_colimit_proved": true, "complex_actual_projector_ind_connection_parameter_colimit_proved": true, "complex_full_Hecke_chartwise_ind_regular_preservation_proved": true, "complex_actual_projector_regular_input_preservation_proved": true, "complex_intrinsic_stack_point_adjunction": { "geometric_ground_field": "Complex numbers", "group_model": "Full operator category of BA is modules over the dual of the full perfect group de Rham cochains, normalized by actual q! restriction", "adjunction": "Intrinsic point !-delta represents actual extraordinary restriction on every unbounded operator target", "point_objects": "Compact and chartwise bounded regular at every complex bundle point, with actual half-twist fibre-line equivalence", "examples": "Multiplicative stabilizer free and constant objects; alpha_p formal relation; nonzero ind-regular rational functions with every closed-point extraordinary fibre zero", "limitations": "No point-generator theorem, enhanced nilpotent generation, full-image regularity or other geometric ground-field comparison is inferred" }, "characteristic_zero_finite_type_stabilizer_smoothness_proved": true, "complex_full_classifying_stack_operator_module_model_proved": true, "complex_intrinsic_stack_point_adjunction_proved": true, "complex_regular_compact_stack_point_objects_proved": true, "complex_enhanced_spectral_stage_comparison": { "geometric_ground_field": "Complex numbers", "enhanced_category": "Full regular-algebra modules are the actual relative tensor category and the existing Hecke-lisse spectral base change", "stages": "Finite equivariant Koszul algebras with conservative continuous pushforwards and continuous linear left and right adjoints", "compact_generation": "The full complex Hecke-lisse category is compactly generated by relative images of enhanced compact generators", "regularity": "Stage monads preserve chartwise ind-regular inputs; full-image regularity follows conditionally from point detection on the full Hecke-lisse category", "examples": "Laurent graph modules, a non-idempotent stage monad, character and degree shifts, and a nonperfect coherent derived residue", "limitations": "Nilpotent singular-support identification, unconditional point detection and full-image regularity, and other geometric ground fields require separate arguments" }, "complex_enhanced_regular_algebra_relative_tensor_comparison_proved": true, "complex_Hecke_lisse_spectral_base_change_equivalence_proved": true, "complex_restricted_actual_Koszul_stage_continuous_linear_adjoints_proved": true, "complex_Hecke_lisse_compact_generation_proved": true, "complex_enhanced_Koszul_stage_regular_input_preservation_proved": true, "complex_reduced_cotangent_morse_test": { "geometric_ground_field": "Algebraically closed characteristic-zero fields for cotangent geometry; complex varieties for the bounded cycle calculation", "transversality": "Zero tangent of the actual fixed cotangent fibre and a smooth Lagrangian source component give a quadratic target test transverse on the source", "Hecke_fibre": "On the smooth open stratum of the reduced bounded support, orbit residue equations identify the actual cotangent fibre with the compatible-Higgs scheme and moving residue", "bounded_model": "An actual bounded constant conormal model has a degree-zero normalized Morse cycle with its orientation line and monodromy (-1)^d", "examples": "Quadratic reduced ramification, cubic nonreduced isolation and a nonzero source conormal with finite support-restricted map", "limitations": "General microlocal coefficient comparison, actual full-category cycle exactness/completion, nilpotent support identification, full-image regularity and geometric field transfer require separate arguments" }, "complex_reduced_cotangent_fibre_quadratic_transversality_proved": true, "complex_open_bounded_Hecke_cotangent_fibre_comparison_proved": true, "complex_bounded_conormal_Morse_cycle_calculation_proved": true, "complex_ind_regular_completion_point_retract": { "ground_field": "Complex numbers; smooth pure-dimensional affine charts for operator resolutions and point adjoints", "completion": "Actual Ind of the bounded regular operator category is recovered by its lower-truncation inverse tower", "point_retract": "Every cohomology object is a filtered union of coherent regular objects with the specified isolated-point closed-carrier decomposition", "cycle_application": "Exact ind-extension of bounded algebraic regular-function cycles; the carrier premise for all coherent heart subobjects and a nonzero bounded point test are retained", "descent": "Smooth normalized pullbacks are t-exact; global stack recovery is asserted only under the explicitly stated truncation and limit evaluation hypotheses", "limitations": "Generic microlocal coefficient/cycle comparison, characteristic-support/RH comparison, nilpotent image, full-image regularity and special enhanced geometric field generation require separate proofs" }, "complex_affine_global_projective_operator_resolutions_proved": true, "complex_affine_ind_regular_truncation_recovery_proved": true, "complex_regular_closed_point_both_adjoints_finite_amplitude_proved": true, "complex_ind_regular_admissible_unbounded_point_retract_proved": true, "complex_ind_regular_algebraic_cycles_exact_extension_proved": true, "complex_generic_microlocal_coefficient_test": { "ground_field": "Complex numbers; bounded weak complex constructible coefficient tests and regular algebraic operator cycles", "coefficient_test": "Smooth conic Lagrangian germ of constant projection rank and a transverse graph dh; arbitrary denominator objects are handled by the actual microlocal stalk test", "carrier": "Analytic isolation of the actual bounded regular cycle gives an isolated component in its reduced algebraic closed carrier", "unbounded_application": "One common smooth microsupport bound near the chosen covector for every coherent heart subobject and every cohomological degree; one nonzero bounded point test", "examples": "Coefficient isomorphism versus monodromy; real halfplane logarithmic strip product; reduced isolated carrier versus a doubled nonreduced fibre scheme", "limitations": "Characteristic-support/RH microsupport equality and the common support hypothesis in the actual Hecke application, nilpotent/full-image equality and special enhanced field generation need separate proofs" }, "complex_bounded_generic_microlocal_coefficient_test_proved": true, "complex_bounded_regular_analytic_isolation_algebraic_carrier_proved": true, "complex_common_microsupport_ind_regular_cycle_retract_proved": true, "complex_scalar_cap_propagation": { "ground_field": "Complex numbers; coherent analytic operator modules on smooth manifolds; coherent algebraic modules for analytification", "scalar_propagation": "Actual compact-cap restriction on one common base neighbourhood gives every real C1 support test", "finite_system_argument": "Uniform support-test amplitude [0,3n+1] and 3n+2 coherent scalar quotient covers", "analytic_extension": "Good order filtration and full Fitting support retained under actual analytification", "examples": "Exact cap and diamond constants; actual first-order solution and paths; a characteristic normal support obstruction", "limitations": "Reverse characteristic inclusion, actual real-kernel solution evaluation, signed solution/de Rham transport and the Hecke common-support premise remain separate" }, "complex_analytic_scalar_cap_propagation_proved": true, "complex_analytic_uniform_support_test_bound_proved": true, "complex_coherent_analytic_solution_microsupport_containment_proved": true, "complex_actual_analytic_good_symbol_extension_proved": true, "complex_matrix_euler_microlocal_test": { "ground_field": "Complex numbers; finite analytic operator cells and arbitrary constant complex Euler matrices", "kernel": "Full DG right-input kernel complex; actual real-kernel morphism and injectivity after derived tensor remain hypotheses", "Euler_model": "Every dimension n>=1, every positive rank and every constant complex matrix, including Jordan blocks and integer resonances; actual support degree1 has dimension r", "transport": "Finite common coefficient algebra D0, genuine R-module cutoff with a D0-linear localized equivalence, and actual finite homotopy equivalence of base-extended resolution complexes", "examples": "Two Jordan support cocycles; pole/ramified pair with zero central stalk; actual derivative integer gauge and constant cone", "limitations": "General characteristic/solution-microsupport equality, actual global kernel comparison/contact transport, signed DR transport, common Hecke support and full nilpotent/image/field scope remain separate" }, "complex_full_kernel_finite_operator_cell_comparison_proved": true, "complex_kernel_degree_zero_derived_tensor_counterexamples_proved": true, "complex_all_matrix_Euler_actual_support_test_proved": true, "complex_matrix_Euler_solution_microsupport_equality_proved": true, "complex_finite_actual_kernel_gauge_transport_criterion_proved": true, "complex_integer_Euler_derivative_gauge_proved": true, "complex_actual_analytic_microlocal_flatness": { "ground_field": "Complex numbers; a nonzero cotangent covector on a smooth complex manifold", "rings": "Actual finite-upper-order factorial analytic E over analytic D; no formal operator replacement", "flatness": "Right D-flatness for all algebraic left modules; coherent input has no higher tensor Tor", "support": "Actual E tensor stalks have exactly the underlying characteristic support off the zero section", "proof": "Faithful mixed coefficients, lifted analytic relations, one-domain factorial series, strict finite free resolutions and finite-lattice zero detection", "limitations": "A fixed E stalk is not faithfully flat on every D module; no general reverse solution-microsupport inclusion, nilpotent theorem or full-image identification follows from this alone" }, "complex_actual_finite_order_microlocal_right_D_flatness_proved": true, "complex_actual_microlocal_characteristic_support_detection_proved": true, "complex_actual_microlocal_extended_presentation_strictness_proved": true, "complex_relative_kernel_cutoff": { "ground_field": "Complex numbers; a fixed nonzero normal covector dRe t and one finite common analytic coefficient diagram", "relative_topology": "Convex ordinary coefficient domains, exact directional section continuation, compact extension, bounded derived unit and real-test counit accuracy", "kernel": "Whole canonical E(G;D_R) with admitted full thick cone and actual round diamond; all complement/collar/density maps retained", "module": "Strict canonical-kernel sheaf F in degree1; finite differential restriction is the ordinary D0 action; G=q^-1 F[-1]", "denominator": "Actual D0-linear concentration and counit roof to O with underlying cones null at p", "limitations": "Finite common-domain realization is explicit; actual base-extended resolution homotopies/contact and faithful infinite-order extension remain required for the general normal-form application" }, "complex_relative_convex_cone_unit_proved": true, "complex_actual_round_domain_kernel_cutoff_proved": true, "complex_finite_D0_linear_microlocal_cutoff_denominator_proved": true, "complex_faithful_finite_solution_transport": { "ground_field": "Complex numbers; a fixed nonzero covector and coherent analytic D modules in the same base coordinates", "scalar_extension": "Full actual B=E-infinity with every-epsilon positive heads and factorial negative tails; faithful exact B/E and right-flat B/D on arbitrary modules", "derived_comparison": "An actual B module isomorphism produces two finite chain maps and both chain homotopies between actual extended finite D resolutions", "kernel_realization": "One finite actual canonical ring diagram plus the proved RKC strict cutoff and D0-linear denominator", "conclusion": "Localized actual solution equivalence; ordinary biholomorphic transport; Euler detection at standard-coordinate whole-lattice FG inputs", "limitations": "No general contact solution equivalence inferred from operator ring maps; full reverse support/signed DR and nilpotent image remain unfinished" }, "complex_actual_B_module_isomorphism_solution_transport_proved": true, "complex_actual_finite_resolution_homotopy_transport_proved": true, "complex_standard_coordinate_Euler_support_detection_proved": true, "complex_ordinary_coordinate_solution_support_transport_proved": true, "complex_legendre_incidence_sheaf_transport": { "ground_field": "Complex numbers; projective dual spaces of complex dimension N at least1", "inputs": "Globally bounded arbitrary vector sheaf complexes; no input constructibility or finite stalk condition", "kernels": "Extension by zero and transposed derived open direct image, both shifted by N, with exact inverse counits", "contact": "Full nonzero incidence conormal with input antipode, identity-induced unit and both proper graph projections", "coefficients": "Strict modules over any constant unital associative complex algebra; bounded finite free Hom is a vector complex", "limitations": "The image of the holomorphic cutoff is an actual transported module; identifying it with the target cutoff remains an extra module-derived comparison. Full reverse solution/characteristic and nilpotent image claims remain unfinished" }, "complex_projective_incidence_bounded_sheaf_equivalence_proved": true, "complex_projective_incidence_nonzero_microsupport_transport_proved": true, "complex_projective_incidence_strict_coefficient_transport_proved": true, "complex_projective_incidence_finite_Hom_transport_proved": true, "complex_holomorphic_incidence_coefficient_maps": { "ground_field": "Complex numbers; compact complex input manifolds, with properly supported local diagonal calculations", "trace": "Full Dolbeault/current coefficient resolutions, normalized (2pi i)^(-N) trace in every degree and ordered input density", "ordinary_maps": "Actual derived density evaluation maps and their ordinary proper composition; all-dimensional diagonal Cauchy identity", "incidence": "Forward closed-to-j! and reverse Rj*-to-closed denominators, followed by exact globally defined finite-pole evaluations and actual inverse time derivatives", "conclusion": "Two explicitly defined vector-derived holomorphic incidence coefficient maps; their one-dimensional Cauchy calibration is an isomorphism", "limitations": "A normalized local vector inverse pair is proved in Sections3.393-3.404. The module-compatible target cutoff identification remains an extra condition; no general contact solution or nilpotent image conclusion follows from vector equivalence alone." }, "complex_proper_holomorphic_trace_realization_proved": true, "complex_proper_density_kernel_composition_proved": true, "complex_diagonal_Cauchy_unit_realization_proved": true, "complex_incidence_vector_coefficient_maps_constructed": true, "complex_vector_incidence_inverse_pair": { "ground_field": "Complex numbers; every projective incidence dimensionN>=1, locally at matched nonzero time covectors", "objects": "Actual bounded sheaf transforms of the holomorphic coefficient objects, with the specified ordinary density evaluations", "descent": "Whole common polynomial connection factor and reduced finite coherent derivative localization; arbitrary torsion without the factor is insufficient", "conclusion": "Two nonzero scalar coefficient products; a single intrinsic reverse normalization gives both local vector inverses and full directional endomorphism transport", "limitations": "No compatible strict cutoff module action or general contact solution-complex transport inferred solely from vector equivalence; nilpotent/full-image and original-field goals remain unfinished" }, "complex_general_incidence_vector_inverse_pair_proved": true, "complex_common_time_factor_coefficient_descent_proved": true, "complex_local_holomorphic_incidence_vector_equivalence_proved": true, "complex_full_directional_holomorphic_End_transport_proved": true, "complex_pure_directional_test_comparison": { "ground_field": "Complex numbers; every incidence dimensionN>=1 at a selected nonzero time covector", "objects": "Actual polynomial Ore algebras and exact directional tests of bounded strict coefficient sheaves", "conclusion": "Pure tested coefficients admit a module-derived comparison when the specified finite generator actions intertwine; all finite Hom cohomology and homotopies are retained. The actual Laurent time actions satisfy the condition.", "action_hypothesis": "Every selected generator must satisfy PDT25 for its prescribed actual canonical-ring realization.", "limitations": "General spatial and whole infinite-order tested-action identities and the untested cutoff-module comparison remain required; no full contact solution, nilpotent image or arbitrary-field conclusion is inferred." }, "complex_ordered_polynomial_incidence_quantization_proved": true, "complex_linear_halfspace_holomorphic_test_concentration_proved": true, "complex_pure_test_operator_diagram_strictification_proved": true, "complex_tested_finite_solution_comparison_criterion_proved": true, "complex_tested_Laurent_time_action_comparison_proved": true, "complex_tested_polynomial_incidence_action": { "ground_field": "Complex numbers; every incidence dimensionN>=1 at matched nonzero time covectors", "objects": "Actual polynomial Weyl algebra localized only at the time derivative; strict cutoff coefficients after the specified pure directional test", "conclusion": "All polynomial generator and word actions intertwine under the fixed ordered parameter-zero incidence quantization. Tested coefficients have an actual module-derived comparison retaining every finite Hom degree, map and homotopy.", "support_scope": "A single linear test is not conservative; the smooth real cubic example has a nonzero conormal with vanishing selected test.", "limitations": "No analytic Taylor-series or whole infinite-order action comparison, untested cutoff-module equivalence, full contact solution/characteristic or nilpotent image conclusion is inferred." }, "complex_tested_polynomial_incidence_action_proved": true, "complex_tested_polynomial_Ore_module_comparison_proved": true, "complex_tested_polynomial_finite_Hom_transport_proved": true, "complex_single_linear_test_nonconservativity_proved": true, "complex_canonical_whole_tested_action": { "ground_field": "Complex numbers; every incidence dimensionN>=1 at matched nonzero time covectors", "objects": "Canonical whole real kernel actions on directional coefficient cohomology; pure output linear test and its actual transported input test", "conclusion": "The retained parameter-zero incidence vector comparison intertwines every whole kernel germ. Strict pure coefficient modules retain all finite Hom degrees and homotopies. An actual full operator-module isomorphism between coherent differential modules gives their specified tested solution comparison.", "limits": "The cutoff-module action comparison, conservativity of the chosen tests and full characteristic-support equality require their respective arguments. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_canonical_whole_kernel_cohomological_action_proved": true, "complex_canonical_whole_tested_incidence_action_proved": true, "complex_canonical_pure_whole_kernel_module_comparison_proved": true, "complex_incidence_tested_coherent_solution_comparison_proved": true, "complex_Euler_incidence_density_shift_test_proved": true, "complex_holonomic_transported_support_test": { "ground_field": "Complex numbers; every complex manifold dimension and every coherent holonomic analytic left differential module", "conclusion": "Unshifted solution microsupport equals characteristic support, including irregular systems and all singular/nonreduced support loci as underlying sets. Actual whole coefficient/test and finite B-resolution maps supply the generic reverse inclusion; density, closedness and a separate connection argument include every remaining point.", "method_scope": "Only endpoint modules need differential-module realizations; intermediate finite contact charts carry bounded tests, pure coefficient heart modules and finite free full-operator complexes.", "limits": "This finite holonomic comparison does not assert the reverse support theorem for arbitrary nonholonomic systems. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_finite_contact_chain_pure_coefficient_transport_proved": true, "complex_generic_characteristic_nonzero_transported_test_proved": true, "complex_holonomic_solution_characteristic_equality_proved": true, "complex_zero_section_solution_support_detection_proved": true, "complex_nonholonomic_free_module_solution_support_proved": true, "complex_signed_de_rham_bounded_support": { "ground_field": "Complex numbers; every complex manifold and bounded coherent holonomic analytic complex", "conclusion": "Intrinsic DR_N(K)=S0(DualK)[N] and both microsupports equal union_i Char(H^iK), as underlying sets, including irregular systems, singular points and zero covectors.", "method_scope": "Actual analytic filtered Ext, density balance, finite-projective evaluation and common pure tests for finitely many cohomology modules.", "limits": "The signed support comparison here requires bounded holonomic input. The arbitrary nonholonomic reverse support theorem is separate. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_analytic_holonomic_dual_characteristic_support_proved": true, "complex_intrinsic_density_de_rham_solution_identity_proved": true, "complex_holonomic_de_rham_characteristic_equality_proved": true, "complex_finite_family_pure_test_proved": true, "complex_bounded_holonomic_solution_support_equality_proved": true, "complex_bounded_holonomic_de_rham_support_equality_proved": true, "complex_algebraic_ind_support_chart": { "ground_field": "Complex numbers; smooth separated finite-type varieties, every closed algebraic conic support; affine charts for actual unbounded differential-module embedding", "conclusion": "Exact algebraic-to-analytic characteristic support and duality; bounded regular and full supported-ind RH equivalences; all-unbounded finite compact factorization criterion; cohomological support equivalence for bounded-below ind-regular objects.", "method_scope": "Both-side PBW coefficient extension, faithful flatness at every cotangent point, full bounded RH functor and actual filtered ind mapping complexes; supported-stage cofiber nullhomotopy and exact finite/upper truncations.", "limits": "No general unbounded-below converse from cohomological support or commutation of ordinary-sheaf infinite support tests is asserted. Full independent nilpotent identification needs supported factorization or inverse-limit compatibility. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_algebraic_analytic_characteristic_support_proved": true, "complex_algebraic_dual_analytification_proved": true, "complex_bounded_supported_regular_RH_equivalence_proved": true, "complex_supported_ind_RH_equivalence_proved": true, "complex_supported_ind_finite_factorization_criterion_proved": true, "complex_bounded_below_regular_ind_cohomological_support_proved": true, "complex_full_unbounded_cohomological_support_ind_generation_proved": false, "complex_noncharacteristic_chart_support": { "ground_field": "Complex numbers; every connected reductive group and smooth projective connected curve, all genera/components; all smooth finite-type affine charts and every actual arrow", "conclusion": "Actual nilpotent cone compatibility/noncharacteristicity, full independent unbounded cohomological support preservation and chart limit; supported regular ind-RH equivalence of actual chart-diagram limits.", "method_scope": "Common-chart section, Higgs deformation pairing, full noncharacteristic finite-module theorem, actual Noetherian unions and finite K-flat graph Koszul strip with differential-linear spectral edges; specified full RH/transfer coherence and half twist.", "limits": "Supported regular ind-descent does not prove regularity or compact supported factorization of every independently nilpotent-supported object. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_common_chart_noncharacteristic_arrows_proved": true, "complex_nilpotent_Higgs_chart_cone_compatibility_proved": true, "complex_unbounded_noncharacteristic_cohomology_pullback_proved": true, "complex_full_nilpotent_support_chart_transition_preservation_proved": true, "complex_full_nilpotent_support_chart_limit_proved": true, "complex_supported_regular_ind_chart_descent_proved": true, "complex_full_nilpotent_regular_ind_RH_identification_proved": false, "complex_fixed_cone_cycle_retract": { "ground_field": "Complex numbers; smooth affine finite-type charts, every unbounded ind-regular complex satisfying the fixed coherent characteristic cone of dimension at most the chart dimension.", "conclusion": "Actual natural isolated-cycle point retract, nonzero when one finite cohomology subobject meets the transverse covector; generic algebraic tests and exact-image deductions.", "method_scope": "AIC characteristic/microsupport equality, common cone, bounded microlocal coefficient calculation, algebraic closed carriers, exact ind-cycles and both point adjunctions with uniform finite amplitude; full unit/counit identities.", "limits": "Regular Ind input and componentwise isolation are explicit premises of this cycle argument. No full supported factorization is inferred from them. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_fixed_characteristic_cone_common_cycle_bound_proved": true, "complex_unbounded_ind_regular_nonzero_cycle_point_retract_proved": true, "complex_generic_characteristic_algebraic_transverse_test_proved": true, "complex_componentwise_cycle_field_support": { "ground_field": "Complex numbers for componentwise actual regular ind-cycles; every characteristic-zero field and arbitrary extension for algebraic operator/characteristic/full unbounded support comparison.", "conclusion": "Actual natural full unbounded isolated-cycle point retract with varying finite-piece neighbourhoods; nonzero with one smooth transverse witness. Arbitrary field extension preserves and reflects the independent cohomological support condition.", "method_scope": "Serre finite-lift argument, exact bounded cycle supports and reduced carriers, actual point adjoints/finite amplitudes/unit and counit; PBW, field-flat good filtration, finite Nakayama at every cotangent prime and containment of every new finite piece in one extended original piece.", "limits": "Regular Ind input remains a cycle premise. No generic-field regular-cycle equivalence or full supported generation is inferred. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_componentwise_isolation_unbounded_cycle_retract_proved": true, "complex_componentwise_isolation_heart_Serre_class_proved": true, "all_char_zero_field_extensions_characteristic_support_proved": true, "all_char_zero_field_extensions_full_unbounded_support_preserved_reflected": true, "complex_isolated_critical_cycle_retract": { "ground_field": "Complex numbers for actual algebraic regular ind-cycles; every field for the bounded perverse isolated stratified-critical cluster count.", "conclusion": "Actual full unbounded point retract under componentwise characteristic isolation, including dh=0; nonzero from one finite witness satisfying isolated stratified criticality on all original adapted strata.", "method_scope": "Actual half-space coefficient map, closed support bound, finite carrier splitting and both point adjoints/finite amplitudes; original whole-cluster restriction-fibre filtration, pure normal Morse coefficients and positive integer intersection counts.", "limits": "The nonvanishing argument requires isolated criticality on every relevant stratum and regular Ind input. Full supported factorization and the other geometric ground fields remain separate. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_critical_differential_componentwise_unbounded_cycle_retract_proved": true, "complex_isolated_stratified_critical_nonzero_cycle_witness_proved": true, "complex_zero_covector_degenerate_isolated_cycle_examples_proved": true, "complex_characteristic_isolated_cycles": { "ground_field": "Every field for bounded finite-rank perverse and bounded constructible sheaf tests; complex numbers for actual algebraic regular ind-cycles.", "conclusion": "Exact positive integer rank formula and nonvanishing under actual microsupport graph isolation, including invisible nonisolated critical strata. Any finite actual characteristic witness detects the full unbounded componentwise ind-regular cycle point retract.", "method_scope": "Relative-conormal zero-fibre isotropy, proper characteristic-value radial exhaustion and signed closed-face bounds; actual real-band restriction, proper open-interval parameter comparison, finite visible Morse cluster and reverse exit filtration, pure normal coefficients and finite perverse truncations.", "limits": "Regular Ind input and componentwise characteristic isolation are premises. PIC.19 gives proper comparison over complex smooth separated finite-type schemes. General relative-section cycles and full independent supported generation remain separate. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_characteristic_isolated_nonzero_cycle_proved": true, "complex_componentwise_unbounded_cycle_detects_characteristic_witness_proved": true, "bounded_isolated_supported_test_perverse_cohomology_comparison_proved": true, "complex_proper_ind_cycle_comparison": { "ground_field": "Complex numbers. Smooth separated finite-type schemes, proper algebraic map, regular function; singular zero fibres represented by ambient closed transfer.", "conclusion": "Actual proper-cycle comparison on the full unbounded regular Ind category, retaining coefficient triangles and monodromy. Actual nonzero source point retract gives a nonzero target cycle retract.", "method_scope": "Bounded flat transfer Spencer and finite affine Cech; scheme compactness and continuous Ind comparison; specified bounded RH/deRham/coefficient maps; actual point composition and retract maps.", "limits": "Regular Ind input is required. UQD gives original-complex universal-family detection with componentwise isolation. General relative sections, other geometric fields and independent nilpotent factorization remain separate. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_proper_regular_ind_cycle_comparison_proved": true, "complex_actual_unbounded_proper_pushforward_ind_comparison_proved": true, "complex_proper_point_retract_cycle_nonvanishing_proved": true, "complex_universal_quadratic_cycle_detection": { "ground_field": "Complex numbers; proper map from a smooth separated finite-type scheme to a smooth affine scheme, fixed nonzero target covector and selected compatible point, full regular Ind input.", "conclusion": "Componentwise isolation of the selected compatible point and one finite actual source characteristic witness give an actual finite target characteristic witness in the full unbounded proper image. Any target cone avoiding that covector, including relative curve lissity, is excluded.", "method_scope": "Global parameter derivatives, product PBW support and full HC/HN closed cycle carrier; both continuous closed-section adjoints with finite amplitudes; actual unbounded unit/counit retract and proper product/cycle/section maps.", "limits": "Regular Ind input and selected-point isolation are premises. RHC supplies the complex regular-input Hecke application; independent nilpotent regularity and other geometric fields remain separate. GFP.55–GFP.56 prove original-complex full Hecke-lisse regularity and nilpotent-support containment in every genus. Equality with all independently nilpotent-supported objects remains a separate converse." }, "complex_universal_quadratic_characteristic_carrier_proved": true, "complex_full_unbounded_product_section_cycle_retract_proved": true, "complex_componentwise_proper_cohomological_lisse_detection_proved": true, "complex_componentwise_proper_characteristic_lower_bound_proved": true, "complex_regular_ind_hecke_input": { "ground_field": "Complex numbers; smooth separated finite-type ambient schemes, embedded singular or nonreduced finite-type source schemes, and smooth affine charts of Bun_G for a smooth projective connected curve and connected reductive group.", "input": "Actual chartwise regular Ind input on the bundle stack; bounded regular kernel and any fixed normalization shift.", "conclusion": "Actual inverse images and regular kernel-weighted bounded Hecke inputs belong to the supported full regular Ind category, on both unbounded tails. Proper ambient images are regular Ind.", "method": "Continuous transfer, finite closed localization and Cech totalization; canonical supported graph comparisons; finite Zariski lifts through high-twist section frames; bounded IC strict support/cosupport comparison.", "limits": "Regular Ind input is required for this preservation theorem; GFP.55 gives it on the entire original-complex Hecke-lisse image. Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_regular_ind_graph_inverse_images_proved": true, "complex_bundle_family_Zariski_atlas_lifts_proved": true, "complex_full_unbounded_Hecke_input_regularity_proved": true, "complex_regular_hecke_characteristic_criterion": { "ground_field": "Complex numbers; every connected reductive group, smooth projective connected curve, genus and bundle component.", "input": "Actual chartwise regular Ind input, with both infinite cohomological tails allowed. Independent cohomological characteristic supports are unions of finite witnesses without closure.", "conclusion": "Nilpotent cohomological support is equivalent to the cohomological zero-curve characteristic bound for every actual one-point IC Hecke family. Nilpotent input gives the stronger nilpotent-times-zero-curve output bound. Categorical Hecke-lissity implies this condition for regular Ind inputs.", "method": "Finite self-nerve witnesses; actual inverse-bound projective source and exact kernel shifts; all-orbit good filtrations and bounded intrinsic universal tests; geometric Cartan/Levi moving residue; full proper output windows and selected-point cycle detection.", "limits": "The regular-input categorical converse is unrestricted in positive genus and locally bounded below in every genus. GFP.55–GFP.56 prove full original-complex Hecke-lisse regularity and nilpotent-support containment. Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_chart_characteristic_horizontality_proved": true, "complex_regular_input_Hecke_nilpotent_curve_upper_bound_proved": true, "complex_regular_input_Hecke_cohomological_characteristic_criterion_proved": true, "complex_regular_Hecke_lisse_nilpotent_support_necessary_condition_proved": true, "complex_external_connection_factors": { "ground_field": "Complex numbers; all genera, bundle components and connected reductive groups.", "input": "For categorical equivalence, chartwise regular Ind objects whose restriction to every smooth affine bundle chart is cohomologically bounded below. Each fixed Hecke output has a lower bound; no common bound over all labels is required.", "conclusion": "Regular complexes bounded below with cohomological curve support lie in the actual external ind-connection image. For the stated locally bounded below regular inputs, nilpotent cohomological support is equivalent to full categorical Hecke-lissity. Their stable colimit-generated class has the same categorical property on both tails.", "method": "Product conormal components, commuting irreducible monodromy evaluation, actual middle extension, full external Hom, finite composition/Postnikov lifts and dualizable chart descent. PBW and the actual curve connection also give relative operator coherence and curve flatness.", "limits": "The categorical converse here is locally bounded below in every genus; PAC27 gives both tails in positive genus. GFP.55–GFP.56 give full original-complex Hecke-lisse regularity and nilpotent-support containment. Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_bounded_regular_external_connection_factorization_proved": true, "complex_relative_curve_operator_coherence_and_flatness_proved": true, "complex_bounded_below_regular_external_connection_factorization_proved": true, "complex_locally_bounded_below_regular_input_categorical_Hecke_criterion_proved": true, "complex_locally_bounded_regular_input_categorical_Hecke_criterion_proved": true, "complex_unbounded_cohomological_curve_to_ind_connection_image_proved": false, "complex_postnikov_affine_curve_coefficients": { "ground_field": "Complex numbers. Actual operator towers and completed curve factors hold in all genera. The unrestricted ind-connection criterion and regular nilpotent full Ran action require positive genus.", "input": "Actual chartwise regular Ind input, with both infinite cohomological tails allowed. Completed operator realization alone is described by ind-connection cohomology and imposes no regularity on an arbitrary coefficient.", "conclusion": "The original affine D(B)-action lifts to completed curve coefficients. Global completed factors descend with all actual maps. In positive genus, nilpotent cohomological support is equivalent to categorical Hecke-lissity for every regular Ind input, and its full Ran action has regular nilpotent coefficients.", "method": "Actual module Postnikov fibre, eventual-constant cohomology, flat open comparison and full descent; completed mapping spaces and colimits; exact affine coefficient sections and the full associative action bar; RSC14 and proper compact coefficient contractions.", "limits": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_operator_Postnikov_effectivity_proved": true, "complex_completed_curve_actual_operator_realization_proved": true, "complex_regular_ind_completed_curve_external_factorization_proved": true, "complex_positive_genus_unbounded_regular_input_categorical_Hecke_criterion_proved": true, "complex_positive_genus_regular_nilpotent_full_Ran_action_proved": true, "complex_enhanced_regular_point_stages": { "ground_field": "Complex numbers and positive genus; every connected reductive group and bundle component. Abstract scalar extension is allowed over every extension of C.", "input": "The full category of chartwise regular Ind objects with independently nilpotent cohomological support, with both infinite cohomological tails.", "conclusion": "The actual restricted spectral action preserves this category. Its enhanced finite Koszul stages have compact point generators, compact also in the full enhanced Hecke-lisse stages. Its inclusion in the full Hecke-lisse category preserves compact objects and has a continuous spectral-linear right adjoint. Relative coefficient and abstract scalar extensions preserve full faithfulness and the actual orthogonal localization.", "method": "Continuous Ran section with LD=Id and full action comparisons; full algebra-action bars; actual point adjunction and finite nilpotent detectors; finite-stage adjoints and generator bars; enriched coend and dualizable coefficient projection formula; balanced tensor adjunction.", "limits": "GFP.25–GFP.52 give the actual geometric realization and point membership of ERP24; GFP.55–GFP.56 give the full original-complex Hecke-lisse regular image in every genus. Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_positive_genus_regular_nilpotent_restricted_spectral_action_proved": true, "complex_positive_genus_regular_nilpotent_enhanced_point_generators_proved": true, "complex_positive_genus_regular_nilpotent_compact_in_Hecke_lisse_proved": true, "complex_positive_genus_regular_nilpotent_continuous_A_linear_coreflector_proved": true, "complex_positive_genus_regular_nilpotent_enhanced_scalar_extension_fully_faithful_proved": true, "complex_geometric_field_point_completion": { "ground_field": "Original complex curve in every genus; every connected reductive group and bundle component. The enhanced point membership proof uses every extension K/C and countable algebraic models with an auxiliary complex embedding.", "input": "Every unbounded original-complex operator object, with both infinite cohomological tails and no regularity or support premise.", "conclusion": "The full Hecke-lisse category is chartwise regular Ind and has the actual enhanced compact point generators. The actual restricted Beilinson projector has this full image and therefore lands in regular Ind objects with independent nilpotent cohomological support on every input.", "method": "All-genus regular Hecke-lisse restricted action; continuous spectral-linear orthogonal triangle; exact nilpotent-supported enhanced image after scalar extension; countable coherent coefficient and IC action models; actual intrinsic point adjunctions; faithful support reflection and all-field point detection.", "limits": "Equality with all independently nilpotent-supported objects and their independent regularity, unrestricted genus-zero cohomological converse, AIC17 full supported factorization, other geometric ground fields and derived tempered Satake retain their separate requirements." }, "complex_all_genus_regular_Hecke_lisse_restricted_spectral_action_proved": true, "complex_all_genus_regular_Hecke_lisse_enhanced_point_generators_proved": true, "complex_original_restricted_stage_new_field_point_membership_proved": true, "complex_full_Hecke_lisse_regular_ind_proved": true, "complex_original_field_full_Hecke_lisse_regular_point_generation_proved": true, "complex_arbitrary_input_restricted_projector_regular_nilpotent_landing_proved": true, "complex_holonomic_curve_factors": { "ground_field": "Complex numbers; every smooth projective connected curve and every genus. Smooth affine base charts and the actual strong smooth-descent class, including the bundle stack.", "input": "Finite arbitrary holonomic modules with zero curve covectors; bounded-below complexes with ind-holonomic cohomology; and arbitrary unbounded such complexes with both infinite tails.", "conclusion": "Simple finite modules are actual exterior products of a simple holonomic base module, possibly irregular, and an irreducible finite connection on the whole curve. Bounded-below objects lift through the full holonomic ind-category tensor ind-connections. With both tails the exact coefficient category is the completed curve category; it equals ind-connections in positive genus.", "method": "Proper relative horizontal Hom and actual fibre evaluation; algebraic holonomic minimal extension and finite Spencer duality; full compact external mapping and extension bars; exact affine coefficient functor, full noncommutative operator-action bar and strong chart descent.", "limits": "Regularity of the arbitrary base factor is not asserted. Unrestricted genus-zero ind-connection membership requires the actual u^2-torsion condition. The full Hecke support transport on independently nilpotent inputs, the converse from all independent nilpotent support, AIC17 supported factorization, other geometric ground fields and derived tempered Satake remain separate." }, "complex_holonomic_zero_curve_finite_external_factorization_proved": true, "complex_holonomic_zero_curve_bounded_below_ind_connection_factorization_proved": true, "complex_holonomic_zero_curve_completed_parameter_both_tails_proved": true, "complex_positive_genus_holonomic_zero_curve_ind_connection_factorization_proved": true, "complex_nilpotent_hecke_upper_transport": { "ground_field": "Complex numbers; every smooth projective connected curve, every connected reductive group and every component.", "input": "The independently defined full cohomological nilpotent support category, including both infinite tails; no regularity premise.", "conclusion": "All genera: actual Hecke output has nilpotent bundle covectors and zero curve covectors, with full completed coefficient lifts and actual action maps. Positive genus: equality of the full independent nilpotent category, full Hecke-lisse category and restricted projector image, and independent chartwise ind-regularity.", "method": "Algebraic graph characteristic formulas, exact smooth and singular-ambient comparisons, actual finite coherent witnesses, IC orbit annihilators, compatible Higgs gluing and reduced nilpotent residue; arbitrary coherent proper characteristic transport and finite unbounded windows; full HCF coefficients and GFP regular image.", "limits": "The unrestricted genus-zero u^2-torsion/restricted-action assertion, AIC17 chartwise supported compact factorization, other geometric ground fields and derived tempered Satake remain separate." }, "complex_independent_nilpotent_Hecke_upper_support_both_tails_proved": true, "complex_independent_nilpotent_Hecke_completed_curve_coefficients_proved": true, "complex_positive_genus_independent_nilpotent_regular_ind_proved": true, "complex_positive_genus_full_nilpotent_restricted_projector_image_proved": true, "complex_genus_zero_restricted_coefficients": { "ground_field": "Complex numbers, every smooth projective connected curve, every connected reductive group and every bundle component.", "input": "The full independently defined cohomological nilpotent support category; no boundedness, compactness or regularity premise.", "conclusion": "In genus zero the full completed Hecke action has actual ind-connection coefficients, with a uniform natural even-central-power nullhomotopy. In every genus the full independent nilpotent category equals the Hecke-lisse restricted-projector image and is chartwise ind-regular.", "method": "Presentable invariant envelopes of arbitrary inputs; full internal loop-module actions; categorical regular framing and the entire odd tensor-derivation complex; finite exterior universal coefficients, actual Hopf descent and full Ran action transport.", "limits": "AIC17 chartwise supported compact factorization and the supported ind-Riemann-Hilbert identification requiring it, other geometric ground fields, derived tempered Satake and the stated additional foundations remain separate. The full completed curve category still differs from ind-connections on P1." }, "complex_genus_zero_completed_Hecke_invariant_envelope_proved": true, "complex_genus_zero_completed_to_restricted_tensor_equivalence_proved": true, "complex_genus_zero_unrestricted_nilpotent_Hecke_ind_connection_proved": true, "complex_genus_zero_universal_even_central_power_nullhomotopy_proved": true, "complex_all_genus_independent_nilpotent_regular_ind_proved": true, "complex_all_genus_full_nilpotent_restricted_projector_image_proved": true }, { "id": "GL-GLC-04", "source": "src/hecke-functors-and-hecke-eigensheaves.md", "proof_obligations": [ "full classical and derived Satake theorems and Hecke descent foundations", "trace and closed-place coefficient formalism" ] }, { "id": "GL-GLC-05", "source": "src/geometric-class-field-theory.md", "proof_obligations": [ "full curve Picard/Poincaré and symmetric quotient geometry", "complete coherent/étale/module-descent foundation chains", "recursive algebra/scheme and adic foundations of the Noetherian finite-cover comparison", "recursive categorical platform for the constructed affine ! and geometric projection proofs", "relative Beilinson–Drinfeld collision and lax-unit factorization" ] }, { "id": "GL-GLC-06", "source": "src/the-gl-1-case-as-an-equivalence-of-categories.md", "proof_obligations": [ "recursive Picard/Hilbert, group projectivity, finite-isogeny and coherent foundations of the constructed dual and universal connection", "recursive local-algebra, ambient Ext and ordinary Picard foundations of the written trace proof", "categorical descent/base change and complete earlier foundation chains", "recursive affine/smooth-descent, coherent-duality, regular-ring and DG-category foundations of the torus zero-support proof", "recursive represented Picard/symmetric-product, perfect cohomology, trace duality, regular-local/Cohen–Macaulay and projective dimension/proper-finiteness foundations of the theta proof" ] }, { "id": "GL-GLC-07", "source": "src/the-unramified-correspondence-for-gl-n.md", "proof_obligations": [ "complete general GL_n vanishing and geometric foundations of the written conditional symmetric-group/rank-induction/adjunction deductions", "fixed-degree finite-diagram quotient realizations, subquotient and ULA properties, and restriction-limit assembly", "full torsion-stack family construction, flag smallness and Laumon/Springer IC identity beyond the written rank-one pullback", "actual stack six-operation, duality, perversity and inverse-limit category foundations", "Picard representability and compact-support line-bundle trace/localization foundations", "Whittaker construction, cleanness, Euler constancy, global descent, Hecke compatibility and cuspidality", "transverse inertia, purity and projective-fibre local-system descent foundations", "finite-field adelic, residue, trace, arithmetic Langlands and L-function proof chains", "characteristic-zero de Rham realization and complete recursive mathematical foundations" ] }, { "id": "GL-GLC-08", "source": "src/opers-critical-level-and-the-beilinson-drinfeld-construction.md", "proof_obligations": [ "recursive root/pinning, split-unipotent, faithful adjoint-group and ordinary bundle-descent foundations; recursive Serre/highest-weight and finite-algebra foundations of invariant polynomiality; full derived oper construction", "non-type-A basic quantum lifts and non-type-A coordinate-equivariant vacuum-center/oper comparison; non-type-A completed punctured-disc center and oper-Poisson comparison with exact dual-form normalization; derived/global Hamiltonian reduction; full chiral/factorization, Satake and derived-family compatibility; geometric/Satake/global central-convention comparison", "half-root, uniformization, localization, nonzero specialization, holonomicity and fusion Hecke proofs", "filtered BD quantization and full-field/reductive-center comparisons", "critical FLE, factorization, Ran, determinant, convergence and IndCoh star/shriek foundations", "recursive fixed-curve Picard/coherent/local-algebra/Ext, stack algebraization and optional analytic comparison" ] }, { "id": "GL-GLC-09", "source": "src/the-spectral-side-local-systems-singular-support-and-indcoh-nilp.md", "proof_obligations": [ "global derived LocSys representability, family perfect cotangent complexes, quasi-smooth atlases and Sing=Arth", "general reductive zero-singular-support IndCoh/QCoh theorem beyond the explicit torus model; recursive affine/smooth-descent, coherent-duality, regular-ring and DG-category foundations of the torus comparison", "general reductive spectral Eisenstein construction, properness, quasi-smoothness, support bounds and exact normalized Levi compatibility", "reducible-locus generation, irreducible zero support, categorical localization, Eisenstein transitivity and semisimple-rank induction", "genuine global nilpotent nonzero support object and strictness, distinct from the explicit local exterior model", "global compact generation and general reductive nilpotence-convention comparison", "Whittaker normalization and qualified classical reducibility/singularity statement", "complete earlier curve, local algebra, stable DG-category and derived stack foundations" ] }, { "id": "GL-GLC-10", "title": "The categorical conjecture and its variants", "source": "src/the-categorical-conjecture-and-its-variants.md", "reader": "GL-GLC-10.html", "proof_obligations": [ "full recursive stable/DG-category, monoidal, bar, adjoint, Yoneda and Ind foundations", "actual nonunital Ran localization, unital/action comparison, Satake and kernel vanishing for all connected reductive groups", "critical/Kac–Moody enhanced Hecke eigen-property, proper Eisenstein enhancement and componentwise cuspidal/chiral generation", "actual vacuum Poincare, exponential/half-twist normalization, compactness, rigidity and complete coefficient identity", "compact eventually-coconnective bound, coarsening equivalence and continuous linear Langlands lift", "global stack point/fibre and restricted-variation base-change comparisons; derived-stack geometric detection after all field extensions", "tempered quotient, compact projection and exact coherent-image theorems with normalized Eisenstein compatibility for G and all Levis", "enhanced torus Fourier equivalence including Cartan inversion, actual vacuum and zero-singular-support theorem", "restricted regular-singular Riemann–Hilbert, Betti specified-functor comparisons and return to original coefficient/base fields", "actual irreducible fibre and normalized multiplicity/perversity consequences, with automorphisms and exceptional hypotheses retained", "finite-field categorical Frobenius trace and its accessible local-term compatibility" ] }, { "id": "GL-GLC-11", "title": "Constructing the Langlands functor (GLC I)", "source": "src/constructing-the-langlands-functor.md", "reader": "GL-GLC-11.html", "proof_obligations": [ "full recursive stable/DG-category, adjoint, coherent Yoneda/action and Ind foundations", "actual normalized half-root, exponential/character duality, vacuum compactness and complete natural Whittaker identity", "actual Ran localization, derived Satake, kernel vanishing, spectral rigidity and perfect-action amplitude", "Lin kernel comparison, nilpotent kernel pairing, geometric Casselman–Shalika and tempered Hecke exactness", "restricted spectral exact/conservative probes, enhanced coefficient/action and all-field base-change comparisons", "truncatability, compact-Dmodule Cousin and affine !-amplitude geometry, global all-field !-fibre criterion", "actual t-exact bounded coarsening equivalence and ordinary-fibre upper detection/projection bounds", "enhanced torus Fourier–Mukai–Laumon/Cartan normalization and zero-singular-support theorem", "nonzero constant full-image example and its bounded source/coarse-vanishing assertions", "Betti quotient-exponential, projector, finite vacuum construction, compact-image and spectral-action theorems", "restricted regular-singular Riemann–Hilbert, field comparisons, geometric detection and tempered/coherent-image/Eisenstein comparisons for all Levis", "eigensheaf regularity/perversity and irreducible fibre/pushforward comparisons; microstalk, perfect restriction, spectral component and scheme-theoretic cycle geometry", "full GLC equivalence and complete earlier mathematical foundation chains" ] }, { "id": "GL-GLC-12", "title": "Kac-Moody localization and the fundamental local equivalence (GLC II)", "source": "src/kac-moody-localization-and-the-fundamental-local-equivalence.md", "reader": "GL-GLC-12.html", "proof_obligations": [ "full critical Kac–Moody, arc-action, coordinate, t-structure and factorization-category constructions", "full Feigin–Frenkel duality, centre and birth-of-opers construction with central and factorization compatibility", "actual affine Skryabin, derived Satake, geometric Casselman–Shalika and pointwise tempered reconstruction", "actual compact preservation, fibrewise-to-factorization comparison, unitality, convergence and collisions", "infinite-type star/shriek IndCoh, placidity, duality, ambient module-category and base-change foundations", "critical localization, infinitesimal Hecke descent, normalized half twist and constant determinant lines", "global monodromy-free oper scalar extension, coherent Hecke property, spectral fibres and localized nonvanishing", "actual local-to-global unitalization and factorization-homology pairing identities", "spectral localization/global-sections full faithfulness, spectral Poincaré zero-support and automorphic tempered images", "full torus unbounded critical FLE, categorical arc equivariance, infinite-dimensional descent and coordinate compatibility", "derived regular-central-character comparison, actual Frenkel–Gaitsgory category and reductive Lie foundations", "unrestricted external fusion comparison with valid hypotheses; no use of the unqualified printed B.11.14 assertion", "complete stable/DG-category, adjoint, coherent Yoneda, localization, derived tensor–Hom, Ind and recursive earlier foundations", "full global GLC equivalence and all original fifteen-lesson mathematical corrections" ] }, { "id": "GL-GLC-13", "title": "Eisenstein series and the reduction to the cuspidal part (GLC III)", "source": "src/eisenstein-series-and-the-reduction-to-the-cuspidal-part.md", "reader": "GL-GLC-13.html", "proof_obligations": [ "actual bundle/local-system stacks, normalized half twists, Drinfeld compactifications, D-module and IndCoh operations, nilpotent-support and geometric adjunctions", "semi-infinite Satake, Jacquet, critical FLE compatibility and coherent local-to-global parabolic comparisons", "full spectral generation including the P=G quasi-coherent contribution, and the geometric extension producing the left adjoint", "miraculous duality, Chevalley involution, determinant normalizations and geometric compactness of the left adjoint", "global linear-kernel representation, tempered compatibility, compact preservation, formal-completion comparison and coherent full faithfulness proving A_G is quasi-coherent", "faithful quasi-coherent embeddings with coherent action and unit compatibility, and geometric conservativity of L_G", "irreducible-open geometry, zero-nilpotent-support comparison, proper-parabolic generation of reducible support and exact support localization", "both compactified constant-term comparisons and the actual line-bundle and supported-unit detection arguments under all proper-Levi conjectures", "root-datum structure of reductive groups, full torus GLC and the irreducible algebra-unit theorem", "extension-stack geometry, any existence assertion beyond the chosen one-form hypothesis, and the recursive algebraic foundations of the linked earlier curve proofs", "stable/DG categories, coherent adjunctions and monads, Yoneda, derived tensor and module categories, localization and recursive earlier mathematical foundations", "the full original fifteen-lesson GLC proof and correction scope" ] }, { "id": "GL-GLC-14", "title": "Ambidexterity, opers and multiplicity one (GLC IV and V)", "source": "src/ambidexterity-opers-and-multiplicity-one.md", "reader": "GL-GLC-14.html", "proof_obligations": [ "actual bundle and de Rham local-system stacks, half twists, D-module and IndCoh operations, nilpotent support and coherent geometric adjunctions", "full proper-Levi and torus conjectures, Eisenstein generation, and the supported-unit comparison for A_G", "Whittaker square, Lin cuspidal duality, Casselman–Shalika/Chevalley compatibility, miraculous duality and determinant-line comparisons", "localization square, critical FLE, common cuspidal support, spectral star/shriek comparisons and the geometric premises of the displayed twist/shift cancellation", "FR nonvanishing and conservativity, Beraldo Ramanujan/tempered comparison, and GL_n full-faithfulness and eigensheaf existence", "generic oper geometry, pseudo-proper presentations, base change, rational maps and coherent unital-Ran/iQCoh comparisons", "classical smoothness of U_G, compatible fibre self-duality and generic-to-global finite étale monodromy", "fundamental groups of de Rham stack components, central character-line classification, local-system topology comparisons and finite-cover descent", "spectral locally complete-intersection/CM theorem, stable-bundle and reducible-local-system codimension estimates with the genus-two A1 exceptions", "vacuum component nonvanishing, scalar H0 endomorphisms, vanishing under nontrivial central translation and two-categorical Fourier–Mukai compatibility", "almost-isogeny, centre, product and low-genus reductions and the classical-group generic-oper theorem", "stable/enriched DG category foundations, Yoneda, coherent adjunctions and duality, derived tensor/Hom and local cohomology, compactness/perfection and smooth descent", "recursive earlier commutative-algebra and rank-one geometric foundations and all retained original fifteen-lesson proof/correction scope" ] }, { "id": "GL-GLC-15", "title": "The proof as a map", "source": "src/the-proof-as-a-map.md", "reader": "GL-GLC-15.html", "proof_obligations": [ "Actual bundle and derived local-system stacks, half twists, D-module/IndCoh operations, nilpotent singular support and coherent geometric adjunctions.", "Whittaker-normalized construction, spectral action, compact eventually-coconnective bounds, corrected critical FLE/factorization and localization comparisons.", "Eisenstein generation, actual global algebra monad with its specified unit, exact open-supported localization and the proper-Levi supported-unit theorem.", "Actual FR nonvanishing/tempered comparisons and conservativity; normalized ambidexterity, generic-opers/Ran geometry, compactness and finite-monodromy bundle theorem.", "Fundamental groups, central-character line decomposition, spectral component topology, CM and codimension estimates, supported-section vanishing with the stated genus and type exceptions.", "Every-component vacuum nonvanishing, vacuum Hom dimensions and twisted vanishings, gerbe/central-Hecke comparisons, and positivity distinct from the upper dimension bound.", "Full categorical torus equivalence, reductive root structure, product/Kunneth comparisons, actual almost-isogeny base change and the separate low-genus/type-A1 results.", "Full/restricted/tempered and de Rham/Betti comparisons, field extensions, compact coherent-image and regular-singular Riemann–Hilbert theorems.", "Actual spectral/eigen-fibre comparison, nonzero fibres and faithful forgetting, irreducible fibre theorem, normalized coefficient, regularity/perversity/residual-gerbe/Peter–Weyl and characteristic-cycle inputs.", "Subsequent positive-characteristic specialization, component/trace theorems, factorization-crystal descent, arithmetic vacuum/orbit comparisons; explicitly conjectural orbit, analytic and local/ramified claims retain their status.", "Stable/DG-category, coherent adjunction, monoidal action, relative tensor, Ind, derived-module, local cohomology and recursive earlier curve/algebra/stack foundations.", "Full geometric Langlands proof, all group and genus reductions, and complete proofs of the mathematical foundations used throughout the course." ] } ], "status": "Draft.", "reference_note": "References credit the human mathematical sources; citations do not replace proofs." }