{ "schema": "research-course-series/v1", "id": "CG-S6", "title": "Complex threefolds from marked torus families", "objective": "Teach the complete S6 construction and its retained refinements through exact arguments and checked prerequisite providers.", "sequence_kind": "Complete eleven-lesson sequence with its included receiving proof companions.", "lessons": [ { "id": "CG-S6-01", "title": "Two finite torus quotients and their line bundles", "source": "src/finite-quotients-and-line-bundles.md", "proof_scope": "Original period matrices, full integral fixed and moving lattices, exact product kernels, free groups of orders nine and eight, and explicit smooth trivializations of holomorphically torsion line bundles.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "FF1-FF22", "FF42-FF46", "Engel Section 3" ] }, { "id": "CG-S6-02", "title": "Normal boundaries and their integral maps", "source": "src/normal-boundaries-and-integral-maps.md", "proof_scope": "Full deck groups, exact central attachment kernel, explicit product-group isomorphism, integral H1 and covering pullback H1 with indices three and four.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "FF30-FF41", "CG-S6-01 normal-line construction" ], "requires": [ "CG-S6-01" ] }, { "id": "CG-S6-03", "title": "Varying finite fillings and every base-change branch", "proof_scope": "All local period germs and their exact inverse parametrization, original covariance, proper finite quotient, complete real-analytic normal-bundle comparison, all normalization branches and branch-independent punctured holomorphic conjugacy.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "FF23-FF41", "candidate_geometry.tex period cocycles and finite-filling proposition", "CG-S6-01", "CG-S6-02" ], "requires": [ "CG-S6-01", "CG-S6-02" ], "source": "src/varying-finite-fillings.md" }, { "id": "CG-S6-04", "title": "Global periods and the line-bundle quotient", "proof_scope": "Global cusp-regular period construction, exact admissible constant range, all fibre line-bundle quotient maps, exact contraction rate, full Weierstrass scale map and explicit rational parametrization.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "108-page reconstruction Sections 2-3", "Engel Sections 1.3 and 2" ], "source": "src/global-periods-and-line-bundle-quotients.md", "requires": [ "CG-S6-01", "CG-S6-03", "LG-MF-04 sections 3-4", "LG-MF-05 sections 1-4" ] }, { "id": "CG-S6-05", "title": "The cusp and the compact threefold", "proof_scope": "Complete unit-dependent cusp model, normalization, properness and compact gluing; rational-section generator, full finite resolutions, exact heights, global translation and every factor in the line-bundle lift comparison.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "108-page reconstruction Sections 4 and 6", "Engel Section 2" ], "source": "src/cusp-and-compact-threefold.md", "requires": [ "CG-S6-01", "CG-S6-02", "CG-S6-03", "CG-S6-04" ] }, { "id": "CG-S6-06", "title": "Integral monodromy and the fundamental group", "proof_scope": "Every original integral monodromy and exterior-power lattice, complete based finite and cusp kernels, full van Kampen proof, exact Engel marking dictionary, and actual additional free affine fillings with their cyclic fundamental groups.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "retained integral-monodromy paper", "108-page reconstruction Section 7", "Engel Sections 5-6" ], "source": "src/integral-monodromy-and-fundamental-group.md", "requires": [ "CG-S6-01", "CG-S6-02", "CG-S6-03", "CG-S6-04", "CG-S6-05" ] }, { "id": "CG-S6-07", "title": "Integral homology and smooth sphere recognition", "proof_scope": "Exact integral homology computation, homotopy equivalence and smooth recognition with proved or exact preceding providers for each topological result.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "108-page reconstruction Sections 7-8", "Engel Sections 4 and 7" ], "provider_search": [ "DG-CHAR Hurewicz and spectral-sequence providers", "existing h-cobordism and homotopy-sphere sources" ], "source": "src/integral-homology-and-sphere-recognition.md", "reader": "CG-S6-07.html", "completed_local_companions": [ "morse-cancellation-with-controlled-support", "spin-six-and-the-framing-comparison", "integral-line-bundle-descent-for-the-affine-fillings", "integral-duality-and-the-specialization-lattices", "cw-models-and-the-first-hurewicz-map", "path-fibres-and-integral-homotopy-equivalences", "belt-sphere-complements-and-the-whitney-disk", "whitney-move-with-controlled-support", "relative-morse-functions-and-original-handles", "rearranging-the-original-framed-handles", "morse-trajectories-and-critical-value-lowering", "removing-the-original-zero-and-top-handles", "creating-an-original-cancelling-handle-pair", "framed-handle-slides-and-index-one-removal", "integral-handle-chains-and-signed-incidences", "higher-index-framed-slides-and-chain-maps", "whitney-moves-in-the-receiving-dimensions", "realizing-framed-handles-and-middle-cancellation", "the-degree-six-milnor-cobar-calculation", "framed-manifolds-and-the-pontryagin-thom-map", "coefficient-rings-and-the-original-degree-six-class", "the-adams-tower-and-exact-inverse-limits", "the-original-adams-tower-and-arithmetic-completion", "the-actual-mod-two-adams-resolution-and-cooperation-maps", "the-exact-characteristic-two-coordinate-calculation", "the-original-projective-coordinate-and-actual-cooperations", "the-actual-mod-two-cooperation-comparison-through-degree-eight", "the-original-odd-prime-cooperations-and-degree-six-adams-calculation", "the-original-integral-stable-group-and-its-completion-maps", "the-original-six-dimensional-framing-invariant", "sphere-twists-and-the-original-smooth-six-sphere-comparison" ], "integral_parity_step": "Closed by the exact smooth line-bundle lift, integer norm equation and retained constant phases; no Wu-class input is needed for this specialization lattice. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "integral_duality_step": "Complete compact-support integer duality, finite generation and perfect free pairing included. Exact covering comparison strengthens the degree-three index to both full image lattices. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "cw_hurewicz_step": "Explicit nested-cube telescope and both manifold comparison maps, including the original punctured cobordism; full coherent-simplex first Hurewicz theorem and the positive sphere class. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "path_fibre_step": "Complete path lifting and homotopy sequence; filtered integer chains and both Serre edges; CW-domain compression and inverse maps for the original degree-one sphere map and two boundary inclusions. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "framed_whitney_disk_step": "Complete original-radius complement diffeomorphism and tangent maps; relative embedded disk avoiding the retained collar strata; full nonorthogonal frame criterion; actual corner boundary, both orientation signs and adapted framed disk for the specified simply connected handle stage. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "supported_whitney_move_step": "Complete collar-germ extension and original corner-disk comparison; exact simultaneous sheet coordinates; full cutoff flow, signs, support and inverse; transport of all original framed tubes with every radius retained; diffeomorphism of attached cobordisms fixing the incoming boundary. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_handle_decomposition_step": "Relative Morse function on the original cobordism with whole fixed boundary collars; full-Hessian coordinate comparison and original metric; exact level maps with their exceptional sets; actual framed handle insertion, complete transverse seam, specified outward rounding and variable-time collar absorption; all critical indices and final outgoing identification retained. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_handle_index_arrangement_step": "Exact comparison of the retained rounded boundary and original surgery collars, with both full endpoint germs and both radial derivatives; compression of the entire next framed tube; supported outward motion through the actual core-complement map; explicit collar extension and literal interchange of disjoint attached pieces; all extreme indices, parameter radii, later frames and final outgoing identification retained. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_morse_trajectory_step": "Entire boundary isotopy realized as descending holonomy with full framing and metric derivatives; exact stable-belt comparison; compact embedded escaping unstable disk; global disk coordinates retaining every original Hessian coefficient; explicit supported critical-value lowering with strictly positive transverse derivative brackets; actual ordered Morse function and its complete comparison to the original handle presentation. The initial and final fields, all boundary collars and both extreme indices are accounted for. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_zero_top_removal_step": "Actual index-one branch arrangement by supported holonomy; stable-level measure-zero proof; exact correspondence between cobordism components and the trajectory graph; complete boundary-to-minimum cancellation and finite zero-index removal; original negative function, all Hessian coefficients, coordinate permutation signs and both collar maps retained in top-index removal; full comparison of the entire handle presentations. Only connectedness is required for this step. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_handle_pair_creation_step": "Complete explicit two-critical-point model, compactly supported cancellation with all derivative bounds, global lower-level parametrization and complete horizontal product with exact inverse, supported single birth path, unique transverse connecting trajectory and compact full truncated unstable set, implantation in the original regular band with all coordinate scales and metric data, and actual lower attaching sphere bounding its framed disk. Supplies index-two/three stabilization in dimensions six and seven. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "original_framed_slide_and_index_one_step": "Complete original based group and finite normal-closure calculation; actual supported framed belt crossing with its negative relator sign and retained conjugator; relative loop isotopy with full ambient and framing extension; simultaneous old critical level; exact original descending map including every Hessian coefficient, radius and variable-time derivative; verified one/two cancellation, full transport of the retained three-handle and finite reversed removal. Proves the stronger surjectivity hypothesis for the actual boundary fundamental-group map. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "current_remaining_smooth_prerequisites": "None left as unproved receiving steps in lesson 7; complete written sphere-twist, gluing, cancellation and original-map comparison are included. Independent review is not claimed.", "original_integral_handle_chain_step": "Complete original relative-handle filtration, singular-homology comparison, signed incidence with retained chart and boundary signs, ordinary integral extension without ambient orientability, full adjacent integer basis changes, unit contraction with every correction term, and complete original-coordinate contractions in receiving dimensions six and seven. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "higher_index_framed_slide_step": "Complete higher-index framed slides, exact supported one-crossing flow, full collar/cobordism/metric maps, filtered singular-prism correction, retained crossing sign and both neighbouring differentials. All integer basis changes are realized by the recorded slides and signed basis comparisons. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "receiving_whitney_step": "Complete receiving Whitney construction for (d,p,q)=(5,2,3),(6,2,4),(6,3,3), with the original complement map, relative embedded disk, full rank-four frame extension through the proved quaternion covering, exact sheet charts, all cutoff derivatives, signed intersections, and original whole framing/cobordism maps. The complete receiving handle reduction is now included in the actual Morse and middle-cancellation companion; The sphere-twist companion completes the original smooth-sphere comparison.", "actual_morse_and_middle_cancellation_step": "Complete original-coordinate Morse realization of modified framed handles, all variable roof and collar derivatives, full original quadratic coefficients, actual stable/unstable traces and exact intersection-to-trajectory bijection, escape and compact support proofs, simultaneous supported middle cancellation, and the finite simply connected h-cobordism product proofs in dimensions six and seven. The sphere-twist companion supplies the complete smooth-sphere comparison.", "degree_six_cobar_step": "Complete all-length algebraic degree-six calculation, with actual operation, convergence and integral comparisons proved in their included companions. The geometric framing companion proves the original framed class zero; the sphere-twist companion supplies the complete smooth-sphere comparison.", "pontryagin_thom_step": "Complete original embedding, normal framing, collapse, inverse bordism/homotopy constructions, suspension and original tangent-frame homotopy. The geometric framing companion now proves the actual class zero and constructs its framed nullbordism.", "coefficient_ring_step": "Complete coefficient/index/weight and Adem presentation maps; original polynomial algebra retained; exact coefficient kernel bound in every length; full original H0^(2,8) polynomial module with no coefficient torsion, complete matrices and exact receiving cycle.", "adams_tower_step": "Complete original tower maps, finite inverse limits, every spectral-sequence page, all-length degree-five cobar vanishing, original coefficient-spectrum construction, all actual stage groups finite with p^(s+1) bound, and convergence to the specified homotopy limit. The arithmetic-completion comparison is now proved in its own companion. The actual cooperation comparison and framed value are now proved in their included companions.", "arithmetic_completion_step": "Complete unit-preserving Adams/Moore equivalence for every bounded-below spectrum, exact finite transition bound, all Moore torsion and derived terms, original unit sign, sharp stage-map exponent and full degree-zero extensions. The actual operation comparison and framed value are now proved in their included companions.", "actual_cooperation_step": "The actual mod-two resolution, cofree comparison and needed polynomial classification through degree eight are proved. The odd-prime companion supplies the original signed comparison and completed degree-six vanishing. The integral comparison is now proved in its included companion. The geometric framing companion now proves the original value zero and framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison.", "projective_coordinate_step": "The original projective morphism and its bijectivity through cooperation degree eight are proved. Both mod-two and odd-prime completed degree-six groups are now calculated. The integral comparison is now proved in its included companion. The geometric framing companion now proves the original value zero and framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison.", "actual_degree_eight_step": "The actual bar comparison, original suspension kernel, full graded cooperation dimensions, nonzero powers of the first actual generator, cooperation isomorphism through degree eight and complete degree-six mod-two Adams calculation are proved. The odd-prime companion now supplies the corresponding vanishing at every odd prime. The integral comparison is now proved in its included companion. The geometric framing companion now proves the original value zero and framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison.", "odd_prime_cooperation_step": "The full original odd-prime bar algebra, actual coefficient-space suspensions, exact cooperation dimensions through degree seven, all-length actual degree-six cobar vanishing and signed actual Adams comparison are proved. Every odd-prime completed sixth sphere group vanishes. The integral comparison is now proved in its included companion. The geometric framing companion now proves the original value zero and framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison.", "integral_stable_comparison_step": "The original integral sphere groups are proved finitely generated through exact chain, coefficient-space, path-fibre and right-suspension comparisons. Every term and transition of the Moore sequence is retained, and the original completion unit identifies pi_6(S) with the two-element 2-completed group. The geometric framing companion now proves the original value zero and framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison.", "geometric_framed_value_and_nullbordism_step": "The actual rank-three framing map, both homotopy defects, original quadratic law, embedded representatives and full index-three reduction are proved. Framed bordism invariance identifies the original degree-six framed group with F2 by its Arf value. The specified Lie-framed product has value one; the original framed homotopy six-sphere has value zero and an actual framed nullbordism. The sphere-twist companion completes the original smooth-sphere comparison, retaining the old framing defect.", "smooth_sphere_comparison": "Complete original tube twist and signed transvection; primitive integral Lagrangians in the actual middle level; new interior gluing with full frame defect; signed slides, Whitney moves and Morse cancellation; actual orientation-preserving diffeomorphism and homotopies to the original maps. All original radii, matrices and tangent comparisons retained.", "general_stable_framing_step": "Complete original tangent-to-normal involution homotopy for every six-dimensional stable frame, using the original stable trivialization on an even-rank bundle, both exact rotations, all inverses and derivatives, and at most one appended identity coordinate. Independent of the smooth-recognition theorem; closes the general framed-bordism receiving comparison." }, { "id": "CG-S6-08", "title": "Canonical sections and the intrinsic fibration", "proof_scope": "Exact Hodge-line extension, evaluation divisor, all signed vertical direct images, full graded ring, recovered fibration and every preserved multiple fibre; four solved exercises.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "analytic_canonical_ring.tex CR1-CR5" ], "source": "src/canonical-divisor-and-intrinsic-fibration.md", "requires": [ "CG-S6-03", "CG-S6-04", "CG-S6-05" ] }, { "id": "CG-S6-09", "title": "Periods and nontrivial deformation", "proof_scope": "Full proper parameter family, exact marked real and Beltrami maps, nonzero fibre and global deformation classes, every possible linear part, exact finite parity obstruction, global even-shift maps and complete quotient parameter.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "antecedents": [ "period_parameter_deformation.tex PD1-PD18", "period_shift_classification.tex" ], "source": "src/periods-and-nontrivial-deformation.md", "requires": [ "CG-S6-03", "CG-S6-04", "CG-S6-05", "CG-S6-06", "CG-S6-08" ] }, { "id": "CG-S6-10", "title": "Non-normal fibres and the vanishing calculation", "proof_scope": "Complete conormal and intrinsic torsion maps, proper test family, full-fibre duality/base change, every analytic twist, full index polynomial, global nonvanishing and exact source corrections; three complete analytical companion chapters and six solved exercises.", "status": "authored_with_included_analytic_proofs_and_exact_algebra_self_checks", "antecedents": [ "108-page reconstruction Section 10", "retained independent test-family paper", "CDP20 original-author TeX", "October status update" ], "source": "src/non-normal-fibres-and-vanishing.md", "requires": [ "CG-S6-01", "CG-S6-05", "CG-S6-06", "CG-S6-07 Sections1–8", "CG-S6-08" ] }, { "id": "CG-S6-11", "title": "Almost complex spheres and the complex-geometry background", "proof_scope": "Full octonion comparison, Nijenhuis and associator maps, complete smooth integrability proof, polynomial Bott generation and sphere restriction, with all receiving characteristic-class proofs included.", "status": "authored_with_complete_proofs_and_exact_algebra_self_checks", "provider_search": [ "DG-CHAR", "compact complex manifolds foundation plan" ], "source": "src/almost-complex-structures-and-integrability.md", "requires": [] } ], "whole_series_complete": true, "gaussian_channel_material": "Retained separately; not a provider for the sphere-recognition theorem.", "review": "Each authored proof is checked on its own terms. A source's stated theorem or a symbolic test is not whole-source certification.", "boundary_repair_20261009": "The smooth line-bundle and cubic comparison is proved in lesson 4. The resolved fibre, section and global lift extension calculations travel together with the singular-fibre work in lesson 5; none is removed from the assignment.", "delivery_order_note": "All eleven lessons and their worked exercises are written. Lesson 7 now includes the full smooth-recognition proof and original-map comparison. The earlier order of delivery is historical; no independent review is claimed." }