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BSO2 integral ring Z[e] separately.", "conditions": "Any commutative coefficient ring A with identity in which2 is a unit; finite n>=0 with BSO0/1 constant; degrees |pi|=4i, |e|=2m. 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Complete existence iff e0 when dimB<=n; higher obstructions retained above that bound. 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Critical-value image null in every target chart; no finite-Cr version claimed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionA LemmaA.1 and TheoremA.2", "retained_anchor": "a-sard-s-theorem-and-the-measure-zero-prerequisite", "reader": "DG-CHAR-13.html#a-sard-s-theorem-and-the-measure-zero-prerequisite" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.cellular-approximation", "aliases": [ "DG-CHAR.homotopy.cellular-approximation" ], "statement": "Full cellular approximation, relative subcomplex and CW-pair versions, by localized affine cell removal and CW homotopy extension.", "conditions": "Hausdorff closure-finite CW complexes with weak topology, arbitrary cells/dimensions; fixed subcomplex map already cellular. 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Compact-stage base requires nested compact exhaustion with interiors covering. Finite universal Grassmannian carrier for compact maps/homotopies. Rank0 has disjoint basepoint convention. No rational/stable Hurewicz conclusion.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionsC/D TheoremsC.1/C.2, LemmasD.1/D.2 and suspensionD.2–D.3", "retained_anchor": "c-thom-spaces-cells-and-homology", "reader": "DG-CHAR-13.html#c-thom-spaces-cells-and-homology" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.smooth.thom-approximation", "aliases": [ "DG-CHAR.smooth.thom-approximation" ], "statement": "Relative fine smooth approximation into a Thom space preserving exactly the collapsed-point preimage.", "conditions": "Positive-rank smooth Euclidean bundle over Hausdorff second-countable manifold without boundary; compact smooth source possibly with boundary. Closed relative set in noncollapsed complement where initial map is smooth near it. 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Positive-rank smooth bundle over Hausdorff second-countable manifold without boundary, compact source/constant regular cylinder ends. Sphere homomorphism for m>=k>=1; oriented bundle plus positive sphere gives oriented bordism, otherwise unoriented. Exact time-first endpoint signs.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionsE/G LemmasE.1/E.2, G.1/G.2 and TheoremG.3", "retained_anchor": "g-transverse-inverse-images-and-the-bordism-map", "reader": "DG-CHAR-13.html#g-transverse-inverse-images-and-the-bordism-map" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.smooth.dimension-controlled-embedding", "aliases": [ "DG-CHAR.smooth.dimension-controlled-embedding" ], "statement": "Compactd-manifold embedding inR2d+1 and relative compact cobordism embedding with arbitrary given boundary collars retained.", "conditions": "Compact smooth manifolds; finite preliminary embedding allows boundary. Absolute2d+1 bound stated without boundary. Wdimensionn+1 compact, boundaryM0/M1 and givenj_i intoRm; m>=2n+2 gives embedding intoI×Rm fixed as product collars, interior maps to interior. Oriented normal compatibility whenW oriented.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionH LemmaH.1 and TheoremsH.2/H.3", "retained_anchor": "h-dimension-controlled-embeddings-including-fixed-boundary-collars", "reader": "DG-CHAR-13.html#h-dimension-controlled-embeddings-including-fixed-boundary-collars" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.bordism.pontryagin-thom", "aliases": [ "DG-CHAR.bordism.pontryagin-thom" ], "statement": "Full oriented/unoriented Pontryagin–Thom isomorphism by normal Gauss collapse and transverse inverse image.", "conditions": "n>=0,k>n+1. Universal CW Thom spaces; compact representatives/homotopies in finite Grassmannian stages. Surjectivity already for k>n, finite ambient parameterp>=n. Canonical collapse orientation tangent first, normal second. No full rational cobordism or unoriented characteristic-number detection claimed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionsI/J/K TheoremI.1, LemmasJ.1/J.2, PropositionJ.3 and TheoremK.1", "retained_anchor": "k-the-full-pontryagin-thom-correspondence", "reader": "DG-CHAR-13.html#k-the-full-pontryagin-thom-correspondence" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.manifold.closed-one", "aliases": [ "DG-CHAR.manifold.closed-one" ], "statement": "Every closed smooth one-manifold is a finite disjoint union of smooth circles; first unoriented bordism groupsN0=F2,N1=0.", "conditions": "Smooth Hausdorff second-countable compact one-manifold without boundary. Finite arc graph and complete flow proof; no noncompact/boundary interval classification claimed. N0 parity uses full mod2 boundary-fundamental identity; RP2 only a nonzero degree2 witness.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "thom-spaces-and-the-pontryagin-thom-construction.md", "source_sha256": "53763fdad17fad6bf5a183d64581aec4dd5b2d284e1ad82f6b564be9b34c2dc5", "proof_location": { "declared_locus": "SectionL LemmaL.2 and ExampleL.3, ExerciseM.8", "retained_anchor": "l-examples-the-real-line-and-the-first-bordism-classes", "reader": "DG-CHAR-13.html#l-examples-the-real-line-and-the-first-bordism-classes" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "DG-CHAR-17", "DG-FND", "D60", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.homology.filtered-spectral-sequence", "aliases": [ "DG-CHAR.homology.filtered-spectral-sequence" ], "statement": "Explicit filtered-chain spectral sequence, representative differentials/page transitions and finite first-quadrant convergence.", "conditions": "Nonnegative abelian chain complex, increasing filtration by subcomplexes, Fp=0 forp<0 and exhaustive; relative quotient homology vanishes for totaldegree

=2; based interval/square lifts and deck action of the universal path cover proved. No general non-CW covering existence claimed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "grassmannians-and-classifying-maps.md", "source_sha256": "c346bc482f49f162882fb0529af135e4f91a8aa109540ea2fef5c125e10f3333", "proof_location": { "declared_locus": "Section8B Theorem8.3 full proof", "retained_anchor": "8b-general-coverings-and-lifted-cw-cells", "reader": "DG-CHAR-03.html#8b-general-coverings-and-lifted-cw-cells" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "D60", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.eilenberg-maclane", "aliases": [ "DG-CHAR.homotopy.eilenberg-maclane" ], "statement": "EM models for finitely generated abelian A, comparison with arbitrary one-group targets, finite-type homology, finite-A rational acyclicity and realization of a first homology homomorphism.", "conditions": "A finitely generated abelian; n>=1 for models/comparison/homology. First-map realization needs n>=2 and an(n-1)-connected CW source. Comparison target path connected with only pi_n=A. No general cohomology representation or uniqueness-of-all-maps theorem claimed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-homotopy-and-the-hurewicz-range.md", "source_sha256": "f8ec2365abaecf637539819149e97d123d4724c479c51d96a6329f50006cbdfd", "proof_location": { "declared_locus": "SectionF LemmasF.3–F.5, TheoremF.6 and LemmaF.7", "retained_anchor": "f-spaces-with-one-homotopy-group", "reader": "DG-CHAR-13C.html#f-spaces-with-one-homotopy-group" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "D60", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.finite-generation", "aliases": [ "DG-CHAR.homotopy.finite-generation" ], "statement": "Every homotopy group of a simply connected integral-homology finite-type CW complex is finitely generated, by finite connected-cover iterations.", "conditions": "Simply connected CW complex; H_j(-;Z) finitely generated in every degree. Each fixed pi_j, j>=1, is finitely generated. No nonsimply-connected extension or infinite tower limit assumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-homotopy-and-the-hurewicz-range.md", "source_sha256": "f8ec2365abaecf637539819149e97d123d4724c479c51d96a6329f50006cbdfd", "proof_location": { "declared_locus": "SectionF TheoremF.8 full proof", "retained_anchor": "f-spaces-with-one-homotopy-group", "reader": "DG-CHAR-13C.html#f-spaces-with-one-homotopy-group" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "D60", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.mod-finite-hurewicz", "aliases": [ "DG-CHAR.homotopy.mod-finite-hurewicz" ], "statement": "Hurewicz finite kernel/cokernel and lower rational homology vanishing when earlier homotopy groups are finite.", "conditions": "Simply connected CW complex with finitely generated integral homology in every degree; n>=2 and pi_i finite for2<=i=1. Polynomial on rank(A) degree-n generators ifn even, exterior ifn odd. Finite groups rationally acyclic. No extension to arbitrary non-finitely-generated subgroups ofQ claimed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-homotopy-and-the-hurewicz-range.md", "source_sha256": "f8ec2365abaecf637539819149e97d123d4724c479c51d96a6329f50006cbdfd", "proof_location": { "declared_locus": "SectionH.1 TheoremH.1 full induction and first-failure proof", "retained_anchor": "h-rational-one-group-spaces-spheres-and-the-hurewicz-range", "reader": "DG-CHAR-13C.html#h-rational-one-group-spaces-spheres-and-the-hurewicz-range" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "D60", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.serre-sphere-finiteness", "aliases": [ "DG-CHAR.homotopy.serre-sphere-finiteness" ], "statement": "Full finiteness of pi_i(S^n) above n with the unique rational even-sphere exception, proved using pullback path fibrations and finite connected covers.", "conditions": "n>=2,i>n. Finite except n even,i=2n-1, where group is Z plus a finite group. Does not identify an integral Hopf generator or calculate finite orders.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-homotopy-and-the-hurewicz-range.md", "source_sha256": "f8ec2365abaecf637539819149e97d123d4724c479c51d96a6329f50006cbdfd", "proof_location": { "declared_locus": "SectionH.2 LemmaH.2 and TheoremH.3 full proof", "retained_anchor": "h-rational-one-group-spaces-spheres-and-the-hurewicz-range", "reader": "DG-CHAR-13C.html#h-rational-one-group-spaces-spheres-and-the-hurewicz-range" }, "consumers": [ "DG-CHAR-13", "DG-CHAR-14", "D60", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.homotopy.stable-hurewicz", "aliases": [ "DG-CHAR.homotopy.stable-hurewicz" ], "statement": "Stable-range ordinary Hurewicz has finite kernel/cokernel; oriented finite Thom comparison with base homology.", "conditions": "Finite(k-1)-connected CW complex,k>2,k<=r<2k-1; also holds with homology finite generation in every degree. Thom consequence: oriented rankk>2 over finite CW base,0<=n=1,p>=0 for cell bounds: base inclusionH_i iso i=2. Finite PT and suspension iso: n>=0,k>n+1,p>=n+1. Last positive line and tangent-first orientation retained; no claim at strict endpoint.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionJ.1–J.3 LemmaJ.1, TheoremJ.2 and PropositionJ.3 full proofs", "retained_anchor": "j-finite-stages-and-the-rational-bordism-rank", "reader": "DG-CHAR-13D.html#j-finite-stages-and-the-rational-bordism-rank" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.bordism.rational-normal-comparison", "aliases": [ "DG-CHAR.bordism.rational-normal-comparison" ], "statement": "Oriented bordism maps to rank-k universal normal homology with finite kernel and cokernel, through a completely specified finite Thom complex.", "conditions": "n>=0,k>max(2,n+1),p>=n+1. Omega_n→pi_(n+k)T_kp→H_(n+k)T_kp→H_nB_kp→H_nBSO(k); actual end isomorphisms, positive oriented Thom class, finite kernel/cokernel and finite-generation hypotheses all proved. No equality of normal and tangent p classes assumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionJ.4 TheoremJ.4 full proof", "retained_anchor": "j-finite-stages-and-the-rational-bordism-rank", "reader": "DG-CHAR-13D.html#j-finite-stages-and-the-rational-bordism-rank" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.bordism.oriented-rank", "aliases": [ "DG-CHAR.bordism.oriented-rank" ], "statement": "Oriented smooth bordism is finitely generated in every nonnegative degree; rank p(r) in degree4r and finite in degrees not divisible by4.", "conditions": "Smooth closed oriented manifolds, possibly disconnected, modulo smooth oriented compact bordism. n>=0; empty partition p(0)=1. No integral torsion-order/Wall-detection claim.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionJ.5 TheoremJ.5 full proof; alternate surjection/independence proof ExerciseL.6", "retained_anchor": "j-finite-stages-and-the-rational-bordism-rank", "reader": "DG-CHAR-13D.html#j-finite-stages-and-the-rational-bordism-rank" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.bordism.rational-polynomial-ring", "aliases": [ "DG-CHAR.bordism.rational-polynomial-ring" ], "statement": "Rational oriented bordism is Q[z_i,i>=1],degz_i=4i, with CP^(2i) generators; tangent Pontryagin numbers give a complete rational coordinate isomorphism in each degree4r.", "conditions": "Graded direct-sum rationalization of smooth oriented bordism. Degree4r projective products indexed by partitionsr; degree0 includes positive point/empty number. All other rational degrees vanish. No assertion of an integral polynomial ring or arbitrary integer number realizability.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionK.1 TheoremK.1 and CorollaryK.2 full proofs", "retained_anchor": "k-projective-generators-and-the-torsion-criterion", "reader": "DG-CHAR-13D.html#k-projective-generators-and-the-torsion-criterion" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.bordism.pontryagin-torsion-criterion", "aliases": [ "DG-CHAR.bordism.pontryagin-torsion-criterion" ], "statement": "Some positive disjoint multiple of a closed oriented manifold bounds iff every top Pontryagin number is zero.", "conditions": "Smooth closed oriented n-manifold,n>=0,possibly disconnected. Non4 degree has automatic condition. Degree0 must include empty-product number, signed point count. Positive-multiple/torsion criterion only; no assertion that M itself bounds when numbers vanish.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionK.2 fraction equivalenceK.6 and TheoremK.3 both directions", "retained_anchor": "k-projective-generators-and-the-torsion-criterion", "reader": "DG-CHAR-13D.html#k-projective-generators-and-the-torsion-criterion" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.bordism.projective-multiple", "aliases": [ "DG-CHAR.bordism.projective-multiple" ], "statement": "A positive integral multiple of every oriented closed manifold is integrally bordant to a signed combination of products of even complex projective spaces; non4 degrees have a multiple bounding.", "conditions": "Smooth closed oriented n-manifold,n>=0. For n=4r use partitionsr,including empty point at0. DenominatorD followed by torsion multipliert gives actual integral equality; negative coefficients reverse orientations. Denominator clearing alone not claimed sufficient.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "rational-oriented-bordism-and-projective-generators.md", "source_sha256": "ee22f48ebada2d6928012ea1c7653a7920e89da02522febf783428dc23dc3f37", "proof_location": { "declared_locus": "SectionK.3 CorollaryK.4 full proof; dimension4/8 computationsK.10–K.13", "retained_anchor": "k-projective-generators-and-the-torsion-criterion", "reader": "DG-CHAR-13D.html#k-projective-generators-and-the-torsion-criterion" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.signature.form-algebra", "aliases": [ "DG-CHAR.signature.form-algebra" ], "statement": "Rational symmetric inertia, isotropic-half vanishing, alternating paired bases and the symmetric/alternating tensor signature formulas.", "conditions": "Finite-dimensional nondegenerate forms over Q; symmetric tensor signatures multiply; the tensor of two alternating forms and its negative have signature zero. No integral congruence classification asserted.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionA LemmasA.1–A.3 and formulasA.1–A.7, all proofs", "retained_anchor": "a-the-algebra-behind-the-signature", "reader": "DG-CHAR-14.html#a-the-algebra-behind-the-signature" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.duality.poincare-lefschetz", "aliases": [ "DG-CHAR.duality.poincare-lefschetz" ], "statement": "Relative right-cap duality for compact oriented manifolds, homological finite generation, rational absolute right-cap duality and the perfect relative/absolute pairing.", "conditions": "Smooth compact oriented W^d, M=boundary, outward convention; disconnected/empty boundary permitted. Relative cap H^r(W,M;R) to H_(d-r)(W;R) for every commutative R. Absolute cap and perfect pairing are asserted over Q. Full collar/interior/excision and cap comparison supplied.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionB.1 LemmaB.1 and SectionB.2 TheoremB.2 full proofs", "retained_anchor": "b-relative-duality-and-the-boundary-pairing", "reader": "DG-CHAR-14.html#b-relative-duality-and-the-boundary-pairing" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.signature.boundary-isotropic-half", "aliases": [ "DG-CHAR.signature.boundary-isotropic-half" ], "statement": "The extendable middle cohomology of an oriented boundary equals its orthogonal complement; its symmetric middle form has signature zero.", "conditions": "W^(2m+1) smooth compact oriented, M=boundary; rational cohomology. Restriction image in H^m(M;Q) is a half-dimensional isotropic subspace. Signature zero conclusion when m is even. Exact outward-sign connecting/cup adjoint identity proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionB.3 LemmaB.3 and PropositionB.4 full proofs", "retained_anchor": "b-relative-duality-and-the-boundary-pairing", "reader": "DG-CHAR-14.html#b-relative-duality-and-the-boundary-pairing" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.signature.bordism-homomorphism", "aliases": [ "DG-CHAR.signature.bordism-homomorphism" ], "statement": "Signature is additive, changes sign under orientation reversal, vanishes on oriented boundaries and is multiplicative in every pair of dimensions; it is an oriented homotopy invariant.", "conditions": "Smooth closed oriented manifolds, possibly disconnected; dimension4k uses the rational middle intersection form, other dimensions have value0. Degree0 is signed point count. Ordinary unital ring map Omega_* to Z, with rational extension; no grading assertion for target. Full Kunneth-orbit product proof includes dimensions2 modulo4.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionC TheoremC.1, all product casesC.4–C.5, and PropositionC.2 full proofs", "retained_anchor": "c-the-signature-as-a-bordism-homomorphism", "reader": "DG-CHAR-14.html#c-the-signature-as-a-bordism-homomorphism" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.genus.multiplicative-sequence", "aliases": [ "DG-CHAR.genus.multiplicative-sequence" ], "statement": "Every formal series1+f_1t+... over a commutative ring determines a unique universal multiplicative sequence, given by stable integral monomial symmetric sums.", "conditions": "Any commutative coefficient ring with unit, including zero divisors; ordinary weight-graded commutative algebras and degree completion. Repeated partition parts do not add a factorial coefficient. Universal two-root-list polynomial identity and coefficientwise independence proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionD.1 TheoremD.1 full proof and formulasD.3–D.6", "retained_anchor": "d-multiplicative-sequences-from-a-power-series", "reader": "DG-CHAR-14.html#d-multiplicative-sequences-from-a-power-series" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.genus.oriented-ring-map", "aliases": [ "DG-CHAR.genus.oriented-ring-map" ], "statement": "A rational multiplicative sequence evaluated on tangent Pontryagin classes defines an ordinary unital genus homomorphism from oriented bordism to Q.", "conditions": "Smooth closed oriented manifolds, all dimensions; weightn class evaluated in dimension4n, other dimensions0, degree0 signed count. Rational Whitney formula removes its integral2-torsion difference. No all-oriented integrality asserted for an arbitrary genus.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionD.2 TheoremD.2 full boundary and product proof", "retained_anchor": "d-multiplicative-sequences-from-a-power-series", "reader": "DG-CHAR-14.html#d-multiplicative-sequences-from-a-power-series" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.signature.hirzebruch-theorem", "aliases": [ "DG-CHAR.signature.hirzebruch-theorem" ], "statement": "The L-sequence of sqrt(t)/tanh(sqrt(t)) evaluates to the signature on every closed oriented4k-manifold; its value is integral and an oriented homotopy invariant.", "conditions": "Smooth closed oriented4k-manifolds, k>=0, possibly disconnected; tangent Pontryagin classes, rational polynomial evaluation. All projective values1 are proved by formal Laurent-residue substitution, then the fully proved rational bordism polynomial basis gives the theorem. No analytic convergence or index theorem assumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionE.1 LemmaE.1, E.2 PropositionE.2 and E.3 TheoremE.3 full proofs; first polynomialsE.2", "retained_anchor": "e-the-formal-l-series-and-the-signature-theorem", "reader": "DG-CHAR-14.html#e-the-formal-l-series-and-the-signature-theorem" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.genus.ahat-sequence", "aliases": [ "DG-CHAR.genus.ahat-sequence" ], "statement": "The formal A-hat sequence and first two polynomials define a rational oriented genus, with explicit projective values demonstrating failure of all-oriented integrality.", "conditions": "Formal series(sqrt(t)/2)/sinh(sqrt(t)/2) over Q. A-hat1=-p1/24, A-hat2=(7p1^2-4p2)/5760; CP2=-1/8, CP4=3/128, CP2xCP2=1/64. No spin/index integrality claim.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "multiplicative-sequences-and-the-signature-theorem.md", "source_sha256": "51a02802fde101f07e1fa72ee135ba5cd9bc6b14be64633b11a96f52a0a6a500", "proof_location": { "declared_locus": "SectionE.4 and ExerciseF.6 complete calculations, using TheoremsD.1–D.2", "retained_anchor": "e-the-formal-l-series-and-the-signature-theorem", "reader": "DG-CHAR-14.html#e-the-formal-l-series-and-the-signature-theorem" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "DG-CHAR-16", "GT-SURG", "GT-TOPM", "DG-SPIN" ], "independent_review": false }, { "id": "DG-CHAR.chains.simplicial-normalization", "aliases": [ "DG-CHAR.chains.simplicial-normalization" ], "statement": "Normalized simplicial chains are naturally a direct summand of unnormalized chains, with explicit projector and actual natural deformation homotopy; relative cochain and cup comparison.", "conditions": "Simplicial F-modules, any field; N_n=intersection ker d_i for iq and P^s x=x^ell when q=2s. Graded-dual cochain ring isomorphism and Theta chain signs explicit.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionsH.2–H.4, H.4–H.11; full tensor interval/norm argument", "retained_anchor": "h-higher-diagonals-and-the-reduced-power-construction", "reader": "DG-CHAR-14B.html#h-higher-diagonals-and-the-reduced-power-construction" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.operations.odd-normalizer-cartan", "aliases": [ "DG-CHAR.operations.odd-normalizer-cartan" ], "statement": "Symmetric normalizer selects the allowable even cyclic indices; negative powers vanish and normalized reduced powers satisfy the Cartan identity.", "conditions": "Odd primeell=2r+1. D_(2a) of degreeq vanishes unless a=rq mod(ell-1). Internal absolute product; relative external products when both subspaces open or either empty with the other arbitrary; relative times absolute internal product. Exact nu and shuffle signs proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionsI.1–I.2, TheoremI.1 and formulasI.1–I.6", "retained_anchor": "i-cartan-suspension-and-the-identity-operation", "reader": "DG-CHAR-14B.html#i-cartan-suspension-and-the-identity-operation" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.operations.odd-connecting-suspension", "aliases": [ "DG-CHAR.operations.odd-connecting-suspension" ], "statement": "Reduced powers commute with relative connecting maps and reduced suspension, through an explicit cochain correction and general leading-factorial calculation.", "conditions": "Ordinary mod-ell singular cohomology of every pair; odd primeell. Formal two-generator calculation handles dependent a,da and da=0. Full alternating correction with zero/negative resolution terms and dual sign transfer supplied.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionI.3 TheoremI.2, formulasI.7–I.15 and all counting/cancellation steps", "retained_anchor": "i-cartan-suspension-and-the-identity-operation", "reader": "DG-CHAR-14B.html#i-cartan-suspension-and-the-identity-operation" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.operations.odd-identity", "aliases": [ "DG-CHAR.operations.odd-identity" ], "statement": "The constructed P0 is the identity on singular cohomology of every space and pair.", "conditions": "All odd primes, arbitrary spaces/subspaces, all nonnegative degrees. Sphere suspension, integer top-cell lifts and maps of prescribed degrees, arbitrary CW skeleton injection/model, simplicial realization and actual relative quotient operation comparison supplied.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionI.4 TheoremI.3 complete proof", "retained_anchor": "i-cartan-suspension-and-the-identity-operation", "reader": "DG-CHAR-14B.html#i-cartan-suspension-and-the-identity-operation" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.characteristic.odd-thom-polynomial", "aliases": [ "DG-CHAR.characteristic.odd-thom-polynomial" ], "statement": "Thom reduced-power classes are natural, stable, orientation independent and Whitney multiplicative; q_i equals the stable symmetric polynomial Q_i(p) with root series product(1+z^r t).", "conditions": "Oriented real finite-rank bundles on every Hausdorff base; F_ell, ell=2r+1 odd. q_i degree4ri and zero if2i>rank. Q_i=K_(ri) for f(z)=1+z^r. Universal odd-field injection and compact-cycle detection extend paracompact classification to all Hausdorff bases.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionsJ.1–J.2, TheoremJ.1, LemmaJ.2, TheoremJ.3 and formulasJ.1–J.8", "retained_anchor": "j-odd-prime-thom-classes-and-wu-s-formula", "reader": "DG-CHAR-14B.html#j-odd-prime-thom-classes-and-wu-s-formula" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.characteristic.odd-wu-formula", "aliases": [ "DG-CHAR.characteristic.odd-wu-formula" ], "statement": "Odd-prime Wu functional classes obey v0=1, v_i=0 when2ell i>d, and the full tangent formula q(TM)=P(v).", "conditions": "Closed oriented smooth Hausdorff second-countable d-manifolds, possibly disconnected; F_ell. v_i degree4ri. Full signed diagonal proof: surviving basis degrees4ri have sign+; normal orientation positive determinant2^d. No Adem/Bockstein premise.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionJ.3 TheoremJ.4, formulasJ.9–J.14, plus projective computationsJ.16–J.18", "retained_anchor": "j-odd-prime-thom-classes-and-wu-s-formula", "reader": "DG-CHAR-14B.html#j-odd-prime-thom-classes-and-wu-s-formula" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.characteristic.odd-homotopy-invariants", "aliases": [ "DG-CHAR.characteristic.odd-homotopy-invariants" ], "statement": "All odd-prime tangent q_i and Wu v_i are preserved by homotopy equivalences; individual Pontryagin classes modulo3 follow.", "conditions": "Homotopy equivalences of closed oriented smooth d-manifolds; orientation-preservation unnecessary, separate component signs cancel. At general odd primes only the specified Q_i(p) conclusion is asserted. No all-prime individual-p claim.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "odd-prime-reduced-powers-and-wu-classes.md", "source_sha256": "5f9ac97cae085d1ffc5e8e93401196a702efb2eb09ef936e7f2af0fddf61a9e7", "proof_location": { "declared_locus": "SectionJ.3 CorollaryJ.5 complete proof, with J.4 and all six exercisesK", "retained_anchor": "j-odd-prime-thom-classes-and-wu-s-formula", "reader": "DG-CHAR-14B.html#j-odd-prime-thom-classes-and-wu-s-formula" }, "consumers": [ "DG-CHAR-14", "DG-CHAR-15", "GT-SURG", "GT-TOPM" ], "independent_review": false }, { "id": "DG-CHAR.PL.simplicial-local-product", "aliases": [ "DG-CHAR.PL.simplicial-local-product" ], "statement": "A simplicial map is a support-preserving product over an open target simplex; generic fibres of finite rational homology manifolds are finite rational homology manifolds.", "conditions": "Locally finite source and target for the product; compact rational n-manifold and standard triangulated k-sphere for generic fibre; rational coefficients; empty fibres allowed; no closed-simplex product claim.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionA, LemmasA.1–A.2 and PropositionA.3 complete proofs", "retained_anchor": "a-generic-simplicial-fibres-and-their-local-homology", "reader": "DG-CHAR-15.html#a-generic-simplicial-fibres-and-their-local-homology" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.signature.all-L-leading-coefficients", "aliases": [ "DG-CHAR.signature.all-L-leading-coefficients" ], "statement": "The coefficient c_j of p_j in every rational L_j is strictly positive; all rational Pontryagin variables are uniquely recovered recursively from the L-classes.", "conditions": "Formal rational multiplicative sequence sqrt(t)/tanh(sqrt(t)); every j>=1; uses actual Riccati, log and integral Newton proofs, with no Bernoulli sign premise.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionB, LemmaB.1 and formulasB.1–B.3 complete proof", "retained_anchor": "b-recovering-pontryagin-classes-from-the-l-classes", "reader": "DG-CHAR-15.html#b-recovering-pontryagin-classes-from-the-l-classes" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.cohomotopy.stable-range-group", "aliases": [ "DG-CHAR.cohomotopy.stable-range-group" ], "statement": "Unbased sphere homotopy classes form a natural abelian cohomotopy group; target degree-d maps act as multiplication by d in this range.", "conditions": "Finite CW X of dimension n; k>=2 and n<2k-1; finite disconnected sources allowed. Wedge-product uniqueness includes endpoint-fixed homotopy compression.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionC.1, LemmasC.1–C.2 and full fold/group/inverse proof", "retained_anchor": "c-sphere-maps-detect-the-rational-cohomology-in-the-required-range", "reader": "DG-CHAR-15.html#c-sphere-maps-detect-the-rational-cohomology-in-the-required-range" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.cohomotopy.rational-sphere-comparison", "aliases": [ "DG-CHAR.cohomotopy.rational-sphere-comparison" ], "statement": "Pullback of the sphere generator identifies cohomotopy tensored with Q with rational cohomology; every integer homomorphism extends to a unique rational cohomology functional.", "conditions": "Finite CW X, k>=2 and dim(X)<2k-1. Factorial degree telescope, based spine, higher cell killing and one-group cellular classification proved. Kernel torsion; no finite-kernel assertion.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionsC.2–C.3, TheoremC.3 and CorollaryC.4 complete proofs", "retained_anchor": "c-sphere-maps-detect-the-rational-cohomology-in-the-required-range", "reader": "DG-CHAR-15.html#c-sphere-maps-detect-the-rational-cohomology-in-the-required-range" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.homology-manifolds.rational-duality", "aliases": [ "DG-CHAR.homology-manifolds.rational-duality" ], "statement": "Right cap with the coherent fundamental cycle gives closed Poincare and both relative and absolute Lefschetz isomorphisms, with a perfect rational relative-absolute pairing.", "conditions": "Finite rational homology n-manifold pair(K,M): sphere links for interior simplices, acyclic K-links at boundary simplices, M a closed rational(n-1)-manifold; coherent signed top cycle as(D.2); no smooth collar assumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionsD.1–D.3, LemmasD.1–D.4 and TheoremD.5, including full flag/star/cap comparison", "retained_anchor": "d-duality-for-finite-rational-homology-manifolds", "reader": "DG-CHAR-15.html#d-duality-for-finite-rational-homology-manifolds" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.homology-manifolds.boundary-signature", "aliases": [ "DG-CHAR.homology-manifolds.boundary-signature" ], "statement": "The middle restriction image in a rational homology boundary equals its orthogonal complement, so every such oriented4i-dimensional boundary has signature zero.", "conditions": "Finite oriented rational homology(4i+1)-manifold pair with exact D.1 boundary conditions and induced fundamental boundary(D.3); i>=0; Q coefficients.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionD.4, TheoremD.6 and full connecting-map adjoint proof", "retained_anchor": "d-duality-for-finite-rational-homology-manifolds", "reader": "DG-CHAR-15.html#d-duality-for-finite-rational-homology-manifolds" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.relative-approximation", "aliases": [ "DG-CHAR.PL.relative-approximation" ], "statement": "Finite PL triangulations admit common refinements, a single PL map can be made simplicial, and continuous maps PL on a subpolyhedron have relative PL approximations; homotopic PL endpoints have an exact-endpoint PL homotopy.", "conditions": "Finite polyhedra; subpolyhedron A; fixed actual map on A. Prism elementary collapses and PL retractions supplied. No arbitrary simultaneous-diagram simplicial claim or smooth triangulation existence claim.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionsE.1–E.2, LemmasE.1–E.3 and complete prism/collapse/relative homotopy proofs", "retained_anchor": "e-finite-pl-refinements-and-relative-approximation", "reader": "DG-CHAR-15.html#e-finite-pl-refinements-and-relative-approximation" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.oriented-generic-fibres", "aliases": [ "DG-CHAR.PL.oriented-generic-fibres" ], "statement": "Generic fibres have induced coherent rational orientations; a PL homotopy with I first has a rational homology fibre pair bounded by the final fibre minus the initial fibre.", "conditions": "Finite closed oriented rational homology n-manifold; PL map to positive standard PL k-sphere; d=n-k>=0; support-preserving target product. Fibre first, target last.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionF.1, LemmaF.1, formulasF.1–F.3 and full boundary proof", "retained_anchor": "f-fibre-signatures-and-the-stable-range-l-class", "reader": "DG-CHAR-15.html#f-fibre-signatures-and-the-stable-range-l-class" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.fibre-signature-invariance", "aliases": [ "DG-CHAR.PL.fibre-signature-invariance" ], "statement": "Generic PL fibre signature is independent of target simplex, triangulation and homotopy representative; in cohomotopy range it is an integer homomorphism.", "conditions": "Finite closed oriented rational homology n-manifold; target standard PL k-sphere, k>=1 and n-k=4i>=0. Homomorphism additionally k>=2, n<2k-1.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionsF.2–F.3, TheoremF.2 and PropositionF.3, using E.4 point movement and D.6 boundary vanishing", "retained_anchor": "f-fibre-signatures-and-the-stable-range-l-class", "reader": "DG-CHAR-15.html#f-fibre-signatures-and-the-stable-range-l-class" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.stable-range-L-class", "aliases": [ "DG-CHAR.PL.stable-range-L-class" ], "statement": "There is a unique rational class ell_i whose sphere-pullback pairings equal generic fibre signatures; it is preserved by PL homeomorphisms, independently of component orientation choices.", "conditions": "Finite closed oriented rational homology n-manifold;4i<(n-1)/2. Class is L-class, not individual p_i. No smooth comparison adopted in this version.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionF.3, TheoremF.4 and formulaF.7 full proof", "retained_anchor": "f-fibre-signatures-and-the-stable-range-l-class", "reader": "DG-CHAR-15.html#f-fibre-signatures-and-the-stable-range-l-class" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.generic-fibre-cap", "aliases": [ "DG-CHAR.PL.generic-fibre-cap" ], "statement": "The included generic fibre fundamental class is exactly right cap of the ambient class with the pulled-back positive sphere generator, and gives its cup-evaluation identity.", "conditions": "Finite closed oriented rational homology n-manifold; PL map to standard positive k-sphere,k>=1,d=n-k>=0; empty fibres allowed; fibre-first target-last orientation.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionF.4, LemmaF.5 and formulasF.8–F.9, including normalized shuffle sign calculation", "retained_anchor": "f-fibre-signatures-and-the-stable-range-l-class", "reader": "DG-CHAR-15.html#f-fibre-signatures-and-the-stable-range-l-class" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.stabilized-L-classes", "aliases": [ "DG-CHAR.PL.stabilized-L-classes" ], "statement": "Products with large spheres define canonical rational L-classes in every degree, independent of sphere dimension, slice and PL triangulation; PL invariance, unit, vanishing, top signature normalization and unrestricted positive-target fibre formula.", "conditions": "Finite closed oriented rational homology n-manifold; all i>=0; choose m>4i and n+m>8i+1. Proved localized degree-one map and double-sphere comparison; formal Pontryagin recovery, with smooth identification still pending.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "combinatorial-l-classes-and-piecewise-linear-fibres.md", "source_sha256": "553ba1eeaa51439ab96bf06cdfe5fa2c2b68dffdb57827ca60e2ec15e3ad54c3", "proof_location": { "declared_locus": "SectionG, LemmasG.1–G.2, TheoremG.3 and CorollaryG.4 full proofs", "retained_anchor": "g-stabilization-beyond-the-cohomotopy-range", "reader": "DG-CHAR-15.html#g-stabilization-beyond-the-cohomotopy-range" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.smooth.regular-fibre-product", "aliases": [ "DG-CHAR.smooth.regular-fibre-product" ], "statement": "Regular values form a dense open set, their compact inverse images have canonical trivial normal quotients, and a small regular-value ball has a smooth fibre-first product trivialization.", "conditions": "Smooth closed M and smooth map to a positive-dimensional sphere. Empty fibres allowed. Gram right inverse and compact-band flow continuation actually proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionI.1, LemmasI.1–I.2 full proofs", "retained_anchor": "i-smooth-regular-fibres-and-their-signature", "reader": "DG-CHAR-15B.html#i-smooth-regular-fibres-and-their-signature" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.smooth.regular-fibre-signature", "aliases": [ "DG-CHAR.smooth.regular-fibre-signature" ], "statement": "The signature of each regular fibre equals the ambient L_i cup pulled-back positive sphere generator evaluated on the fundamental class; independence of regular value and homotopy.", "conditions": "Smooth closed oriented n-manifold, target S^k with k>=1 and n-k=4i>=0. Fibre first, normal/target last; Q L-classes, integral sphere generator. Empty fibres treated.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionI.2, TheoremI.3, formulasI.6–I.9 full proofs", "retained_anchor": "i-smooth-regular-fibres-and-their-signature", "reader": "DG-CHAR-15B.html#i-smooth-regular-fibres-and-their-signature" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.sphere-annular-retraction", "aliases": [ "DG-CHAR.PL.sphere-annular-retraction" ], "statement": "The straight annulus around a simplex boundary has an explicit finite PL retraction fixing that boundary, with a uniform displacement bound; ambient PL approximations become sphere-valued and homotopic.", "conditions": "Standard polyhedral k-sphere Q boundary of simplex centred at origin, 0=1, fibre dimension nonnegative and divisible by four for signature. Does not prove existence of t.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionJ.4, TheoremJ.5 complete local/global interpolation and oriented normal-graph proof", "retained_anchor": "j-comparison-for-a-given-smooth-compatible-triangulation", "reader": "DG-CHAR-15B.html#j-comparison-for-a-given-smooth-compatible-triangulation" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.smooth-L-Pontryagin-comparison", "aliases": [ "DG-CHAR.PL.smooth-L-Pontryagin-comparison" ], "statement": "For every given finite smooth-compatible triangulation, combinatorial L-classes equal pulled-back smooth Hirzebruch L-classes, and recovered rational Pontryagin classes equal the smooth rational ones in every degree.", "conditions": "Smooth closed oriented manifold with a given finite smooth-compatible triangulation. Strict-range cohomotopy comparison followed by explicit product triangulation and stabilization. General triangulation existence and integral extension not asserted.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionJ.5, TheoremJ.6 and formulasJ.10–J.12 full proof", "retained_anchor": "j-comparison-for-a-given-smooth-compatible-triangulation", "reader": "DG-CHAR-15B.html#j-comparison-for-a-given-smooth-compatible-triangulation" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.PL.orientation-cover-descent", "aliases": [ "DG-CHAR.PL.orientation-cover-descent" ], "statement": "Rational combinatorial classes descend uniquely from the orientation cover of a given smooth-compatible triangulated closed manifold, retain smooth comparison and are PL natural without an orientation premise.", "conditions": "Given finite smooth-compatible triangulation of a smooth closed manifold. Canonical local-top-homology orientation cover, lifted simplices, orientation independence and rational two-sheet transfer; dimension zero separately trivial.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionJ.5, CorollaryJ.7 full proof", "retained_anchor": "j-comparison-for-a-given-smooth-compatible-triangulation", "reader": "DG-CHAR-15B.html#j-comparison-for-a-given-smooth-compatible-triangulation" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.lens.rational-cohomology", "aliases": [ "DG-CHAR.lens.rational-cohomology" ], "statement": "A free cyclic weighted sphere quotient is a smooth closed oriented lens space; every-sheet singular transfer identifies its rational cohomology with sphere deck invariants.", "conditions": "r>=2,d>=1, all weights coprime to r; quotient of S^(2d-1) in C^d. Rational groups only in degree0 and odd top degree.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionK.1, PropositionK.1 and transferK.2 full proof", "retained_anchor": "k-lens-spaces-and-the-rational-integral-distinction", "reader": "DG-CHAR-15B.html#k-lens-spaces-and-the-rational-integral-distinction" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.lens.tangent-Pontryagin-formula", "aliases": [ "DG-CHAR.lens.tangent-Pontryagin-formula" ], "statement": "The lens tangent bundle plus a radial trivial line is the real bundle of the sum of character complex lines; its exact integral Pontryagin class is product(1+q_j^2*t^2).", "conditions": "Weighted lens quotient asK.1; eta quotient character zeta, t=c1(eta), rt=0. Exact underlying-complex identity, no general two-torsion Whitney qualification; degree truncation retained.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionK.1, PropositionK.2, stable splitting and formulaK.4 full proof", "retained_anchor": "k-lens-spaces-and-the-rational-integral-distinction", "reader": "DG-CHAR-15B.html#k-lens-spaces-and-the-rational-integral-distinction" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.lens.equal-weight-integral-torsion", "aliases": [ "DG-CHAR.lens.equal-weight-integral-torsion" ], "statement": "Equal-weight lens spaces have even groups Z/r generated by pulled-back projective powers; the character line is pulled-back tautological dual. L5_5 has nonzero integral p1=3*t^2 and zero rational image.", "conditions": "Equal weights,d>=2,1<=j<=d-1; Hopf quotient identified with unit gamma tensor-r, Euler=-r*u, integral Gysin proof. This is torsion loss, not failed integral PL invariance.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "smooth-fibres-triangulation-comparison-and-lens-spaces.md", "source_sha256": "0f38b1101ea7ed1203ed0e2f26c15d6a4a239aa534d06b4b147c4de53eb11335", "proof_location": { "declared_locus": "SectionK.2, PropositionK.3 and formulasK.6–K.8 full proof", "retained_anchor": "k-lens-spaces-and-the-rational-integral-distinction", "reader": "DG-CHAR-15B.html#k-lens-spaces-and-the-rational-integral-distinction" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.quaternionic-line", "aliases": [ "DG-CHAR.exotic.quaternionic-line" ], "statement": "The underlying right-i complex quaternionic line has unit sphere total space diffeomorphic to S7, Euler generator u and integral p1=-2u.", "conditions": "Base S4 oriented by its primitive Euler class; fibre complex orientation for right multiplication by i. Explicit two-chart sphere diffeomorphism and Gysin/Chern normalization.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "Lemma A.1, formulasA.1–A.3 complete proof", "retained_anchor": "a-1-the-quaternionic-line-and-its-integral-normalization", "reader": "DG-CHAR-16.html#a-1-the-quaternionic-line-and-its-integral-normalization" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.clutch-additivity", "aliases": [ "DG-CHAR.exotic.clutch-additivity" ], "statement": "Degree-four oriented characteristic numbers are additive under pointwise clutch-map multiplication and negate under inversion.", "conditions": "Based continuous g:S3→SO4; c in integral H4BSO4. Explicit normalized projection plane model, reduced suspension degree, pinch chain and based product/concatenation homotopy.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "Lemma A.2, formulasA.4–A.6 complete proof", "retained_anchor": "a-2-additivity-for-clutching-with-an-explicit-classifying-map", "reader": "DG-CHAR-16.html#a-2-additivity-for-clutching-with-an-explicit-classifying-map" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.prescribed-four-plane-classes", "aliases": [ "DG-CHAR.exotic.prescribed-four-plane-classes" ], "statement": "The smooth bundle xi_hj has e=(h+j)u and p1=2(j-h)u; every integer pair (e,k) with k congruent2e modulo4 is constructed explicitly.", "conditions": "Single normalization of A.1; h,j arbitrary integers; fibre orientation and base reflection signs retained.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "SectionA.3, formulasA.7–A.10 and TheoremA.3 full proof", "retained_anchor": "a-3-the-right-factor-and-all-prescribed-pairs", "reader": "DG-CHAR-16.html#a-3-the-right-factor-and-all-prescribed-pairs" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.four-plane-classification", "aliases": [ "DG-CHAR.exotic.four-plane-classification" ], "statement": "Every smooth oriented real four-plane bundle on S4 is smoothly isomorphic to a unique xi_hj; its e and p1 determine it and satisfy the exact parity congruence.", "conditions": "Oriented sphere/base generator of A.1. Full hemisphere clutch classification, quaternionic two-sheet cover, pi3 degree computation and orientation-preserving smooth approximation.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "PropositionA.4 complete proof, formulaA.11", "retained_anchor": "a-4-why-this-family-includes-every-oriented-four-plane-bundle", "reader": "DG-CHAR-16.html#a-4-why-this-family-includes-every-oriented-four-plane-bundle" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.disk-tangent-relative-square", "aliases": [ "DG-CHAR.exotic.disk-tangent-relative-square" ], "statement": "The disk bundle tangent is pi*(TS4+xi); with Euler number1 its relative form is (1), signature1 and relative first-Pontryagin square k^2.", "conditions": "W=D(xi_hj), e=1, p1=k*u; base first/fibre last ambient orientation, outward-first boundary. Full disk/zero-pair Thom comparison and positive local evaluation.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "SectionB, formulasB.1–B.5 complete proofs", "retained_anchor": "b-the-disk-bundle-and-its-relative-square", "reader": "DG-CHAR-16.html#b-the-disk-bundle-and-its-relative-square" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.extremum-normal-form", "aliases": [ "DG-CHAR.exotic.extremum-normal-form" ], "statement": "A nondegenerate minimum or maximum has a smooth squared-norm normal form with the corresponding sign.", "conditions": "Smooth real function near an extremum on an n-manifold. Positive Hessian, integral Taylor matrix, smooth Cholesky and proved inverse-function theorem.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "LemmaC.1 complete proof, formulaC.1", "retained_anchor": "c-1-the-local-normal-form-at-an-extremum", "reader": "DG-CHAR-16.html#c-1-the-local-normal-form-at-an-extremum" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.two-critical-point-sphere", "aliases": [ "DG-CHAR.exotic.two-critical-point-sphere" ], "statement": "A closed connected n-manifold with exactly two nondegenerate critical points is homeomorphic to Sn, diffeomorphic off one point and obtained by gluing two smooth disks.", "conditions": "n>=1; actual compact-band flow inverse and uniform collapsed-boundary continuity supplied. No boundary gluing extension over a disk presumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "TheoremC.2 complete proof, formulasC.2–C.4", "retained_anchor": "c-2-the-sphere-criterion-including-the-missing-endpoint", "reader": "DG-CHAR-16.html#c-2-the-sphere-criterion-including-the-missing-endpoint" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.quaternionic-sphere-function", "aliases": [ "DG-CHAR.exotic.quaternionic-sphere-function" ], "statement": "For h+j=1 the global quaternionic sphere-bundle function has exactly two critical points with Hessians plus/minus I7; Euler-number minus one follows by base reflection.", "conditions": "Explicit xi_hj transition q^h*v*q^j; second-chart a=uprime*vprime^-1, strictly positive real derivative. Sphere homeomorphism uses the proved C.2 criterion.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "SectionC.3 complete chart/derivative/Hessian proof, formulasC.5–C.13", "retained_anchor": "c-3-quaternionic-clutching-and-the-global-function", "reader": "DG-CHAR-16.html#c-3-quaternionic-clutching-and-the-global-function" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.relative-pairing", "aliases": [ "DG-CHAR.exotic.relative-pairing" ], "statement": "A filling of M7 with integral H3=H4=0 has a unique degree-four relative lift and a nondegenerate rational symmetric relative intersection form.", "conditions": "Given compact oriented smooth W8 with outward boundary M; integer vanishing, double/MV/cup extension, finite generation and rational coefficient comparison all proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "SectionD.1 complete proof, formulasD.1–D.5", "retained_anchor": "d-1-the-relative-square-and-its-nondegenerate-pairing", "reader": "DG-CHAR-16.html#d-1-the-relative-square-and-its-nondegenerate-pairing" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.filling-gluing", "aliases": [ "DG-CHAR.exotic.filling-gluing" ], "statement": "Gluing W0 to -W1 across this boundary subtracts signatures and relative p1-square evaluations.", "conditions": "Two given smooth oriented fillings of the same M7 with integer H3=H4=0; smooth collar tangent restrictions and orthogonal relative extensions, no arbitrary-boundary additivity theorem assumed.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "LemmaD.1 complete proof, formulasD.6–D.7", "retained_anchor": "d-2-gluing-two-different-fillings", "reader": "DG-CHAR-16.html#d-2-gluing-two-different-fillings" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.mod-seven-invariant", "aliases": [ "DG-CHAR.exotic.mod-seven-invariant" ], "statement": "The residue lambda=2q(W)-sigma(W) modulo7 is filling independent, diffeomorphism invariant with orientation preserved, and negates under orientation reversal; standard S7 has value0.", "conditions": "Closed oriented M7 with integer H3=H4=0 and a given smooth compact filling W8. Uses proved closed eight-dimensional signature polynomial; no general bounding theorem adopted.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "TheoremD.2 complete proof, formulasD.8–D.10", "retained_anchor": "d-3-the-congruence-and-the-exotic-sphere", "reader": "DG-CHAR-16.html#d-3-the-congruence-and-the-exotic-sphere" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.exotic.exotic-seven-sphere", "aliases": [ "DG-CHAR.exotic.exotic-seven-sphere" ], "statement": "S(xi_-1,2) is homeomorphic to S7 and has lambda1, hence admits no diffeomorphism to the standard sphere with either orientation behaviour.", "conditions": "Boundary orientation of explicit disk bundle; Euler1,p1=6u,q=36,signature1. Full two-critical-point topology and filling invariant proofs used; no PL assertion.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "milnors-exotic-seven-spheres.md", "source_sha256": "0d445478337f16040f81b95ca2ef4ce8195ff7005922c26087d597a5f3fe48ef", "proof_location": { "declared_locus": "TheoremD.3 complete proof, formulaD.11; ExerciseD.5 full solution", "retained_anchor": "d-3-the-congruence-and-the-exotic-sphere", "reader": "DG-CHAR-16.html#d-3-the-congruence-and-the-exotic-sphere" }, "consumers": [ "DG-CHAR-15", "DG-CHAR-16", "GT-SURG" ], "independent_review": false }, { "id": "DG-CHAR.cw.de-rham-comparison", "aliases": [ "DG-CHAR.cw.de-rham-comparison" ], "statement": "De Rham cohomology identifies naturally and multiplicatively with singular real/complex cohomology, including open-pair cones, smooth-singular comparison and positive orientation/integration evaluation.", "conditions": "Finite-dimensional Hausdorff second-countable smooth manifolds without boundary. Exact strong convexity and all inversion/ODE prerequisites read; explicit locally finite good cover, two double complexes and finite-degree eliminations. No global triangulation premise.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "PartA full proof, formulasA.1–A.16", "retained_anchor": "a-differential-forms-and-the-cohomology-comparison", "reader": "DG-CHAR-17.html#a-differential-forms-and-the-cohomology-comparison" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.invariant-differentiation", "aliases": [ "DG-CHAR.cw.invariant-differentiation" ], "statement": "Invariant polarized contractions satisfy the covariant exterior differentiation rule for arbitrary form degrees; evaluating a homogeneous invariant polynomial on curvature gives a global closed form.", "conditions": "Column-frame convention d+A, F=dA+A wedge A. GL(r,C), or SO(r) with oriented metric frames and compatible connection. Polarization normalized by1/k!.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "LemmaB.1 and closedness in TheoremB.2, formulasB.1–B.7", "retained_anchor": "b-1-evaluate-an-invariant-polynomial-on-forms", "reader": "DG-CHAR-17.html#b-1-evaluate-an-invariant-polynomial-on-forms" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.transgression", "aliases": [ "DG-CHAR.cw.transgression" ], "statement": "For a degree-k invariant polynomial, the global CS primitive k integral_0^1 p(alpha,Ft,...,Ft) dt has derivative P(F1)-P(F0), giving connection independence and smooth pullback naturality.", "conditions": "Two connections on the same bundle; for SO both compatible with the same metric. Affine path; difference transforms by conjugation, dot F=D alpha and Bianchi proved.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "TheoremB.2 full proof, formulasB.8–B.10; ExerciseB.3 general and trace-square solutions", "retained_anchor": "b-1-evaluate-an-invariant-polynomial-on-forms", "reader": "DG-CHAR-17.html#b-1-evaluate-an-invariant-polynomial-on-forms" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.normalized-forms", "aliases": [ "DG-CHAR.cw.normalized-forms" ], "statement": "The forms det(I+iF/2pi), trace exp(iF/2pi), and (-1)^j c2j of a complexified real connection are closed and connection independent; the precise metric/general p1 distinction is proved.", "conditions": "Finite-dimensional base; formal series truncated by form degree. 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Rank zero/odd real Euler scope included.", "conditions": "Smooth oriented metric bundle, compatible connection. Exact open relative pair E,radius>1 and radial homotopy equivalence to deleted-zero pair; positive fibre evaluation. Odd Euler2-torsion proved by-I.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "SectionC.4 full proof, formulasC.18–C.20; ExercisesC.3–C.4 full solutions", "retained_anchor": "c-4-support-and-the-topological-normalization", "reader": "DG-CHAR-17.html#c-4-support-and-the-topological-normalization" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.characteristic-identification", "aliases": [ "DG-CHAR.cw.characteristic-identification" ], "statement": "Normalized curvature determinant and trace exponential represent the earlier integral Chern classes and Chern character after real coefficient change; complexification gives Pontryagin classes.", "conditions": "Every smooth finite-rank bundle on the stated bases; arbitrary connection. Line Euler normalization and smooth complete flag pullback with all-coefficient injectivity proved/read; unitary forms are real.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "TheoremD.1 complete proof, formulasD.1–D.4", "retained_anchor": "d-identify-the-integral-classes-real-images", "reader": "DG-CHAR-17.html#d-identify-the-integral-classes-real-images" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.all-complex-invariants", "aliases": [ "DG-CHAR.cw.all-complex-invariants" ], "statement": "Every complex GL-conjugation-invariant polynomial is a polynomial in elementary coefficients; its curvature class is that polynomial in (-2pi i)^j c_j.", "conditions": "Complex matrices; complete upper-triangular shrinking argument with epsilon^(j-i), owned symmetric-polynomial theorem. Even form entries commute.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "LemmaD.2 full proof, formulasD.5–D.6", "retained_anchor": "d-identify-the-integral-classes-real-images", "reader": "DG-CHAR-17.html#d-identify-the-integral-classes-real-images" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.tautological-line-curvature", "aliases": [ "DG-CHAR.cw.tautological-line-curvature" ], "statement": "The induced Fubini–Study connection on the tautological CP1 line has F=dbar z wedge dz/(1+|z|^2)^2 and Chern number-1 in the complex orientation.", "conditions": "Unit frame(1,z)/sqrt(1+|z|^2), Hermitian first slot antilinear; second-chart unit phase |z|/z.", "licence": "CC0-1.0", "authorship": { "writer": "GPT-6.1 Sol (OpenAI)", "effort": "ultra", "independent_review": false }, "source": "connections-curvature-and-characteristic-forms.md", "source_sha256": "2865cd253340de7bb776c3b730d1f32acaa76c7aebfc73c02302e9a1a5abc116", "proof_location": { "declared_locus": "SectionD.3 and ExerciseD.3 full computation, formulasD.7–D.9", "retained_anchor": "d-3-the-tautological-line-on-mathbb-cp-1", "reader": "DG-CHAR-17.html#d-3-the-tautological-line-on-mathbb-cp-1" }, "consumers": [ "DG-CHAR-17", "DG-FND" ], "independent_review": false }, { "id": "DG-CHAR.cw.gauss-bonnet", "aliases": [ "DG-CHAR.cw.gauss-bonnet" ], "statement": "For a closed oriented Riemannian2m-manifold, the integral of Pf(F/2pi) for Levi-Civita curvature is the Euler characteristic.", "conditions": "Positive metric; actual compatible Levi-Civita provider proof read. Proved metric Euler form, de Rham oriented integration and full owned signed diagonal/Euler-number theorem. 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