ExampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Two Milnor spheres with distinct congruence invariants

Example

Assume the Axiom of Choice and countable choice. For $(h, j) = (1, 0)$ we have $h + j = 1$ and $k = h - j = 1$ , so $λ (M_{1, 0}) = (h - j)^{2} - 1 = 1 - 1 = 0 (m o d 7),$ while for $(h, j) = (2, - 1)$ we have $h + j = 1$ and $k = h - j = 3$ , so $λ (M_{2, - 1}) = (h - j)^{2} - 1 = 9 - 1 = 8 \equiv 1 (m o d 7) .$ The displayed values follow from The Milnor sphere $M_{2, - 1}$ is homeomorphic but not diffeomorphic to $S^{7}$ . Both manifolds are homeomorphic to $S^{7}$ by The Milnor homotopy seven-spheres are homeomorphic to $S^{7}$ , and the invariant is preserved under orientation-preserving diffeomorphism and negated under orientation reversal by The Milnor lambda invariant is well defined modulo seven, so no diffeomorphism between them exists in either orientation. Thus $M_{1, 0}$ and $M_{2, - 1}$ are two closed smooth seven-manifolds with distinct congruence invariants.

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Sources

John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956), 399-405 (standard reference, not scraped)