Alphabeta Math
RemarkRemark: Literature-sourcedProof: Not applicablePipeline-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.

The cap-product order is fixed by the AT convention, not minted here

Remark

This page mints no independent cap-product or cup-product sign convention. The cap product is the cohomology-first, front-evaluation operation of Cap product with cohomology written first with its relative form Relative cap products with quotient domains displayed, and The geometric intersection number is the Poincare-dual cup pairing is stated in exactly that convention. The geometric factor order (first factor A, second factor B, as in The local oriented intersection sign) is authoritative: if a source writes the dual pairing with the opposite valuation order, the identity must be read through the sign of Intersection number under factor interchange, I(B,A)=(−1)abI(A,B). No third convention is to be introduced at authoring time; a mismatch is a sign error in the translation, not a choice.

Depends on

Used by

Dependency tree · two levels

48 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources