Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Well-definedness of the Mayer–Vietoris connector

Statement

The class [∂u] defining the Mayer–Vietoris connector is independent of the small-chain decomposition and of the chosen cycle representative.

Facts & Assumptions

Given: Two decompositions z=u+v=u′+v′ of a cover-small cycle.

Proof

technique · direct
1.1givenalgebra

Exactness of the chain sequence gives u−u′=w and v−v′=−w for some w∈Cn(U∩V;G).

2.1step 1.1algebra∎

Hence ∂u−∂u′=∂w, so the two overlap cycles give the same homology class. Changing z by a boundary yields the same calculation one degree higher.

Depends on

Used by

Cited to discharge well-definedness by Mayer–Vietoris connecting class.

Dependency tree · two levels

4 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