Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-04
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.

FALSE: the cochain and crossed-homomorphism definitions of first cohomology agree automatically

Statement

The inhomogeneous cochain definition and the crossed-homomorphism definition of first cohomology agree automatically, so no separate comparison is needed.

Facts & Assumptions

Given: The two degree-one definitions of first cohomology.

[L1]

The inhomogeneous model uses the explicit differential d1(f)(g,h)=f(g)+g⋅f(h)−f(gh) (First group cohomology via inhomogeneous one-cocycles).

[L2]

Their agreement is a theorem proved by explicit identification (The inhomogeneous one-cocycle model agrees with crossed homomorphisms in degree one).

Refutation

technique · direct
1.1

Fact [L1] shows that the cochain model comes with a specific degree-one differential whose sign and action conventions matter.

givenL1
2.1

Fact [L2] is therefore not vacuous: one must check that the cocycle equation and principal cocycles match the crossed-homomorphism formulas. So the claim that no comparison proof is needed is false.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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