Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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 factor set is independent of the section as a function

Statement

Changing the section leaves the factor set unchanged as a function G×G→M.

Facts & Assumptions

Given: The direct-product extension 0→Z→Z×C2→C2→0 with trivial action.

[L1]

Changing the section changes the factor set by a coboundary (Changing the section changes the factor set by a coboundary).

Refutation

technique · direct
1.1

Let s(1)=(0,1) and s(t)=(0,t). Its factor set is 0. Let s′(1)=(0,1) and s′(t)=(1,t). Then s′(t)2=(2,1), so the new factor set satisfies fs′(t,t)=2.

givenalgebra
2.1

Step 1.1 shows fs′≠fs as functions. This is consistent with [L1], which says only that the two factor sets differ by a coboundary and therefore define the same cohomology class.

L1step 1.1
3.1

Hence the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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