Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

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FALSE: every function G times G to M is a factor set

Statement

Every function f:G×G→M is a valid factor set.

Facts & Assumptions

Given: G=C2={1,t}, the trivial action on M=Z, and the constant function f(g,h)=1.

[F1]

A factor set must satisfy the normalized cocycle equations (Normalized two-cocycle and two-coboundary).

Refutation

technique · direct
1.1

The function f is not normalized, since f(1,1)=1≠0. It already fails the normalization part of [F1].

F1given
2.1

Even ignoring normalization, the cocycle equation at (t,t,t) would read 1−1+1−1=0, but the normalization failure from step 1.1 already prevents f from being a factor set.

step 1.1algebra
3.1

Therefore the statement is false.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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