ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

A point retains the second gauge stabilizer

Example

Let $K$ consist of one vertex. Take $G = Z_{2}$ , $A = Z_{3}$ , trivial action and $β = 0$ , with unit phase weight. There is one background object, two distinct raw $1$ -automorphisms and three $2$ -automorphisms of each. The higher gauging sum is $3 / 2$ . Identifying arrows by their effects on the singleton object set would give $1$ ; retaining $G$ but discarding $2$ -arrows would give $1 / 2$ .

Example references

Finite 2-group homotopy gauging and cochain gluing

Facts & Assumptions

Given: the one-vertex complex and finite data of the Example.

[F1]

Finite 2-group homotopy gauging and cochain gluing gives point mass $∣ A ∣ / ∣ G ∣$ and the full gauge normalization.

Verification

1.1givenF1algebra

There are no edges or faces, so the only background is empty. A raw $1$ -arrow has one freely chosen $G$ -label and no edge labels, giving two arrows. A $2$ -arrow preserves that label and is any of the three vertex $A$ -labels, since its coboundary has no edges. Thus $π_{1} = Z_{2}$ and $π_{2} = Z_{3}$ .

2.1givenF1algebra∎

The higher mass is $3 / 2$ by [F1]. The quotient by induced object functions has only the identity, of mass $1$ ; the ordinary groupoid with only the two raw $G$ -arrows has automorphism order $2$ , of mass $1 / 2$ . These distinct values demonstrate the retained higher stabilizer.

Depends on

Finite 2-group homotopy gauging and cochain gluing