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.

Split two-group interval gluing with a nontrivial action

Example

Let $G = Z_{2}$ , $A = Z_{3}$ , $ρ_{1} (t) = - t$ , and $β = 0$ . Let $K$ be the ordered interval with vertices $0 < 1 < 2$ and edges $01, 12$ , decomposed into those two edges with intersection the vertex $1$ . Give every field unit weight. Each edge and the full interval have higher gauging sum $3 / 2$ . Each relative edge amplitude at the unique boundary field is $1$ . The gluing contraction has boundary mass $3 / 2$ and gives $3 / 2$ again; discarding that second gauge stabilizer would change the contraction.

Example references

Finite 2-group homotopy gauging and cochain gluing

Facts & Assumptions

Given: the split finite data, interval decomposition and unit weights.

[F1]

Finite 2-group homotopy gauging and cochain gluing gives $c (K) = ∣ A ∣^{V - E} / ∣ G ∣^{V}$ , relative factor $c (K) / c (L)$ , and higher gluing.

Verification

1.1givenF1algebra

A single edge has $V = 2, E = 1$ , hence $c = 3 / 4$ and two edge-label backgrounds, giving $Z = 3 / 2$ . The full two-edge interval has $V = 3, E = 2$ , hence $c = 3 / 8$ and four independently chosen edge-label backgrounds, again giving $Z = 3 / 2$ . There are no faces or tetrahedra imposing further conditions.

2.1givenF1algebra∎

The shared point has $c = 3 / 2$ and one background. Each relative edge has two extensions, with relative normalizer $(3 / 4) / (3 / 2) = 1 / 2$ , so its amplitude is $1$ . The boundary contraction is $(3 / 2) \cdot 1 \cdot 1 = 3 / 2$ , equal to the full sum. The sign action remains in the gauge transformations; the count uses the complete higher formula, which applies to this nontrivial action.

Depends on

Finite 2-group homotopy gauging and cochain gluing