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.

The normalized sum on a one-object finite groupoid

Example

Let $G = C_{2} = {1, s}$ with $s^{2} = 1$ . Let $B G$ be the category with one object $*$ , with morphisms $Hom (*, *) = G$ and composition given by multiplication in $G$ . Give its unique object the weight $W (*) = 1$ . The normalized gauging sum of this finite background groupoid is $1 / 2$ .

Example references

Finite background groupoid and cocycle-weighted gauging data

Facts & Assumptions

Given: The one-object category $B G$ and the constant weight $W (*) = 1$ .

[F1]

Finite background groupoid and cocycle-weighted gauging data: the normalized gauging sum is the sum over isomorphism classes of the weight divided by the order of the object's automorphism group.

Proof

1.1givenF1construct

The unique object of $B G$ has automorphism group $G$ , which has order two, and there is exactly one isomorphism class.

2.1givenF1step 1.1algebra∎

Substituting $W (*) = 1$ and $∣ {Aut}_{B G} (*) ∣ = 2$ into the normalized sum gives $Z (B G, 1) = 1 / 2$ .

Depends on

Finite background groupoid and cocycle-weighted gauging data