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 homotopy-pullback gluing factor for the two-element group

Example

Let $G = C_{2} = {1, s}$ with $s^{2} = 1$ , and let $B G$ be the category with one object $*$ , morphism set $Hom (*, *) = G$ , and composition given by group multiplication. Let both restriction functors $B G \to B G$ be identity functors. Give both pieces and the glued groupoid the constant weight $1$ , and define the glued groupoid to be the homotopy pullback $B G \times_{B G}^{h} B G$ . Its normalized gauging sum is $1 / 2$ . Each homotopy fiber of an identity restriction over the unique boundary object has normalized integral $1$ , so the gluing formula gives the same value.

Example references

Finite background groupoid and cocycle-weighted gauging data

Finite cocycle-weighted groupoid gauging and gluing

Facts & Assumptions

Given: The group $C_{2}$ , the groupoid $B G$ with one object and morphism group $C_{2}$ , two identity restriction functors, and the constant action weights.

[F1]

Finite background groupoid and cocycle-weighted gauging data: a finite groupoid's normalized sum weights each isomorphism class by the reciprocal automorphism-group order.

[F2]

Finite cocycle-weighted groupoid gauging and gluing: for composable finite groupoid data with a homotopy-pullback comparison equivalence and a factorizing invariant weight, the glued sum is the boundary sum divided by the boundary automorphism orders.

[F3]

Homotopy pullback of finite groupoids: a homotopy-pullback object is a pair of source objects together with an isomorphism between their images; a morphism is a compatible pair of arrows.

Proof

1.1givenF3algebra

The objects of $B G \times_{B G}^{h} B G$ are the two elements $α \in C_{2}$ . A morphism $(u, v)$ from $α$ to $α^{'}$ satisfies $v α = α^{'} u$ . For each of the two choices of $u$ , there is exactly one $v$ satisfying this equation, so the groupoid is connected and each object's automorphism group has order two.

2.1givenF1F3step 1.1algebra

Its groupoid cardinality is therefore $1 / 2$ . For the homotopy fiber of an identity functor $B G \to B G$ over its unique object, the objects are the two isomorphisms $η \in C_{2}$ ; between any ordered pair $η, η^{'}$ there is exactly one compatible arrow, so this fiber is equivalent to the terminal groupoid and has cardinality one.

3.1givenF1F2step 2.1algebra∎

The comparison functor is the identity onto the declared glued homotopy pullback, and the constant weight factors. By [F2], $Z = (1 \cdot 1) / ∣ {Aut}_{B G} (*) ∣ = 1 / 2$ , in agreement with step 2.1.

The homotopy-pullback gluing factor for the two-element group

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels