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.

Interval sewing with the finite gauge-volume factor

Physical scope

Exact boundary-subcomplex sewing calculation in the adopted finite G = Z/2 model with trivial A; no continuum or empirical claim.

Example

Let $K$ be the ordered path with vertices $0 < 1 < 2$ and edges $01, 12$ , split into $K_{1} = [0, 1]$ and $K_{2} = [1, 2]$ with common subcomplex $Σ = {1}$ , inside the relative state-sum convention for the finite model with $G = Z / 2$ and $A = {0}$ . The increasing vertex order orients the edges, and the two induced boundary orientations at the shared vertex cancel. Each edge has background amplitude $1 / 2$ ; its relative amplitude at a fixed shared-point background is $1$ ; the shared point has gauge-volume weight $1 / 2$ . The sewing contraction gives $1 / 2$ , equal to the amplitude of the full path.

The diagram labels the ordered edge subcomplexes and their common vertex. The exact finite counts below use the stated simplicial data and do not rely on the picture.

Example references

finite model and boundary conventions

finite background gauge-volume and sewing theorem

Facts & Assumptions

Given: the two-edge path, its split at the shared vertex, the finite $G = Z / 2$ model, and unit background response.

[F1]

Finite two-group lattice spin state-sum model: the background fields are flat $G$ edge cochains, $A$ is trivial, and the spin partition weight is $1$ for every background.

[F2]

Gauge invariance and gluing in the finite symmetry model: $c (J) = 2^{- V_{J}}$ converts the higher sum to the background-object sum; relative amplitudes divide by $c (Σ)$ and glue with the seam factor $∣ π_{2} ∣ / ∣ π_{1} ∣$ .

[A1]

Adopted finite two-group gauging model: the unit-response finite spin model and its relative state-sum convention are adopted.

Verification

Given: the ordered path $K$ , its two one-edge pieces, and $Σ = {1}$ .

1.1givenA1F1F2algebra

In the adopted model, for one edge $(V, E) = (2, 1)$ , so $c = 1 / 4$ , and its two possible flat edge backgrounds give amplitude $2 (1 / 4) = 1 / 2$ . For the full path $(V, E) = (3, 2)$ , so $c = 1 / 8$ , and its four independent edge backgrounds give amplitude $4 (1 / 8) = 1 / 2$ . The normalized spin factor is $1$ for each background.

2.1givenA1F1F2step 1.1algebra

For the shared point, $c (Σ) = 1 / 2$ and there is one background. A fixed boundary background on an edge has two edge-label extensions, while $c (K_{i}) / c (Σ) = (1 / 4) / (1 / 2) = 1 / 2$ ; each relative edge amplitude is therefore $1$ .

3.1givenA1F1F2step 1.1step 2.1algebra∎

The unique seam component has $∣ π_{2} ∣ / ∣ π_{1} ∣ = 1 / 2$ . The contraction is $1 \cdot 1 \cdot (1 / 2) = 1 / 2$ , agreeing with the direct full-path count in step 1.1 and retaining the gauge automorphism factor.

Interval sewing with the finite gauge-volume factor

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels