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.

Point normalization in the finite symmetry model

Physical scope

One-vertex boundary evaluation in the adopted finite G = Z/2 model with trivial A and normalized spin measure; no continuum or empirical claim.

Example

Let $K$ consist of one vertex and no higher-dimensional simplices in the relative state-sum convention for the finite model with $G = Z / 2$ and $A = {0}$ . There is one background object, its first automorphism group has order $2$ , and its second automorphism group is trivial. The normalized spin partition weight for the unique background is $1$ , and the normalized background-gauging amplitude is $1 / 2$ .

Example references

selected finite model

finite point formula

Facts & Assumptions

Given: the one-vertex boundary complex and the adopted normalized spin model.

[F1]

Finite two-group lattice spin state-sum model: spin configurations are vertex labels in $Z / 2$ , each of weight $2^{- V}$ ; the background gauge group has $G = Z / 2$ and trivial $A$ .

[F2]

Gauge invariance and gluing in the finite symmetry model: the background amplitude is the higher sum with component mass $∣ π_{2} ∣ / ∣ π_{1} ∣$ .

[A1]

Adopted finite two-group gauging model: the finite spin model and unit background response are adopted assumptions.

Verification

Given: one vertex, no edges, and no face labels.

1.1givenF1A1algebra

There is one flat background object. Its raw vertex gauge labels are $G = Z / 2$ and its $2$ -gauge group is $A = {0}$ , so $∣ π_{1} ∣ = 2$ and $∣ π_{2} ∣ = 1$ . The two spin assignments each have weight $1 / 2$ , hence their total per-background partition weight is $1$ .

2.1givenA1F2step 1.1algebra∎

Under the adopted model the sole background component contributes $1 \cdot ∣ π_{2} ∣ / ∣ π_{1} ∣ = 1 / 2$ to the finite gauging sum.

Point normalization in the finite symmetry model

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels