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.

A finite twisted-local-coefficient background and gauge shift

Example

Let $K$ be the ordered tetrahedron $0 < 1 < 2 < 3$ , let $G = Z_{2}$ under addition, and let $A = Z_{3}$ under addition. Define the action by $ρ_{0} (t) = t$ and $ρ_{1} (t) = - t$ , and take $β = 0$ . From vertex labels $x_{0} = 0, x_{1} = 1, x_{2} = 0, x_{3} = 1$ in $Z_{2}$ , set $a_{i j} = x_{j} - x_{i}$ and $B_{i j k} = 0$ . These data form a finite 2-group background. For the 1-form gauge parameter with $λ_{01} = 1$ and all other edge labels zero, and with all vertex labels $g_{i} = 0$ , the transformed fields satisfy $a^{'} = a$ , $B_{012}^{'} = B_{013}^{'} = 1$ , and $B_{023}^{'} = B_{123}^{'} = 0$ ; they form a gauge-equivalent background.

Facts & Assumptions

[F1]

Finite 2-group background cochains: flat edge labels satisfy $a_{i j} a_{j k} = a_{i k}$ , the twisted tetrahedron equation is $ρ (a_{01}) B_{123} - B_{023} + B_{013} - B_{012} = β (a_{01}, a_{12}, a_{23})$ , and a 1-form parameter changes $B$ by $δ_{a} λ$ .

Verification

Given: the groups, action, cocycle, vertex potential, and gauge parameter specified in the Example.

1.1givenF1algebra

The edge labels are $a_{01} = 1$ , $a_{12} = 1$ , $a_{23} = 1$ , $a_{02} = 0$ , $a_{13} = 0$ , and $a_{03} = 1$ , so each triangle satisfies $a_{i j} + a_{j k} = a_{i k}$ . Since $β = 0$ and $B = 0$ , the tetrahedron equation also holds.

1.2givenF1algebra

With $g_{i} = 0$ , the 0-form correction is zero and $δ_{a} λ$ has values $1$ on $[012]$ and $[013]$ and $0$ on $[023]$ and $[123]$ : on the first two faces its formula is $ρ (a_{01}) λ_{1 k} - λ_{0 k} + λ_{01} = 1$ , and the other two faces contain only zero edge labels.

2.1F1step 1.1step 1.2algebra∎

The transformed tetrahedron equation is $ρ (a_{01}) B_{123}^{'} - B_{023}^{'} + B_{013}^{'} - B_{012}^{'} = 0 - 0 + 1 - 1 = 0 = β (a_{01}, a_{12}, a_{23})$ . Thus the displayed nonzero face cochain is a valid gauge transform of the zero cochain.

Depends on

Finite 2-group background cochains

A finite twisted-local-coefficient background and gauge shift

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 1 level