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 nonzero finite Postnikov cocycle background

Example

Let $G = A = Z_{2}$ under addition with trivial action, and let $β (g, h, k) = g h k$ in $Z_{2}$ . On the ordered tetrahedron $K = Δ^{3}$ with vertices $0 < 1 < 2 < 3$ , take vertex labels $x_{0} = 0, x_{1} = 1, x_{2} = 0, x_{3} = 1$ and edges $a_{i j} = x_{j} - x_{i}$ . Put $B_{012} = 1$ and $B_{013} = B_{023} = B_{123} = 0$ . Then $β$ is a normalized $3$ -cocycle representing a nonzero class, and $(a, B)$ is a finite $2$ -group background. Under the $0$ -form gauge labels $g_{0} = 1$ and $g_{1} = g_{2} = g_{3} = 0$ , with all $1$ -form labels zero, the transformed edge labels are $a_{01}^{'} = 0, a_{12}^{'} = 1, a_{23}^{'} = 1, a_{02}^{'} = 1, a_{13}^{'} = 0, a_{03}^{'} = 0$ and all four transformed face labels vanish; these transformed data also form a background.

Incidence diagram

The drawing records only the vertices and edges of $Δ^{3}$ ; edge labels use the increasing vertex order. The face labels are $B_{012} = 1$ and $B_{013} = B_{023} = B_{123} = 0$ . All algebraic checks are given independently in the Verification.

Facts & Assumptions

[F1]

Finite 2-group background cochains: the group coboundary of $c : G^{2} \to A$ is $(δ c) (g, h, k) = c (h, k) - c (g h, k) + c (g, h k) - c (g, h)$ when the action is trivial; a normalized $3$ -cocycle satisfies the five-term $4$ -variable equation; and backgrounds obey $δ_{a} B = a^{*} β$ .

Verification

Given: the finite groups, cocycle formula, ordered tetrahedron, vertex potential, face values, and gauge labels specified in the Example.

1.1givenF1algebra

The function $β (g, h, k) = g h k$ vanishes if any input is $0$ . Its $3$ -cocycle coboundary on $(w, x, y, z)$ is $x y z - (w + x) y z + w (x + y) z - w x (y + z) + w x y$ modulo $2$ ; after expansion every monomial occurs twice, so this is $0$ .

1.2givenF1algebra

The edge values are $a_{01} = a_{12} = a_{23} = a_{03} = 1$ and $a_{02} = a_{13} = 0$ , so they are differences of the stated vertex potential and are flat. On the tetrahedron, $δ_{a} B = B_{123} - B_{023} + B_{013} - B_{012} = - 1 = 1 = β (1, 1, 1)$ in $Z_{2}$ .

1.3givenF1algebra

With $g = (1, 0, 0, 0)$ and $λ = 0$ , the transformation formula gives $a_{i j}^{'} = g_{i} + a_{i j} - g_{j}$ and $ζ_{012} = - β (1, 1, 1) = - 1 = 1$ ; each other $ζ_{i j k} = 0$ because its three arguments include $0$ . Therefore $B_{012}^{'} = 1 + 1 = 0$ and the other face labels remain zero, as asserted.

2.1givenF1step 1.1algebra

If $β = δ c$ for any $2$ -cochain $c : Z_{2}^{2} \to Z_{2}$ , evaluation at $(0, 1, 1)$ gives $c (0, 0) = c (0, 1)$ , while evaluation at $(1, 0, 1)$ gives $c (0, 1) = c (1, 0)$ . Thus $(δ c) (1, 1, 1) = c (1, 0) - c (0, 1) = 0$ , contradicting $β (1, 1, 1) = 1$ . Hence the displayed cocycle class is nonzero.

3.1F1step 1.3algebra∎

The transformed tetrahedron has $δ_{a^{'}} B^{'} = 0$ , while $β (a_{01}^{'}, a_{12}^{'}, a_{23}^{'}) = β (0, 1, 1) = 0$ . Thus the transformed cochains satisfy the required equation.

Depends on

Finite 2-group background cochains