ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Explicit physical electric/occupation basis state

Example

Take two vertices $v_{1}, v_{2}$ , one oriented link $e : v_{1} \to v_{2}$ , one fermion mode at each vertex, and background $b_{v_{1}} = 1, b_{v_{2}} = 0$ . Let $ψ$ be the tensor of the zero-electric Fourier vector and the occupation vector with the $v_{1}$ mode occupied and the $v_{2}$ mode empty. This is a concrete nonzero state in the fixed-background physical sector.

Facts & Assumptions

Given: The conditional finite-graph CAR model, one oriented link, and the chosen backgrounds.

[F1]

Gauss operators, gauge group and nonempty physical projection: for the conditional compact-rotor/CAR data, a normalized Fourier/occupation basis vector lies in $H_{p h y s} = ⋂_{x} \ker G_{x}$ exactly when every integer Gauss label $γ_{x}$ is zero, where $γ_{x}$ is outgoing link label minus incoming link label minus site occupation plus $b_{x}$ .

Verification

Given: [F1] and the stated graph, occupation, and link labels.

1.1F1given

The vector $ψ$ is a normalized Fourier/occupation basis vector: its link label is $n_{e} = 0$ , its site occupations are $n_{v_{1}} = 1, n_{v_{2}} = 0$ , and these occupations lie in the finite fermion Fock basis.

2.1F1step 1.1algebra∎

At $v_{1}$ , $γ_{v_{1}} = n_{e} - (n_{v_{1}} - b_{v_{1}}) = 0 - (1 - 1) = 0$ . At $v_{2}$ , $γ_{v_{2}} = - n_{e} - (n_{v_{2}} - b_{v_{2}}) = 0 - (0 - 0) = 0$ . Thus every Gauss operator annihilates $ψ$ , so $ψ \in H_{p h y s}$ and the physical projection is nonzero.

Depends on

Gauss operators, gauge group and nonempty physical projection

Explicit physical electric/occupation basis state

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources