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)

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Charged expectation vanishes by gauge averaging

Example

Take a two-vertex graph with one link $e : v_{1} \to v_{2}$ , one fermion mode per vertex and zero backgrounds. The zero-electric, zero-occupation vector $Ω$ is physical. The bounded creation operator $O = c_{v_{1}, 1}^{†}$ has nonzero gauge charge at $v_{1}$ , so its expectation in any physical vector, including $Ω$ , vanishes.

Facts & Assumptions

Given: The conditional finite-graph CAR model, the gauge action, and the Axiom of Choice required by the time-dependent Ward supplier.

[A1]

The Axiom of Choice is the principle that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

Exact regulated gauge and infinitesimal Ward identities: every bounded gauge-charge eigenoperator with nonzero charge has zero expectation in a physical vector; time evolution preserves gauge charge under AC. This current A-page item depends on the still-draft QFT-MF01 CAR supplier through its graph-model dependencies.

Verification

Given: [A1], [F1], and the zero-background two-vertex state described above.

1.1A1F1given

For $Ω$ , $n_{v_{1}} = n_{v_{2}} = 0$ and $E_{e} = 0$ , so both Gauss eigenvalues vanish. The CAR creation operator is bounded and nonzero; the gauge law gives $U (α) O U (α)^{†} = e^{- i α_{v_{1}}} O$ , so its charge vector is $q_{v_{1}} = - 1, q_{v_{2}} = 0$ .

2.1F1step 1.1algebra∎

The nonzero charge in step 1.1 forces $⟨ Ω, O Ω ⟩ = 0$ by the finite-regulator selection rule, even though $O Ω$ is a nonzero one-fermion vector. This demonstrates that the selection rule concerns expectations in the physical sector, not the vanishing of the charged operator itself.

Charged expectation vanishes by gauge averaging

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources