Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-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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Exact Ward/Gauss predictions in adopted regulator

Physical scope

Exact finite-graph selection, infinitesimal, contact and regional charge/flux deductions under the adopted rotor/CAR model, full AC and explicit operator domains, plus a separate conditional continuum implication. It constructs no continuum representation and uses no empirical premise.

Statement

Assume the finite-graph model of Specified finite spatial compact U(1) gauge regulator and the Axiom of Choice. Write $U (t) = e^{- i t H / ℏ}$ for the unitary group of Finite-regulator physical Gauss-sector existence and dynamics, and let $H_{p h y s} = ⋂_{x} \ker G_{x}$ be its nonzero physical sector.

For a bounded operator $O$ on $H_{k i n}$ , define its gauge charge $q = (q_{x})_{x \in V} \in Z^{V}$ by $U (α) O U (α)^{†} = e^{i \sum_{x} q_{x} α_{x}} O (α \in T^{V}) .$ For each normalized $ψ \in H_{p h y s}$ , $⟨ ψ, O ψ ⟩ = 0$ when $q \neq 0$ . A finite product of bounded gauge-charge eigenoperators has zero expectation whenever the sum of its charge vectors is nonzero. If $O$ maps $D (G_{x})$ into itself and $[G_{x}, O]$ extends to a bounded operator, then $⟨ ψ, [G_{x}, O] ψ ⟩ = 0$ . These statements also hold for $O (t) = U (t)^{†} O U (t)$ ; the finite-regulator evolution preserves gauge charge and the domains $D (G_{x})$ .

For a fixed vertex $x$ , distinct real insertion times $t_{1}, \dots, t_{n}$ with $n \geq 1$ , bounded operators $O_{j}$ that map $D (G_{x})$ into itself, and bounded commutators $[G_{x}, O_{j}]$ , define ordinary time ordering by placing the factor with larger time to the left. With $O_{j} (t_{j}) = U (t_{j})^{†} O_{j} U (t_{j})$ and $G_{x} (t) = U (t)^{†} G_{x} U (t)$ , the following identity holds as a distribution in $t$ between normalized physical/domain vectors: $\frac{d}{d t} ⟨ ψ, T ⁣ [G_{x} (t) \prod_{j = 1}^{n} O_{j} (t_{j})] ψ ⟩ = \sum_{j = 1}^{n} δ (t - t_{j}) ⟨ ψ, T ⁣ ([G_{x}, O_{j} (t_{j})] \prod_{i \neq j} O_{i} (t_{i})) ψ ⟩ .$ The products are evaluated on the stated common domains and $G_{x}$ is bosonic, so no grading sign is inserted.

For a region $R \subseteq V$ , define its boundary links by $\partial R = {e \in E : exactly one endpoint of e lies in R},$ with $ϵ (e, R) = 1$ for a link directed out of $R$ and $- 1$ for a link directed into $R$ . On $D_{R} = H_{p h y s} \cap ⋂_{e \in \partial R} D (E_{e})$ , put $Q_{R}^{l a t} = \sum_{x \in R} (n_{x} - b_{x} I)$ . If bounded $O$ preserves $H_{p h y s}$ , maps each boundary $D (E_{e})$ into itself, and $E_{e} O ϕ = O E_{e} ϕ$ for every $ϕ \in D (E_{e})$ and every $e \in \partial R$ , then $[Q_{R}^{l a t}, O] ψ = 0$ for every $ψ \in D_{R}$ . Consequently, an operator that changes $Q_{R}^{l a t}$ by a fixed nonzero charge cannot satisfy all these preservation and boundary-commutation hypotheses while acting nontrivially on that domain.

There is also a separate conditional continuum implication; it does not assert that a continuum representation exists. Let $H_{c}$ be a Hilbert space and $D \subseteq H_{c}$ a common invariant dense domain. For real $f, g_{k} \in C_{c}^{\infty} (R^{3})$ , let $ρ (f), E_{k} (g_{k}) : D \to D$ be linear assignments and put $E (g) = \sum_{k = 1}^{3} E_{k} (g_{k})$ for real $g = (g_{1}, g_{2}, g_{3}) \in C_{c}^{\infty} (R^{3}; R^{3})$ . Fix $ε_{0} > 0$ and assume $ρ (f) = - ε_{0} \sum_{k = 1}^{3} E_{k} (\partial_{k} f) (f \in C_{c}^{\infty} (R^{3})) .$ Let $K \subset R^{3}$ be compact. Suppose bounded $O$ maps $D$ into itself and $E (g) O ψ = O E (g) ψ$ for every $ψ \in D$ whenever $supp g$ is disjoint from $K$ . For integers $R \geq R_{0}$ , suppose real $χ_{R} \in C_{c}^{\infty} (R^{3})$ equals $1$ on a neighborhood of $K$ , $supp \nabla χ_{R}$ is disjoint from $K$ , and, with $Q_{R}^{c o n t} : = ρ (χ_{R}) = - ε_{0} E (\nabla χ_{R})$ , there is $Q : D \to D$ such that $Q_{R}^{c o n t} ψ \to Q ψ$ and $Q_{R}^{c o n t} O ψ \to Q O ψ$ for every $ψ \in D$ . Then $[Q, O] ψ = 0$ on $D$ . If also $[Q, O] ψ = q O ψ$ on $D$ for a fixed $q \neq 0$ , boundedness of $O$ implies $O = 0$ on $H_{c}$ . This clause is conditional on the displayed common-domain, Gauss, locality and limit hypotheses only.

Facts & Assumptions

Given: The adopted finite-graph model, full AC, the nonzero physical sector and its unitary evolution, the exact finite-regulator Ward suppliers, and, separately, the continuum hypotheses in the Statement.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. It supplies the choice premise used by the finite-graph evolution and Ward suppliers.

[F1]

Finite-regulator physical Gauss-sector existence and dynamics: under the adopted fixed finite model and full AC, $H_{p h y s}$ is nonzero and the exact self-adjoint Hamiltonian generates a strongly continuous unitary group preserving that sector. This is the preceding item on this page and is conditional on its stated draft mathematical inputs.

[L1]

Exact regulated gauge and infinitesimal Ward identities: for bounded gauge-charge operators, physical-state expectations vanish at nonzero total charge; under the stated $D (G_{x})$ and bounded-commutator hypotheses the infinitesimal matrix element vanishes; the exact evolution preserves charge and those domains. The current carrier is draft with a complete conditional proof and a Step-1 ready record, not independent proof acceptance.

[L2]

Time-ordered Ward contact relation: for pairwise distinct times and domain-preserving bounded insertions with bounded $G_{x}$ commutators, the time-ordered contact identity holds distributionally; the current carrier is a draft with a complete conditional proof and a Step-1 ready record.

[L3]

Gauss law and charged localization obstruction: on the finite graph, a bounded physical-sector-preserving operator that preserves each boundary electric domain and commutes with its electric operator there has zero regional-charge commutator on the common physical/electric domain. Its separate continuum clause requires exactly the common-domain, Gauss, locality and convergence assumptions listed in the Statement. This current draft carrier has a complete conditional proof and a Step-1 ready record.

Proof

Given: [A1], [F1], [L1]–[L3], and the separate continuum assumptions in the Statement.

1.1A1F1given

By [F1], under the adopted model and full AC the physical sector is a nonzero Hilbert subspace and $U (t)$ is its exact unitary evolution. Thus the physical vectors and time-evolved insertions in the finite clauses are those of the declared regulator, not an assumed continuum state space.

1.2A1L1givenalgebra

For any normalized physical $ψ$ and gauge-charge eigenoperator with $q \neq 0$ , the gauge transformations fix $ψ$ while multiplying the operator by its charge character. Choosing a vertex $x$ with $q_{x} \neq 0$ and $α_{x} = π / q_{x}$ makes that character $- 1$ , so its expectation must vanish. For a finite product, inserting $U (α)^{†} U (α) = I$ between factors adds their charges; the same argument applies if their total charge is nonzero. The exact gauge transformation, domain and evolution hypotheses are those in [L1], so the finite selection and infinitesimal identities apply also to the stated $O (t)$ .

1.3L2givenalgebra

The insertion-time and domain hypotheses in the Statement match [L2]. Since $G_{x}$ is gauge invariant under the exact evolution, $G_{x} (t) = G_{x}$ ; between successive distinct $t_{j}$ the ordinary time-ordered product is constant, and at $t_{j}$ its jump is the same ordered product with $G_{x} O_{j} (t_{j}) - O_{j} (t_{j}) G_{x}$ in place of the adjacent factors. The distributional derivative of a piecewise constant function is the sum of its jumps times $δ (t - t_{j})$ , which gives the displayed contact relation.

1.4L3givenalgebra

The finite-graph clause of [L3] uses the same region $R$ , regional charge $Q_{R}^{l a t}$ and domain $D_{R}$ as the Statement. Its hypotheses are precisely boundedness of $O$ , preservation of $H_{p h y s}$ , preservation of every boundary $D (E_{e})$ , and commutation with $E_{e}$ on that domain; all are assumed here. Hence [L3] gives $[Q_{R}^{l a t}, O] ψ = 0$ for every $ψ \in D_{R}$ . For $R = \emptyset$ , $Q_{R}^{l a t} = 0$ , so the conclusion also holds directly. If the same commutator equals $q O ψ$ with $q \neq 0$ , then $O ψ = 0$ for every vector in this common domain. This excludes only nontrivial action there; it does not infer that $O$ is zero on the full finite Hilbert space.

2.1L3givenalgebra∎

For the separate continuum clause, the support of $\nabla χ_{R}$ is disjoint from $K$ , so the assumed locality gives $E (\nabla χ_{R}) O ψ = O E (\nabla χ_{R}) ψ$ . The smeared Gauss identity yields $Q_{R}^{c o n t} O ψ = O Q_{R}^{c o n t} ψ$ . Taking the two stated limits and using boundedness of $O$ gives $Q O ψ = O Q ψ$ , hence $[Q, O] ψ = 0$ on $D$ . If also $[Q, O] ψ = q O ψ$ with $q \neq 0$ , then $O$ vanishes on dense $D$ and, being bounded, vanishes on all of $H_{c}$ . No existence of such a continuum model is inferred.

Exact Ward/Gauss predictions in adopted regulator

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources