LemmaStatement: Literature-sourcedProof: 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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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.

Finite/formal gauge covariance and matrix Ward identity

Statement

Let $K (z; ζ) = K_{0} (ζ) + z V (ζ)$ and $S (z; ζ)$ be the finite-dimensional formal matrices of Finite matrix formal kernel and inverse series, with $d \geq 1$ , $K_{0} (ζ)$ invertible, and both coefficient matrices covariant under the vertex phase angles of Finite-graph U(1) phases for matrix kernels. Then $S$ is a two-sided inverse of $K$ in $M_{d} (C ⟦ z ⟧)$ , and $S (z; ζ^{θ}) = D_{θ} S (z; ζ) D_{θ}^{- 1}$ coefficientwise. For a real vertex tuple $θ$ , put $Θ : = {diag}_{x \in X} (θ_{x} I_{r})$ , $ζ_{s} : = ζ^{s θ}$ , and $D_{s} : = D_{s θ} = e^{i s Θ}$ . Write $K_{s} : = K (z; ζ_{s})$ and $S_{s} : = S (z; ζ_{s})$ , and define $δ K : = {\frac{d}{d s} K_{s} ∣}_{s = 0}$ and $δ S : = {\frac{d}{d s} S_{s} ∣}_{s = 0}$ coefficientwise along this differentiable phase path. The variations satisfy $δ K = i [Θ, K], δ S = i [Θ, S] = - S (δ K) S .$ For ordinary finite matrices $A, B \in G L_{d} (C)$ , the corresponding resolvent identity is $B^{- 1} (B - A) A^{- 1} = A^{- 1} - B^{- 1}$ . These are formal or finite algebraic statements; they assert no convergence of a perturbation series, continuum limit, or loop Ward identity.

Facts & Assumptions

Given: $d \geq 1$ , an invertible finite base $K_{0}$ , link matrices $V (ζ)$ , the phase action $D_{θ}$ , covariance of both coefficient matrices, and the coefficientwise formal-series operations.

[A1]

The coefficient matrices are covariant for every vertex phase angle; along the path $θ (s) = s θ$ , they vary differentiably by conjugation. The base is invertible. These are hypotheses of the Statement, not conclusions of the proof.

[F1]

Finite matrix formal kernel and inverse series: $K = K_{0} + z V$ and $S = \sum_{n \geq 0} (- z K_{0}^{- 1} V)^{n} K_{0}^{- 1}$ are defined in the formal power-series ring $C ⟦ z ⟧$ ; each coefficient of $S$ is a finite matrix.

[F2]

Finite-graph U(1) phases for matrix kernels: $ζ_{e}^{θ} = e^{i (θ_{s (e)} - θ_{t (e)})} ζ_{e}$ and $D_{θ} = {diag}_{x} (e^{i θ_{x}} I_{r})$ ; the hopping blocks are $T_{e} ζ_{e}$ .

[F3]

Charged gauge data and SI-natural conventions: the continuum charged-field phase is $e^{i g α (x)}$ . The finite graph angle of [F2] is independent; identifying it numerically with $g α_{x}$ requires a separately assigned real vertex tuple and does not choose a spacetime embedding. This convention is distinct from the phase action used by the finite algebraic proof.

Proof

1.1givenF1algebra

Put $B : = K_{0}^{- 1} V$ . The coefficient of $z^{n}$ in $S$ is $(- B)^{n} K_{0}^{- 1}$ , a finite matrix, so the series defines an element of $M_{d} (C ⟦ z ⟧)$ without an analytic convergence assumption.

1.2A1F1F2F3algebra

Each link block transforms as $T_{e} ζ_{e}^{θ} = e^{i θ_{s (e)}} (T_{e} ζ_{e}) e^{- i θ_{t (e)}}$ ; summing links in each block gives $V (ζ^{θ}) = D_{θ} V (ζ) D_{θ}^{- 1}$ . With [A1], the assumed covariance of $K_{0}$ gives the same conjugation for each coefficient of $K$ ; [F3] supplies the separate continuum phase convention, without identifying graph vertices with spacetime points.

2.1A1F1step 1.1algebra

Since $K = K_{0} (I + z B)$ , multiplication gives $K S = K_{0} (I + z B) \sum_{n \geq 0} (- z B)^{n} K_{0}^{- 1}$ . In the product $(I + z B) \sum_{n \geq 0} (- z B)^{n}$ , the constant coefficient is $I$ and the two contributions to every positive-degree coefficient cancel; hence $K S = I$ .

2.2A1F1step 1.1algebra

Also $K_{0}^{- 1} K = I + z B$ , so $S K = \sum_{n \geq 0} (- z B)^{n} (I + z B) = I$ by the same coefficientwise cancellation. Thus $S$ is the two-sided formal inverse.

2.3F1step 1.2algebra

Substituting the conjugated $K_{0}, V$ into the defining series for $S$ conjugates each finite coefficient by $D_{θ}$ , proving $S (z; ζ^{θ}) = D_{θ} S (z; ζ) D_{θ}^{- 1}$ .

3.1F1F2step 2.1step 2.2givenalgebra

Use the path $ζ_{s} = ζ^{s θ}$ , $D_{s} = D_{s θ}$ and the definitions of $K_{s}, S_{s}, δ$ in the Statement. Differentiate $K_{s} S_{s} = I$ coefficientwise at $s = 0$ . This gives $(δ K) S + K (δ S) = 0$ , hence $δ S = - S (δ K) S$ using $K^{- 1} = S$ .

4.1A1step 2.1step 2.2step 3.1givenalgebra

From $D_{s} = e^{i s Θ}$ , direct differentiation gives $δ K = i [Θ, K]$ . Therefore $- S i [Θ, K] S = - i S Θ K S + i S K Θ S = - i S Θ + i Θ S = i [Θ, S]$ .

5.1givenalgebra∎

For ordinary invertible finite matrices $A, B$ , expand the product: $B^{- 1} (B - A) A^{- 1} = B^{- 1} B A^{- 1} - B^{- 1} A A^{- 1} = A^{- 1} - B^{- 1}$ . This is exactly the resolvent identity in the Statement.

Finite/formal gauge covariance and matrix Ward identity

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources