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)

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.

Qualified formal/tree gauge-current identities

Physical scope

Conditional coefficientwise covariance for finite formal matrices and decoupling of the specified conserved tree currents in the adopted QED convention. No loop-product, all-orders Ward, renormalization, continuum scattering, or nonperturbative claim is made.

Statement

Use the coefficient convention of Explicit formal QED vertex/free-line convention and formal ring of Finite matrix formal kernel and inverse series. Let $ζ$ denote finite graph link data, $z$ a formal indeterminate, and $K (z; ζ) = K_{0} (ζ) + z V (ζ) \in M_{d} (C ⟦ z ⟧)$ with $d \geq 1$ and $K_{0} (ζ) \in G L_{d} (C)$ . For each vertex-angle tuple $θ$ , let $ζ^{θ}$ and the invertible diagonal matrix $D_{θ}$ be the finite gauge action and phase matrix of Finite-graph U(1) phases for matrix kernels. Assume the coefficient matrices satisfy $K_{0} (ζ^{θ}) = D_{θ} K_{0} (ζ) D_{θ}^{- 1}, V (ζ^{θ}) = D_{θ} V (ζ) D_{θ}^{- 1} .$ Let $S (z; ζ)$ be the coefficientwise formal inverse specified by Finite matrix formal kernel and inverse series. Then $S (z; ζ^{θ}) = D_{θ} S (z; ζ) D_{θ}^{- 1}, \det K (z; ζ^{θ}) = \det K (z; ζ) .$ For a real vertex tuple $θ$ , put $Θ = {diag}_{x \in X} (θ_{x} I_{r})$ , $ζ^{s} : = ζ^{s θ}$ , and $D_{s} : = D_{s θ} = e^{i s Θ}$ . For any coefficient family $A_{s} = A (z; ζ^{s})$ , write $δ A : = {\frac{d}{d s} A_{s} ∣}_{s = 0}$ coefficientwise. Assume also that $K_{0} (ζ^{s})$ and $V (ζ^{s})$ vary differentiably along the gauge orbit $ζ^{s}$ . Coefficientwise differentiation then gives $δ S = i [Θ, S] = - S (δ K) S .$ These are finite/formal identities only; the formal variable is not assigned a numerical value and no convergence is asserted.

For the tree-current clause, take $μ > 0$ and four future-shell momenta $p, p^{'}, q, q^{'}$ with $h (p, p) = h (p^{'}, p^{'}) = h (q, q) = h (q^{'}, q^{'}) = μ^{2}$ , in inverse-length units, and put $k = p - p^{'} = q^{'} - q$ . Let $J_{1}^{a} = \overset{ˉ}{u} (p^{'}) γ^{a} u (p), J_{2}^{b} = \overset{ˉ}{u} (q^{'}) γ^{b} u (q),$ where $\overset{ˉ}{u} = u^{†} β$ and the spinors satisfy the corresponding positive-energy Dirac equations $(p̸ - μ) u (p) = 0$ , $(̸ p^{'} - μ) u (p^{'}) = 0$ , $(q̸ - μ) u (q) = 0$ , $(̸ q^{'} - μ) u (q^{'}) = 0$ and their Dirac-adjoint left equations, in the $h = - η$ convention of Explicit formal QED vertex/free-line convention. For a finite scalar $c (k)$ , set $Δ D_{a b} (k) = c (k) k_{a} k_{b}$ . Then $J_{1}^{a} Δ D_{a b} (k) J_{2}^{b} = 0.$ For either displayed conserved current, the contraction with a covector $ϵ_{a}$ is unchanged by $ϵ_{a} \mapsto ϵ_{a} + ρ k_{a}$ for finite $ρ \in C$ . If $c (k)$ is a rational coefficient containing $1 / h (k, k)$ , require $h (k, k) \neq 0$ at the selected momentum; a distributional $+ i 0$ boundary value is not a pointwise finite $c (k)$ . The adopted off-shell fermion coefficient has denominator $h (p, p) - μ^{2}$ and is used only when that denominator is nonzero. No internal-line singular product is used here. The tree clause is limited to these explicit finite contractions and does not assert gauge independence for arbitrary diagrams.

Facts & Assumptions

Given: The finite formal kernel and phase-covariance hypotheses in the Statement, and the four on-shell spinors with the stated Dirac equations.

[F1]

Explicit formal QED vertex/free-line convention: $η = diag (- 1, 1, 1, 1)$ , $h = - η$ , ${γ^{a}, γ^{b}} = 2 h^{a b} I_{4}$ , $p̸ = γ^{a} h_{a b} p^{b}$ , and $\overset{ˉ}{u} = u^{†} β$ . The adopted electron vertex is $- i e_{n} γ^{a}$ ; the fermion coefficient uses $h (p, p) - μ^{2} \neq 0$ , and the pointwise photon coefficient uses $h (k, k) \neq 0$ . The $+ i 0$ notation is a boundary-value convention, not a definition of singular distribution products.

[F2]

Finite matrix formal kernel and inverse series: $C ⟦ z ⟧$ uses coefficientwise Cauchy multiplication; for invertible $K_{0}$ the inverse candidate is $S = \sum_{n \geq 0} (- z K_{0}^{- 1} V)^{n} K_{0}^{- 1}$ , with finite matrix coefficients, and $\det$ is defined by the Leibniz formula. The current batch-45 manifest lists this draft definition, whose Step-1 record is escalated because its QFT-MG01 gauge-data input has no current readiness receipt; this use remains conditional.

[F3]

Finite-graph U(1) phases for matrix kernels: a link transforms as $ζ_{e}^{θ} = e^{i (θ_{s (e)} - θ_{t (e)})} ζ_{e}$ and the vertex phase matrix is diagonal. The finite graph phase data are the hypotheses used here.

[F4]

Dirac Clifford matrices and energy projections and Scalar future mass shell, invariant measure and test restriction: $\overset{ˉ}{u} = u^{†} β$ , $p̸ = γ^{0} p^{0} - \sum_{j} γ^{j} p_{j}$ , and a future-shell momentum has $h (p, p) = μ^{2}$ , $μ > 0$ . These fix the spinor notation and shell convention. The Dirac-projection draft has an escalated Step-1 record; the scalar-shell definition has a ready item record but its containing QFT-MF02 page edge remains open. The spinor use is conditional on those exact conventions.

[L1]

Finite/formal gauge covariance and matrix Ward identity: if the finite formal coefficients $K_{0}, V$ are covariant and $K_{0}$ is invertible, the formal inverse is two-sided and covariant coefficientwise; along $D_{s} = e^{i s Θ}$ it obeys $δ S = i [Θ, S] = - S (δ K) S$ . Its inspected draft proof is complete for this conditional matrix interface. The current batch-45 manifest lists the lemma, and its Step-1 record is escalated through the QFT-MG01 Charged gauge data and SI-natural conventions prerequisite, which has no current readiness receipt.

[L2]

Gauge-longitudinal kernel decoupling for explicit tree currents: for two future-shell positive energy Dirac currents with transfers $k$ and $- k$ , the on-shell equations give $k_{a} J_{1}^{a} = k_{b} J_{2}^{b} = 0$ ; a finite longitudinal kernel $c (k) k_{a} k_{b}$ therefore contracts to zero, and a pure-gauge external polarization shift leaves each such current contraction unchanged. Its inspected draft file and complete conditional proof are present in the current batch-45 manifest. Its Step-1 record is escalated because Off-shell free Dirac Ward telescoping and tree currents is escalated through the QFT-MF03 Dirac definition and the QFT-MG01 formal-matrix chain; this use remains conditional.

[F5]

For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ : for square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ . This is a published mathematical supplier and applies to $C ⟦ z ⟧$ .

The coefficientwise gauge-covariance equalities hold for every $θ$ , and $K_{0}, V$ are differentiable along the selected phase orbit. These are hypotheses, not conclusions of this theorem.

Proof

Given: [F1]–[F5], [L1]–[L2], and the Statement's formal matrix and on-shell spinor hypotheses.

1.1F2algebra

Put $B = K_{0}^{- 1} V$ . The coefficient of $z^{n}$ in $S = \sum_{n \geq 0} (- z B)^{n} K_{0}^{- 1}$ is finite. In $(I + z B) \sum_{n \geq 0} (- z B)^{n}$ , the constant coefficient is $I$ and the two terms in every positive degree cancel. Since $K = K_{0} (I + z B)$ , this proves $K S = I$ . The same coefficient cancellation in $\sum_{n \geq 0} (- z B)^{n} (I + z B)$ proves $S K = I$ . At $z = 0$ this reduces to $S (0) = K_{0}^{- 1}$ .

1.2F1F4L2givenalgebra

For the first current, $k̸ = p̸ - ̸ p^{'}$ , so the right equation for $u (p)$ and the left equation for $\overset{ˉ}{u} (p^{'})$ give $k_{a} J_{1}^{a} = \overset{ˉ}{u} (p^{'}) (p̸ - ̸ p^{'}) u (p) = μ \overset{ˉ}{u} (p^{'}) u (p) - μ \overset{ˉ}{u} (p^{'}) u (p) = 0$ . For the second current its transfer is $q - q^{'} = - k$ ; the same calculation gives $(- k)_{b} J_{2}^{b} = 0$ , hence $k_{b} J_{2}^{b} = 0$ .

2.1F2F3L1step 1.1algebra

The hypotheses match [L1]'s finite formal matrix interface. Replacing $K_{0}, V$ by their conjugates in the series of step 1.1 conjugates each coefficient of $S$ by $D_{θ}$ , because $(D_{θ} K_{0} D_{θ}^{- 1})^{- 1} = D_{θ} K_{0}^{- 1} D_{θ}^{- 1}$ and matrix products associate. Hence $S (z; ζ^{θ}) = D_{θ} S (z; ζ) D_{θ}^{- 1}$ . With $K_{s} = K (z; ζ^{s θ})$ , $S_{s} = S (z; ζ^{s θ})$ and $δ$ as defined in the Statement, differentiating $K_{s} S_{s} = I$ coefficientwise at $s = 0$ gives $δ S = - S (δ K) S$ ; differentiating $D_{s} = e^{i s Θ}$ gives $δ K = i [Θ, K]$ , and substitution with $K S = S K = I$ yields $δ S = i [Θ, S]$ . These operations use only formal coefficients.

3.1F2F3F5step 2.1algebra

By step 2.1 and [F5], $\det K (z; ζ^{θ}) = \det (D_{θ} K (z; ζ) D_{θ}^{- 1}) = \det (D_{θ}) \det (K (z; ζ)) \det (D_{θ}^{- 1})$ . The finite matrices $D_{θ}$ and $D_{θ}^{- 1}$ are inverse, so [F5] gives $\det (D_{θ}) \det (D_{θ}^{- 1}) = 1$ . This proves determinant invariance coefficientwise.

4.1F1F4L2step 1.2givenalgebra∎

The longitudinal contraction factors as $J_{1}^{a} Δ D_{a b} J_{2}^{b} = c (k) (k_{a} J_{1}^{a}) (k_{b} J_{2}^{b}) = 0$ by step 1.2. The polarization change is $ρ k_{a} J^{a} = 0$ for either current by step 1.2. These are finite algebraic contractions under the stated denominator condition; no $+ i 0$ distribution product appears.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Quantum Field Theory, complete author-hosted Cambridge lecture notes, Chapter 6: Quantum Electrodynamics, §§6.4.1–6.5