Physics theoremStatement: 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)

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.

Conditional classical charged gauge equations and current

Physical scope

The smooth local classical Maxwell–Dirac branch on an open flat Minkowski domain, with compactly supported variations and the stated SI signs; it does not assert a global coupled PDE solution, an interacting quantum representation, or a pointwise quantum current.

Statement

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the selected free-field correspondence in Explicit charged Dirac/Maxwell local coupling model. Let $U$ be an open subset of flat Minkowski spacetime with $η = diag (- 1, 1, 1, 1)$ and length coordinates $x^{0} = c t$ . Let $A_{S I, a} : U \to R$ be a supplied $C^{\infty}$ covector potential, let $ψ : U \to C^{4}$ be a $C^{\infty}$ commuting spinor, and take a selected real charge $q_{S I}$ and mass $m_{S I} > 0$ with the conventions of Charge, canonical photon field and observer units and Charged gauge data and SI-natural conventions. Use independent compactly supported smooth variations of $A_{S I}$ and of both real and imaginary components of $ψ$ . If $(A_{S I}, ψ)$ is stationary for the adopted classical action, then

$(i γ^{a} D_{a} - μ) ψ = 0, \partial_{a} F_{S I}^{a b} = - μ_{0} j^{b}, \partial_{a} j^{a} = 0,$

where $D_{a} = \partial_{a} - i q_{S I} A_{S I, a} / ℏ$ , $μ = m_{S I} c / ℏ$ , and $j^{a} = q_{S I} c \overset{ˉ}{ψ} γ^{a} ψ$ . In laboratory components this current is $ρ_{q} = q_{S I} ψ^{†} ψ$ and $J_{q} = q_{S I} c ψ^{†} α ψ$ , so $j = (c ρ_{q}, J_{q})$ . With $[ψ] = m^{- 3 / 2}$ , $[q_{S I}] = C$ , and $[c] = m s^{- 1}$ , the charge density $ρ_{q}$ has unit $C m^{- 3}$ , while $J_{q}$ and every component of $j^{a}$ have unit $A m^{- 2}$ ; here $x^{0} = c t$ , so $j^{0} = c ρ_{q}$ has the same component unit as $J_{q}^{i}$ . Together with $F = d A$ on the supplied potential domain, the field equation and Bianchi identity have exactly the smooth SI Maxwell signs in The microscopic vacuum Maxwell equations: smooth and weak branches. The current and spinor here are classical smooth commuting fields; this statement is not a construction of a pointwise quantum composite operator.

Facts & Assumptions

Given: Countable Choice for the selected free-field representations; the adopted smooth local Maxwell–Dirac action; the actual QFT-MG01 Dirac variation interface; the SI tensor conventions; and the stated smoothness and compact-support hypotheses.

[A1]

Explicit charged Dirac/Maxwell local coupling model: the selected model adopts the smooth local Maxwell–Dirac action with $D_{a} = \partial_{a} - i q_{S I} A_{S I, a} / ℏ$ , uses independent compactly supported variations, and asserts no interacting continuum representation or global PDE existence.

[A2]

Countable Choice is assumed for the selected free QFT representations. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A3]

The supplied fields $A_{S I, a}, ψ$ and gauge functions are smooth on the open set $U$ , and stationarity is tested against every compactly supported smooth variation of the real potential and complex spinor components.

[F1]

Charge, canonical photon field and observer units: $q_{n} = q_{S I} \sqrt{μ_{0} c / ℏ}$ , $a_{a} = A_{S I, a} / \sqrt{μ_{0} ℏ c}$ , $q_{n} a_{a} = q_{S I} A_{S I, a} / ℏ$ , and $μ = m_{S I} c / ℏ$ .

[F2]

The electromagnetic two-form, its observer decomposition and the four-current: $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , and a supplied potential gives $F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}$ .

[F3]

Framework-local SI units, the vacuum constants and the exact/measured status of their values: $ε_{0} = (μ_{0} c^{2})^{- 1}$ , $q_{S I}$ carries C, charge density carries $C m^{- 3}$ , and current density carries $A m^{- 2}$ .

[F4]

The microscopic vacuum Maxwell equations: smooth and weak branches: the smooth SI branch has $div E = ρ / ε_{0}$ , $curl E = - \partial_{t} B$ , $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ , and $div B = 0$ .

[F5]

Dirac Clifford matrices and energy projections: $γ^{0} = β$ , $γ^{i} = β α_{i}$ , $\overset{ˉ}{ψ} = ψ^{†} β$ , $β^{†} = β$ , $β^{2} = I_{4}$ , and the matrices $α, β$ are Hermitian.

[F7]

Potential representations, gauge transformations and the Lorenz and Coulomb conditions: the SI potential convention is $A_{a} \mapsto A_{a} + \partial_{a} χ$ ; no reachability or global gauge-fixing claim is part of that definition.

[F6]

A smooth bump between concentric Euclidean balls: every open Euclidean ball contains a nonnegative smooth compactly supported function that is positive on a smaller ball.

[F8]

Quantum field quantities and compatible SI normalizations: spacetime coordinates have length units, $x^{0} = c t$ with $t$ in seconds and $c$ in $m s^{- 1}$ ; physical energy is obtained from inverse-length wave number by $E = ℏ c ω$ ; the canonical spinor-field dimension is $[ψ] = m^{- 3 / 2}$ .

[L1]

Dirac gauge covariance, action and conserved current (draft carrier; current Step 1 decision escalated through its QFT QED-gauge-data and Dirac-projection prerequisites: under Countable Choice, for smooth real natural potential $a_{a}$ and smooth commuting spinor on $R^{4}$ , compactly supported variations of the symmetrized Dirac density give $(i γ^{a} D_{a} - μ) ψ = 0$ and its adjoint; the density is invariant under the stated gauge transformation; on solutions $\partial_{a} (g \overset{ˉ}{ψ} γ^{a} ψ) = 0$ , with natural current $g \overset{ˉ}{ψ} γ^{a} ψ$ . Its exact Statement uses the free QFT gauge data and supplies the corresponding SI Dirac Hamiltonian density. The carrier does not state the Maxwell-field variation used below; that is derived from [A1]. The local test-function argument is repeated below on $U$ , so no global extension of fields is assumed.

Proof

Given: [A1]–[A3], [F1]–[F8], and the conditional Dirac interface [L1].

1.1A1F1F2F7F8algebra

The action uses length coordinates $x^{0} = c t$ by [F8]. Put $S_{0} = \sqrt{μ_{0} ℏ c}$ , $a_{a} = A_{S I, a} / S_{0}$ , and $g = q_{S I} \sqrt{μ_{0} c / ℏ}$ . Then $g a_{a} = q_{S I} A_{S I, a} / ℏ$ by [F1], so the matter covariant derivative in [A1] is exactly $D_{a} = \partial_{a} - i g a_{a}$ in [L1]. For an SI gauge function $χ$ , put $α = χ / S_{0}$ ; then $g α = q_{S I} χ / ℏ$ . Thus the SI transformation of [A1] becomes $a_{a} \mapsto a_{a} + \partial_{a} α$ and $ψ \mapsto e^{i g α} ψ$ . Differentiating the phase shows $D_{a}^{'} ψ^{'} = e^{i g α} D_{a} ψ$ ; commuting smooth partial derivatives give $F^{'} = F$ .

1.2A1A2A3F5F6L1givenalgebra

Countable Choice [A2] is carried for the separately adopted free-field correspondence; the classical local variation below itself requires no choice operation. Let $R_{D} = (i γ^{a} D_{a} - μ) ψ$ . The compact-support first-variation identity in [L1] is local: for every $ζ \in C_{c}^{\infty} (U; C^{4})$ , integration by parts in the symmetrized density gives $\frac{δ S_{D}}{ℏ} = 2 Re \int_{U} ζ^{†} β R_{D} d^{4} x .$ Indeed, the derivative terms use $\int_{U} \overset{ˉ}{ψ} γ^{a} D_{a} ζ = - \int_{U} (D_{a} \overset{ˉ}{ψ}) γ^{a} ζ$ and $- \int_{U} (D_{a} \overset{ˉ}{ζ}) γ^{a} ψ = \int_{U} \overset{ˉ}{ζ} γ^{a} D_{a} ψ$ ; there are no boundary terms because $ζ$ is compactly supported. If $R_{D} (x_{0}) \neq 0$ , continuity gives a ball $B \subset U$ on which $R_{D}$ is nonzero. Choose $ρ$ from [F6] supported in $B$ and positive on a smaller ball, and set $ζ = ρ β R_{D}$ . By [F5], $ζ^{†} β R_{D} = ρ ∣ R_{D} ∣^{2}$ , so the first variation is $2 \int_{U} ρ ∣ R_{D} ∣^{2} d^{4} x > 0$ , contradicting stationarity. Therefore $R_{D} = 0$ on $U$ , and taking its adjoint gives the adjoint Dirac equation. This is a local consequence of compact variations, not an existence claim.

2.1F1F5L1step 1.1step 1.2algebra

Multiplying the Dirac equation by $\overset{ˉ}{ψ}$ gives $i \overset{ˉ}{ψ} γ^{a} D_{a} ψ = μ \overset{ˉ}{ψ} ψ$ ; multiplying its adjoint by $ψ$ gives $- i (D_{a} \overset{ˉ}{ψ}) γ^{a} ψ = μ \overset{ˉ}{ψ} ψ$ . Subtracting the equations gives $(D_{a} \overset{ˉ}{ψ}) γ^{a} ψ + \overset{ˉ}{ψ} γ^{a} D_{a} ψ = 0$ . Expanding $D_{a} \overset{ˉ}{ψ} = \partial_{a} \overset{ˉ}{ψ} + i g a_{a} \overset{ˉ}{ψ}$ and $D_{a} ψ = \partial_{a} ψ - i g a_{a} ψ$ cancels the two gauge-potential terms, leaving $\partial_{a} (\overset{ˉ}{ψ} γ^{a} ψ) = 0$ . The natural current in [L1] is $j_{n}^{a} = g \overset{ˉ}{ψ} γ^{a} ψ$ . By step 1.1, $S_{0} g = μ_{0} q_{S I} c$ , so the SI current associated to $\partial_{a} F_{S I}^{a b} = - μ_{0} j^{b}$ is $j^{a} = (S_{0} / μ_{0}) j_{n}^{a} = q_{S I} c \overset{ˉ}{ψ} γ^{a} ψ$ and is conserved because the conversion factor is constant. Using [F5], $j^{0} = q_{S I} c ψ^{†} ψ$ and $j^{i} = q_{S I} c ψ^{†} α_{i} ψ$ , hence $j = (c ρ_{q}, J_{q})$ with the displayed SI charge and current densities.

3.1A1F1F2F3F4F5F6step 2.1algebra

Write the natural action density from [A1] as $- \frac{1}{4} f_{a b} f^{a b} + L_{D}$ . For a real compactly supported variation $δ a_{b}$ , antisymmetry of $f^{a b}$ gives $δ (- \frac{1}{4} f_{a b} f^{a b}) = - f^{a b} \partial_{a} δ a_{b} .$ The matter variation is $δ L_{D} = \frac{i}{2} [\overset{ˉ}{ψ} γ^{b} (- i g δ a_{b}) ψ - (i g δ a_{b} \overset{ˉ}{ψ}) γ^{b} ψ] = g \overset{ˉ}{ψ} γ^{b} ψ δ a_{b} .$ Integrating the Maxwell term by parts and using compact support therefore gives $\frac{δ S}{ℏ} = \int_{U} (\partial_{a} f^{a b} + g \overset{ˉ}{ψ} γ^{b} ψ) δ a_{b} d^{4} x .$ By [F5], $\overset{ˉ}{ψ} γ^{0} ψ = ψ^{†} ψ$ and $\overset{ˉ}{ψ} γ^{i} ψ = ψ^{†} α_{i} ψ$ are real. If a component $G^{b} = \partial_{a} f^{a b} + g \overset{ˉ}{ψ} γ^{b} ψ$ were nonzero at a point, continuity would give a ball in $U$ on which it has fixed nonzero sign. Choose a nonnegative bump $ρ$ there using [F6] and vary only $a_{b}$ by $δ a_{b} = ρ$ . The displayed variation would then be nonzero, contrary to stationarity. Thus every $G^{b}$ vanishes pointwise and $\partial_{a} f^{a b} = - g \overset{ˉ}{ψ} γ^{b} ψ$ . Since $F_{S I}^{a b} = S_{0} f^{a b}$ and $S_{0} g = μ_{0} q_{S I} c$ , this is exactly $\partial_{a} F_{S I}^{a b} = - μ_{0} j^{b}$ . For $b = 0$ , [F2] gives $\partial_{a} F^{a 0} = - div E / c$ , so the source equation becomes $div E = μ_{0} c^{2} ρ_{q} = ρ_{q} / ε_{0}$ by [F3]. For $b = i$ , [F2] gives $\partial_{a} F^{a i} = c^{- 2} \partial_{t} E_{i} - (curl B)_{i}$ , which yields $curl B = μ_{0} J_{q} + c^{- 2} \partial_{t} E$ . Finally, $F = d A$ and commuting partial derivatives give $\partial_{[a} F_{b c]} = 0$ , whose spatial components are $div B = 0$ and $curl E = - \partial_{t} B$ . These are the smooth equations in [F4], derived conditionally from the adopted action; they are not evidence that the model describes nature.

4.1F3F5F8givenalgebra∎

By [F8], $[ψ] = m^{- 3 / 2}$ , and [F3] gives $[q_{S I}] = C$ , so $[ρ_{q}] = [q_{S I} ψ^{†} ψ] = C m^{- 3}$ . Since $[c] = m s^{- 1}$ and the matrices $α$ are dimensionless by [F5], $[J_{q}] = C m^{- 2} s^{- 1} = A m^{- 2}$ . Therefore $[j^{0}] = [c ρ_{q}] = A m^{- 2}$ , and each spatial component $j^{i} = J_{q}^{i}$ has the same unit. These are units of the classical current in length coordinates $x^{0} = c t$ , not units of a quantum operator product.

Conditional classical charged gauge equations and current

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources