PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Necessary Ward consistency of coupled sources

Statement

Let $M$ be a smooth oriented, time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ with metric-selected connection $\nabla$ . Let $F \in Ω^{2} (M)$ , $j \in Γ (T M)$ , and $T_{r e n} \in Γ ({Sym}^{2} T^{*} M)$ be smooth, with $μ_{0}$ a positive real constant, $κ$ a nonzero real constant, and $Λ$ a constant real. Define $T_{a b}^{E M} : = \frac{1}{μ_{0}} (F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d}), G_{a b} : = {Ric}_{a b} - \frac{1}{2} R g_{a b} .$ Assume the mathematical field equations $d F = 0, \nabla_{a} F^{a b} = - μ_{0} j^{b}, G_{a b} + Λ g_{a b} = κ (T_{a b}^{E M} + T_{r e n, a b}) .$ Then $\nabla_{a} j^{a} = 0, \nabla^{a} T_{r e n, a b} = F_{b c} j^{c} .$ These are necessary conditional consequences of the displayed equations and match the contract-admissibility identities of the source-contract Definition; in this item, “Ward consistency” means exactly these two divergence identities. In SI length-coordinate bookkeeping, $[F] = W b m^{- 2}$ , $[j] = A m^{- 2}$ , $[T_{r e n}] = J m^{- 3}$ , $[Λ] = m^{- 2}$ , and $[κ] = s^{2} k g^{- 1} m^{- 1}$ ; the units are formal and no measured value is used. The result asserts no quantum-state construction, renormalized-source existence, or well-posed semiclassical evolution.

Statement references

Conditional semiclassical Einstein–Maxwell source contract

Sourced Maxwell sign and current conservation

Curved electromagnetic stress divergence

The second Bianchi identity and the vanishing divergence of the Einstein tensor

Facts & Assumptions

Given: The smooth geometry and fields, positive $μ_{0}$ , nonzero real constant $κ$ , constant real $Λ$ , and all three conditional field equations in the Statement.

[F1]

Sourced Maxwell sign and current conservation: for smooth $F, j$ and $μ_{0} > 0$ , the hypotheses $d F = 0$ and $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ imply $\nabla_{a} j^{a} = 0$ .

[F2]

The second Bianchi identity and the vanishing divergence of the Einstein tensor: for the metric-selected connection and $G_{a b} = {Ric}_{a b} - \frac{1}{2} R g_{a b}$ , $\nabla^{a} G_{a b} = 0$ for each $b$ .

[F3]

Curved electromagnetic stress divergence: under the same smoothness and sourced Maxwell sign, $\nabla^{a} T_{a b}^{E M} = - F_{b c} j^{c}$ for the displayed Maxwell Hilbert tensor.

[F4]

Conditional semiclassical Einstein–Maxwell source contract defines contract-admissibility by exactly the two identities to be derived; it does not assume the coupled equations or construct the source maps.

[A1]

$κ \in R ∖ {0}$ and $Λ \in R$ are constants. These hypotheses are required to divide by the coupling and to eliminate the cosmological term on taking the divergence; if $κ$ varied, differentiating $κ (T^{E M} + T_{r e n})$ would also produce $(\nabla^{a} κ) (T_{a b}^{E M} + T_{r e n, a b})$ .

Proof

1.1givenF1

The first two field equations in the Statement are exactly the hypotheses of [F1]. Applying that supplier gives $\nabla_{a} j^{a} = 0$ . This is a model deduction conditional on the sourced Maxwell equation, not a measured conservation law.

1.2givenF2A1algebra

Take $\nabla^{a}$ of the Einstein equation. By [F2], $\nabla^{a} G_{a b} = 0$ ; metric compatibility gives $\nabla^{a} (Λ g_{a b}) = 0$ because $Λ$ is constant. Therefore $0 = κ (\nabla^{a} T_{a b}^{E M} + \nabla^{a} T_{r e n, a b}) .$ Since $κ \neq 0$ , the two stress divergences sum to zero.

2.1givenF3F4step 1.2algebra∎

Substitute [F3] into step 1.2: $0 = - F_{b c} j^{c} + \nabla^{a} T_{r e n, a b}$ . Rearranging gives $\nabla^{a} T_{r e n, a b} = F_{b c} j^{c}$ , which is the second contract-admissibility condition in [F4]. Together with step 1.1 this proves both claimed identities. No existence, uniqueness, renormalization prescription, state regularity, or evolution theorem is used.

Remarks

The divergence identities are necessary compatibility checks only. They do not determine the renormalization prescription or rule out state-dependent nonlocality and higher-curvature terms. If such terms are included, they can alter differential order; order reduction is a separately justified approximation, not an algebraic consequence of this proposition.

Necessary Ward consistency of coupled sources

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources