CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Unscaled electric vectors fail the transformed Gauss law

Statement refuted

The assignment $E = \overset{ˉ}{E}$ always preserves the source-free electric divergence constraint under $h = ψ^{4} \overset{ˉ}{h}$ .

Statement refuted references

Exact conformal electrovacuum constraint transformation

Facts & Assumptions

Given: The Euclidean manifold $Σ = R^{3}$ with Cartesian coordinate $x$ , seed metric $\overset{ˉ}{h} = δ$ , positive conformal factor $ψ = e^{x}$ , and constant seed vector $\overset{ˉ}{E} = \partial_{x}$ . Set $\overset{ˉ}{B} = B = 0$ and use $h = ψ^{4} δ$ .

[F1]

Exact conformal electrovacuum constraint transformation: the divergence-preserving geometric field is $E = ψ^{- 6} \overset{ˉ}{E}$ , and $D_{i} E^{i} = ψ^{- 6} {\overset{ˉ}{D}}_{i} {\overset{ˉ}{E}}^{i}$ .

Counterexample

1.1givenF1algebra

The seed vector $\overset{ˉ}{E} = \partial_{x}$ has Euclidean divergence ${\overset{ˉ}{D}}_{i} {\overset{ˉ}{E}}^{i} = \partial_{x} (1) = 0$ , so it satisfies the source-free seed Gauss condition. The function $ψ = e^{x}$ is smooth and positive on all of $R^{3}$ .

2.1givenstep 1.1algebra

If instead one assigns the unscaled vector $E = \overset{ˉ}{E}$ , then $\sqrt{\det h} = ψ^{6} = e^{6 x}$ and the coordinate divergence formula gives $D_{i} E^{i} = (\sqrt{\det h})^{- 1} \partial_{i} (\sqrt{\det h} E^{i}) = e^{- 6 x} \partial_{x} (e^{6 x}) = 6$ . Thus the claimed universal preservation fails: this smooth field violates $D_{i} E^{i} = 0$ everywhere.

3.1F1step 1.1step 2.1algebra∎

The correct conformal field is $E = ψ^{- 6} \overset{ˉ}{E} = e^{- 6 x} \partial_{x}$ , for which step 1.1's seed identity gives $D_{i} E^{i} = e^{- 6 x} {\overset{ˉ}{D}}_{i} {\overset{ˉ}{E}}^{i} = 0$ . Hence the counterexample isolates the missing vector weight and uses no singular source at a removed point.

Depends on

Exact conformal electrovacuum constraint transformation