A Negative Answer to Harrach's Question on the Potential Map in the Unified Eddy-Current Formulation

In the unified variational formulation of the parabolic-elliptic eddy-current equations due to Arnold and Harrach, the electric field is written as E = A + ∇φ_A, where A is divergence free and φ_A solves div(σ∇φ_A) = −div(σA). Harrach asked whether the potential map can be defined for general nonnegative bounded conductivity. We give a negative answer for H¹_loc potentials. A cyclic necklace of touching balls, with indicator conductivity or a smooth conductivity positive exactly on the open balls, admits a smooth, compactly supported, divergence-free field for which the potential equation has no H¹_loc solution. A solid torus with conductivity positive almost everywhere but vanishing on one cross-section gives the same conclusion. With zero initial data, corresponding smooth divergence-free forcing yields no eddy-current solution in L²(0,T;W(curl)). The weighted current σ(A + ∇φ_A), however, is defined for every nonnegative conductivity by an orthogonal projection, and the unified variational formulation remains uniquely solvable and uniformly coercive. Scope: the obstruction concerns H¹_loc potentials and the stated variational solution class. The eddy-current equivalence and nonexistence conclusions are proved with zero initial data; this is not a classification of every conductivity. Sketches in Remarks 7.1 and 7.2 are not used in the proved theorems. The ingredients include classical capacity and touching-conductor energy arguments. No absolute priority claim is made. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Author: Alper Ferudun, Mercury Software GmbH. Corpus identifier: OWR-11578-001 (Oberwolfach Report 11/2012, p. 631, B. Harrach). Paper page: https://eulersolve.org/papers/owr-11578-001/