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CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)
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Raw edge sampling is not defined on every H(curl) class

Physical scope

A counterexample about representative-dependent line sampling of arbitrary real H(curl) equivalence classes along an interior segment, including interior tetrahedral-mesh edges. It does not deny the well-defined polynomial edge degrees of freedom of the Whitney space or the existence of separate smoothed projections under their mesh hypotheses. No measurement or continuum approximation claim is made.

Statement refuted

Let Omega be a nonempty open subset of R3 and let e be a closed straight segment of positive length contained in Omega, with an orientation and unit tangent t_e. (This includes any mesh edge contained in the interior of a tetrahedral mesh.) The false claim is that for every real H(curl;Omega) equivalence class [u], the value integral_e u dot t_e ds is independent of the representative and therefore defines raw line sampling on all of H(curl;Omega).

Facts & Assumptions

Given: A closed straight segment e of positive length contained in a nonempty open subset Omega of R3, its chosen unit tangent t_e, and the real L^2-based graph-space convention.

[F1]

The real H(curl) graph space on a polyhedron: H(curl;Omega) consists of L^2 equivalence classes whose distributional curl is represented by an L^2 class; changing a representative on a three-dimensional null set changes neither the class nor its distributional curl.

[F2]

PEC Whitney edge-face Maxwell semidiscretization and Lowest-order Whitney and Nedelec de Rham spaces: the semidiscrete finite-element definition uses oriented edge circulations only on the cellwise polynomial Whitney space, where the local polynomial representatives define those moments; it does not extend the moment to arbitrary H(curl) classes.

[F3]

Uniformly bounded commuting projection for lowest-order Nedelec spaces: if Omega is additionally a nonempty bounded Lipschitz polyhedron and {T_h} is a shape-regular family of matching affine tetrahedral meshes with h=max_{K in T_h} diam(K) tending to zero, then, assuming Countable Choice, a separate smoothed projection maps H_0(curl;Omega) uniformly boundedly into N_h^1 intersect H_0(curl;Omega) and converges strongly in graph norm; this is not the raw edge-sampling rule refuted here.

Counterexample

Given: The edge e and graph space above.

1.1givenF1construct

Let u_0 be zero everywhere. Define a second measurable representative u_1 to equal t_e on e and zero on Omega minus e. A line segment has three-dimensional Lebesgue measure zero, so u_0 and u_1 represent the same zero L^2 class. Their distributional curls are both the zero distribution by the integral definition in [F1]; hence they represent the same element of H(curl;Omega).

2.1givenF2step 1.1algebra

Using the displayed representatives, the oriented edge integral of u_0 is zero, while that of u_1 is integral_e |t_e|^2 ds=length(e)>0. Therefore this integral is not a function of the H(curl) equivalence class, which refutes the universal raw edge-sampling claim. This does not affect the polynomial Whitney moments in [F2].

3.1F3step 1.1step 2.1∎

The counterexample concerns only unregularized line restriction. Under the distinct hypotheses in [F3], a smoothed relative commuting projection is available; its construction is not an edge-by-edge evaluation of the arbitrary H(curl) representative. Thus it supplies no missing edge trace to the refuted map and is consistent with steps 1.1-2.1.

Depends on

Dependency tree · 2 levels

Sources