CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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The polar helicity frame is singular while its projector is smooth

Physical scope

Conditional classical Maxwell-Lorentz vacuum model in the smooth source-free nonzero-Fourier-mode regime stated by the supplier, with no extra approximation. The example distinguishes a coordinate-dependent phase frame from the smooth helicity projection; it makes no empirical claim, quantization assertion, or claim of a global nonvanishing polarization basis.

Statement refuted

For the helicity line on $S^{2}$ , the fact that the spherical unit frame $v_{σ} = (e_{θ} + i σ e_{ϕ}) / \sqrt{2}$ is undefined at the coordinate poles implies that the projection family $P_{σ} (n)$ is itself undefined or nonsmooth there.

Statement references

Helicity eigenlines over the direction sphere

Facts & Assumptions

Given: The direction sphere $S^{2}$ , $σ \in {+ 1, - 1}$ , and the projector and line bundle of Helicity eigenlines over the direction sphere.

[F1]

Helicity eigenlines over the direction sphere defines $P_{T} (n) z = z - n (n \cdot z)$ and $P_{σ} (n) = \frac{1}{2} (P_{T} (n) + σ i n \times)$ for every $n \in S^{2}$ and proves that its range is a smooth line bundle with local smooth unit frames.

[F2]

Curvature and the obstruction to a global polarization basis proves that the line has no globally smooth unit frame. This is an obstruction to one global phase basis, not a singularity of the projection family.

Counterexample

Given: The facts [F1] and [F2].

1.1F1algebra

Since $n \cdot n = 1$ , the formula in [F1] is equivalently $P_{σ} (n) = \frac{1}{2} (I - n n^{T} + σ i N_{n})$ , where $N_{n} z = n \times z$ . Each matrix entry is a polynomial in the real coordinates of $n$ , so $n \mapsto P_{σ} (n)$ is smooth on all of $S^{2}$ , including both poles.

2.1F1F2step 1.1algebra∎

The local formula for $v_{σ}$ uses $ϕ$ , which is not a coordinate at either pole, while the projection matrix from step 1.1 is defined and smooth there. By [F1], projecting a fixed vector with nonzero projection at a given point and normalizing it gives a smooth local unit frame near that point. Thus local polarization sections remain smooth; [F2] rules out only a single nowhere-zero unit frame on the entire sphere. The displayed false implication is refuted.

The polar helicity frame is singular while its projector is smooth

Statement refuted

Statement references

Facts & Assumptions

Counterexample

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