LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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No global continuous real polarization frame

Statement

Assume Countable Choice. For $n \in S^{2} \subset R^{3}$ , let $E_{n} = n^{⊥} = Ran (I - n n^{T})$ be the real transverse plane. There do not exist continuous maps $e_{1}, e_{2} : S^{2} \to R^{3}$ such that, for every $n$ , $(e_{1} (n), e_{2} (n))$ is an orthonormal basis of $E_{n}$ . A continuous real orthonormal polarization frame would in particular give a continuous nowhere-zero tangent vector field on $S^{2}$ .

Facts & Assumptions

Given: Countable Choice; the transverse fiber is $E_{n} = n^{⊥}$ ; and the published even-sphere tangent-field theorem.

[A1]

Assume Countable Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The real range of $P_{T} (n) = I - n n^{T}$ is exactly $n^{⊥}$ . Positive transverse photon one-particle space

[F2]

For every positive even $d$ , there is no continuous nowhere-zero tangent vector field on $S^{d}$ . No nowhere zero tangent vector field on an even sphere

Proof

Given: [A1], [F1], and [F2].

1.1givenF1algebra

If such a frame existed, then $e_{1} (n) \in n^{⊥}$ for every $n$ by [F1], and $∣ e_{1} (n) ∣ = 1$ by orthonormality. Therefore $e_{1}$ is a continuous nowhere-zero tangent vector field on $S^{2}$ .

2.1A1F2step 1.1∎

Since $2$ is a positive even integer, [F2] forbids the vector field produced in step 1.1. This contradiction proves the nonexistence of the global frame.

No global continuous real polarization frame

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources