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

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Normalized polarization is undefined at darkness

Physical scope

A zero-amplitude classical ensemble in the two-component coherency-matrix model. It demonstrates a boundary of a normalization definition; it is not a reported dark-detector observation, a noise model, or an empirical claim.

Statement refuted

For every square-integrable two-component classical ensemble, the quotient $P = \sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} / S_{0}$ defines a real degree of polarization.

Statement references

Classical two-component coherency matrix

Facts & Assumptions

Given: The one-point probability space $Ξ = {σ_{0}}$ with $P ({σ_{0}}) = 1$ , and the square-integrable two-component complex amplitude $Z (σ_{0}) = 0$ .

[F1]

Classical two-component coherency matrix: $Γ = E [Z Z^{†}]$ , $S_{0} = Γ_{11} + Γ_{22}$ , and the normalized degree is defined only when $S_{0} > 0$ .

[A1]

$Z (σ_{0}) = 0$ on the declared one-point probability space.

Counterexample

1.1givenF1A1algebra

From [A1], $Z Z^{†} = 0$ at the sole sample point, so [F1] gives $Γ = 0$ and $S_{0} = S_{1} = S_{2} = S_{3} = 0$ .

2.1step 1.1F1algebra∎

The proposed quotient therefore has denominator $S_{0} = 0$ and is undefined; the universal statement is false, even though the unnormalized cone inequality holds as $0 \geq 0$ .

Remarks

This is a domain-of-definition counterexample. It makes no claim about a real detector's dark counts, background, resolution, or uncertainty.

Depends on

Classical two-component coherency matrix

Dependency tree · 2 levels

Sources

José J. Gil, Polarimetric Characterization of Light and Media, §2.1–2.6, author-revised 2020 version of Eur. Phys. J. Appl. Phys. 40 (2007)