Wigner entropy below vacuum: physical counterexamples and stability limits

Positive Wigner functions can have less Shannon entropy than the vacuum, even arbitrarily close to the vacuum state. We construct physical counterexamples and identify the competition that controls their entropy: the relative entropy to the vacuum phase-space density can exceed twice the mean photon number. A two-level family exhibits a finite window in which coherence lowers entropy while preserving global Wigner positivity. A complementary construction repairs a remote negative tail with an exponentially small thermal admixture; an explicit mixing weight of 2 × 10−21 preserves a rigorously established entropy decrease. Optimisation over all one-mode Wigner-nonnegative states gives the sharp low-energy scale Eγ/[ln(1/E)]β, with γ ≃ 0.7412 and β ≃ 0.5861. Because γ < 1, tensor products can have vanishing total energy and trace distance from vacuum while their entropy deficit diverges. We derive the exact half-transmission threshold for universal Shannon-entropy recovery and connect it to companion results showing residual non-Gaussian structure. The counterexamples reveal distinct controls on entropy: coherence sets the local descent, mode number amplifies it, and loss restores the vacuum bound. Version history. This is an expanded revision of arXiv:2609.13312v1, submitted on 10 September 2026 under the title "Physical Counterexamples to the Wigner Shannon Entropy Conjecture". That version already presented physical Wigner-positive counterexamples, finite-support families, a sixth-order one-photon-core construction, minimum thermal-repair asymptotics and the sharp one-mode low-energy deficit scale. The present revision develops the dimensional and loss-dependent consequences, supplies explicit rational entropy-sign certificates and a smaller thermal-repair example, and connects these results to the companion studies cited in the manuscript.