Distinct stability laws for Wigner entropy and non-Gaussian structure

Proximity to the vacuum need not stabilise phase-space entropy. We construct Wigner-positive states whose total mean photon number and trace distance from vacuum vanish as the mode number m grows, while their Shannon entropy deficit diverges. Their distance from the Gaussian convex hull also vanishes. Near Rényi order two, α=2-δ with δ↓0, the mode scale for a fixed positive deficit is Θ(δ⁻¹log(1/δ)) under any fixed positive total-energy budget. Pure loss provides a dimension-independent control. The exact recovery threshold is half transmission for every 0<α≤1 and η=α/2 for 1<α<2; above it, deficits remain unbounded at fixed total energy. Along the recovery boundary, a distinct bounded-energy family retains a fixed positive trace distance from the closed Gaussian convex hull. Preserving a fixed positive part of this gap by Gaussian processing of one complete copy requires Θ(m) output modes when actual success probability is bounded below independently of m. A finite single-mode source model connects entropy recovery and structural survival to loss-dependent measurement predictions with explicit error and raw-trial budgets. Entropy and non-Gaussian structure thus obey distinct stability laws. Version 1.4 — 28 September 2026. This 87-page revision completes the pure-loss recovery boundary for all Shannon and Rényi orders 0<α<2: ηc(α)=1/2 for 0<α≤1 and ηc(α)=α/2 for 1<α<2. It adds a complete low-order proof, establishes unbounded entropy deficits above half transmission at fixed or vanishing total energy, and extends the structural-survival statement along the full recovery boundary. The abstract, introduction, theorem statements, Methods, Supplementary Information and loss phase diagram are updated together. The PDF contains the main article, complete proofs and references. Related counterexample paper. Wigner entropy below vacuum: physical counterexamples and stability limits, doi:10.5281/zenodo.22941099, is an expanded revision of arXiv:2609.13312v1 (10 September 2026). The present paper develops the dimensional, loss-dependent and structural stability consequences.