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, $\alpha=2-\delta$ with $\delta\downarrow0$, the mode scale for a fixed positive deficit is $\Theta(\delta^{-1}\log(1/\delta))$ under any fixed positive total-energy budget. Pure loss provides a dimension-independent control: for every $1\le\alpha<2$, it restores the vacuum entropy bound at transmissivity $\eta\le\alpha/2$, whereas above this threshold 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 $\Theta(m)$ 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 1.3 — 28 September 2026. This 86-page revision establishes the exact pure-loss recovery boundary throughout Shannon and Rényi orders $1\le\alpha<2$. It adds the exact unrestricted absolute-Wigner-norm growth rate and finite-mode, energy-constrained upper bounds that exclude a prescribed entropy deficit above the recovery boundary. Complete proofs connect these bounds to the quantitative figures. The revised figures also explain near-vacuum entropy divergence, structural survival on the recovery boundary, thermal repair of a remote negative tail, and finite-source loss scans with explicit error and raw-trial budgets. The PDF contains the main article, Methods, Supplementary Information 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.