Distinct stability laws for Wigner entropy and non-Gaussian structure

Proximity to the vacuum need not stabilise phase-space entropy. We construct states with nonnegative Wigner functions whose total mean photon number and trace distance from vacuum both vanish as the number of modes $m$ grows, while their Shannon entropy deficit relative to vacuum diverges. Their distance from the Gaussian convex hull therefore vanishes as the deficit grows. Near Rényi order two, $\alpha=2-\delta$ with $\delta\downarrow0$, matching bounds place the mode scale for a fixed positive deficit at $\Theta(\delta^{-1}\log(1/\delta))$ under any fixed positive total-energy budget. Pure loss marks a second boundary: it restores the Shannon vacuum bound for every input at transmissivity $\eta\le1/2$, while at each fixed $\eta>1/2$, Wigner-positive inputs of fixed total energy have unbounded deficits as $m$ grows. Under a fixed exponential photon-number moment bound, the output Wigner $L^\alpha$ norm is also bounded by the vacuum value at $\eta=\alpha/2$ in a mode-independent window near order two. Entropic recovery does not imply Gaussianisation: after half loss, a separate bounded-energy family retains a fixed positive trace-distance gap from the trace-norm-closed convex hull of Gaussian states. With actual success probability bounded below independently of $m$, the minimum number of output modes needed to preserve a fixed positive gap by Gaussian processing of one complete copy is $\Theta(m)$. A finite single-mode model predicts entropy recovery under loss while a non-Gaussian witness remains positive, with explicit sampling budgets. Dimensional amplification, loss-induced entropic recovery and residual non-Gaussian structure obey distinct stability laws. Version 1.2 — 24 September 2026. This 76-page revision adds three complete supplementary proofs: (i) recovery at transmissivity η = α/2 throughout 1 < α < 2 for the complete vacuum–single-photon sector, including arbitrary coherences and mixtures, with an explicit mode-independent gap; (ii) a same-state decomposition of the remaining Rényi entropy loss into a defective probability affinity, with variational and two-copy representations; and (iii) a uniform exclusion bound for low photon cores in the entropy-weighted norm, allowing entangled ancillas, together with a conditional complexity bound for the normalised actual rare tail. The abstract and figures are unchanged. The sharp full-entropy crossover constant and the unrestricted-input Rényi loss boundary remain open. 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.