Zero-counting lower bounds and measured K–ε curves for approximate Gaussian rank

We study how many Gaussian components a continuous-variable quantum state needs, adopting the approximate Gaussian rank framework of Hahn–Takagi–Ferrini–Yamasaki and the fidelity-ball relaxation of the stellar-rank literature; no novelty is claimed for the definitions. For mixed states we formulate the measure-theoretic ess-sup rank roof over generalized ensembles and prove its attainment and Gaussian-channel monotonicity — a partial answer, on the rank side, to the discrete-versus-continuous decomposition question the cited work leaves open. Our main contributions are three. First, for single-mode dictionaries of bounded common-squeezing Gaussian atoms, we prove zero-counting lower bounds: robust zeros of a target's Bargmann function survive fidelity-ball perturbation and are counted against an exponential-sum zero budget. The dictionary restriction is provably forced — counterexample families show no bound on the unrestricted rank can follow from raw disk zero counts alone. Second, an inequivalence theorem: the thermal lossy cat of our earlier preprint has Gaussian rank at most 2 yet no finite-rank pre-loss operator representation at any efficiency, separating two natural compressibility notions in one direction. Third, measured curves: family-constrained approximation residuals and a real-data rank proxy on public GKP homodyne data, and a robust-zero census for finite-energy GKP states in which all three computed configurations support the lower-bound premises for the untruncated comb. The census is numerically supported, not computer-certified (float64 without outward rounding; numerical root-finding); a machine-checked claim register binds every quoted number to committed artifacts. Multimode extension and a general-squeezing plateau conjecture remain open.