ENTROPIX / MESA Soft Regulated Entanglement Entropy in Open Quantum Systems: A Framework for Bounded Variance and UV Control

We present an exploratory framework for regulated entanglement entropy in open quantum systems, utilizing soft Gaussian regulators to achieve bounded variance and controlledUV suppression. Standard hard cutoffs yield power-law and logarithmic divergences withscheme dependent finite parts. Soft regulators (Gaussian form factors e−(kϵ)2, common intensor networks and lattice smoothing) exponentially screen high-momentum modes, yielding an effective expansion:Sreg(ϵ) = S0 +bϵ2 +O(ϵ4),where S0 is the physical entropy and b is a dimensionless coefficient. Across numericalmodels (amplitude damping channels, regulated harmonic chains), we find |b| ≈ 0.5–0.75,consistent with the theoretical prediction from Gaussian variance σ2 = 1/4, which gives theexpected range |b| ∈ [0.5,0.75] from boundary corrections.The quadratic form is removable (ϵ → 0 recovers S0) and practically useful for numericalstability. Preliminary checks demonstrate variance reduction of approximately 80% compared to hard cutoffs and consistency with monotonic export in open system flows. Theapproach remains scheme dependent and exploratory, applicable primarily to sub criticalor damped regimes where logarithmic terms are softened. Explicit falsifiability criteria areprovided throughout.