ENTROPIX/MESA Regulated Hyperscaling Violating Entanglement Entropy: An Exploratory Framework for Emergent Geometry, Information Flow, Open Quantum Systems, and Black Hole Evaporation

We propose an exploratory, information theoretic framework centered on soft regulatedhyperscaling violating entanglement entropy in open quantum systems. Gaussian form factors e−(kϵ)2 yield a removable quadratic expansion Sreg(ϵ) = S0+bϵ2+O(ϵ4) with |b| ≈ 0.50.75, providing bounded variance, monotonic export, and controlled UV suppression. Thisform aligns structurally with holographic renormalization in hyperscaling violating (HV) geometries, where subtracted integrals isolate universal finite parts (≈ 0.40091 for α = 2). Thestability coefficient γ ≈ 2|b| ≈ 1.0–1.2 emerges consistently from Gaussian variance andexport extremization. A constrained variational principle the Master Entropy Selection Algorithm (MESA) selects this functional uniquely under monotonic export and stationarityconditions (full elimination proof with stability checks provided). Downstream applications suggest mechanisms for emergent spacetime (Fisher-Rao metric with detailed geometry proofs), open system irreversibility via Lindblad master equation, cored galactic profiles(pseudo-isothermal benchmark), scale dependent couplings (compounded-exponential form),warped dimension stabilization, and exploratory quantum extremal surface (QES) phasetransitions in black hole evaporation (expanded stability proofs and high-precision graphs;minimal alternative to replica wormholes in AdS/CFT). The framework remains speculativein stronger extensions but offers sharp, falsifiable predictions in holography proxies, precisiontests, and analog systems