Emergent Arrow of Time from Finite Observers in a Closed Reversible Computational Universe: A Proof-of-Concept Phase Transition Framework

When the universe is modeled as a closed system governed by strictly reversible microscopic evolution, the origin of macroscopic irreversibility and the arrow of time remains a profound foundationalquestion. This paper introduces a discrete, closed computational universe model incorporating afinite internal observer. Observation is treated relationally: bidirectional coupling strictly altersboth the observer and the environment. Due to finite memory capacity, updating internal recordsinevitably involves irreversible overwriting. We parameterize this minimal penalty as an irreversiblebit cost (Cmeas). By defining a sustainability order parameter, netB(λ), as the net balance betweena phenomenological proxy for mutual information gain and this bit cost (where λ is the internaldynamic coupling strength), we translate the abstract notion of existence into a quantifiable phasetransition. Critical scanning precisely locates break-even points, revealing an exceptionally preciselinear scaling law, λ∗ ≈ a · Cmeas + b. Furthermore, embedding this bookkeeping into a resourcescalar with a density of states Ω(E) leads to the spontaneous statistical emergence of the conjugateparameter β, strictly satisfying ∆Emin ≈ β−1∆S. As a proof-of-concept, our central conclusionposits that the arrow of time is not a global property of the universe, but rather operationally corresponds to the unidirectional, perspectival bit efflux strictly required by a finite observer to maintaina correctable physical boundary.