The Arithmetic Arrow of Time: Entropy as Deterministic Information Loss in a Finite Alphabet

Abstract

The Second Law of Thermodynamics dictates that entropy in an isolated system always increases (), giving rise to the "Arrow of Time." In standard physics, this is a statistical phenomenon: systems evolve toward macrostates with a higher number of microstates (Boltzmann). We propose a radical alternative: the Arrow of Time is not statistical, but arithmetic and deterministic. In a universe governed by a finite computational alphabet (Modulo-9), every mathematical operation (division, non-linear transformation) produces results that exceed the hardware's finite precision. These results must be rounded or truncated to fit the discrete state space. The discarded "residual bits" represent a permanent, deterministic loss of information. This irreversible truncation is the fundamental mechanism of entropy generation. We provide the mathematical proof of this "Residual Bit Loss," demonstrate its computational simulation, link it directly to via Landauer’s Principle, and unify it with the Fisher Collapse, the Menger Sponge void fraction, and Gravitational Wave echoes. Finally, we present falsifiable predictions regarding quantum computing decoherence floors and quantized black hole evaporation steps.