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The paper investigates the role of transcendental constants in time-irreversibility and their effects on physical computation.

June 9, 2026Open Access

Irrational Numbers and the Arrow of Time: Algebraic Irreversibility and the Origin of Randomness in Physical Computation

Key Points

The paper investigates the role of transcendental constants in time-irreversibility and their effects on physical computation.
Proposes a theoretical framework combining transcendental constants and the Finite-Information State Hypothesis (FISH).
Analyzes the consequences of information loss in physical systems governed by irrational numbers.
Explores coarse-grained entropy and its implications for the arrow of time.
Identified that each truncation incurs information loss, contributing to non-decreasing coarse-grained entropy.
Established that representational constraints on physical law shift the explanation of the arrow of time.
Demonstrated that the incompatibility between irrational constants and finite information leads to a directed arrow of time.

Abstract

This is a preprint of the manuscript: "Irrational Numbers and the Arrow of Time: Algebraic Irreversibility and the Origin of Randomness in Physical Computation" Abstract:We argue that a previously underexplored source of time-irreversibility lies in the incompatibility between transcendental constants embedded in physical law and the finite information capacity of any bounded physical system. Every physical law contains dimensionless irrational constants such as π. Transcendence alone does not break time-reversal symmetry in exact equations; the irreversibility arises from the combination with the Finite-Information State Hypothesis (FISH), grounded in the Bekenstein bound. Any physically realizable instantiation of laws containing transcendental constants necessarily incurs irreducible information loss at each step. Each truncation is a many-to-one Voronoi projection on state space; iterated projections induce a generically non-decreasing coarse-grained entropy—a structural tendency that generates a directed arrow. The framework shifts the explanatory basis of the arrow of time from statistical typicality to representational constraints on physical law, and situates itself alongside Price, Callender, Prigogine, Gisin, and Castagnino and Lombardi. The manuscript has been submitted to a peer-reviewed journal and is currently under review. Keywords: arrow of time, irrational numbers, transcendental numbers, algebraic irreversibility, Bekenstein bound, finite-information reals, chaotic amplification, non-normality, coarse-grained entropy, prime infinitude, representational constraints

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