Toward a Stochastic Number System Emergent Arithmetic and Fluctuating Peano Axioms via Ising like Continuous Cellular Automata

This paper introduces a conceptual framework for a Stochastic Number System, where numbers and foundational arithmetic operations are not treated as static, deterministic scalars, but as emergent statistical phenomena arising from global thermodynamic and axiomatic constraints over a continuous complex field. Inspired by Ising spin interactions and continuous cellular automata, we formulate a toy model utilizing a Gaussian Free Field (GFF) to demonstrate how fundamental mathematical truths (such as 1 + 1 = 2) dissolve into Gaussian probability distributions under thermal and quantum fluctuations. In this framework, classical deterministic arithmetic is naturally recovered as a zero-temperature (T -> 0) ground state limit through a mechanism analogous to quantum decoherence. By bridging mathematical logic, statistical physics, and quantum foundations, this work provides novel theoretical insights into Planck-scale quantum gravity, the thermodynamic limits of computation (Landauer's Principle), and stochastic axiomatics.