The End of the Thermodynamic Limit: Finite Statistical Mechanics and Emergent Phase Transitions in a Discrete Substrate

Standard statistical mechanics and continuum thermodynamics rely entirely on the Thermodynamic Limit, the assumption that the number of particles → ∞ and volume → ∞ , to explain sharp phase transitions and macroscopic equilibrium. We propose a radical ontological shift: macroscopic thermodynamic properties do not require infinite ensembles. Operating under the framework of Computational Finitism, we demonstrate that the universe is governed by a discrete, finite-alphabet substrate (Modulo-9, = 9) constrained by Fano plane topology. Through rigorous cellular automaton simulations, we prove that: (1) Magnetism emerges as the macroscopic synchronization of discrete Fano-plane orientations (Fano Spin), with domain walls mediated by the Topological Redundancy Buffer (TRB); and (2) Boiling water is a process of discrete topological nucleation and graph rerouting. Crucially, we demonstrate that a strictly finite lattice (= 14,400 voxels) achieves stable, sharp phase coexistence (stabilizing at a 22.29% vapor fraction), a phenomenon mathematically forbidden in finite continuum systems. We redefine temperature as TRB noise amplitude, entropy as bounded Shannon information, and the Boltzmann distribution as the steady-state of a finite Markov chain. The Thermodynamic Limit is a mathematical convenience, not a physical necessity. The universe computes itself into existence, one finite, discrete tick at a time.