Discrete Coarse-Graining, Scale Invariance, and the First-Principles Derivation of the Speed of Light from Computational Finitism From Planck-Scale Pixels to Macroscopic Physics

The reconciliation of discrete quantum mechanics with continuous general relativity remains the paramount challenge in theoretical physics. Standard approaches either assume continuity from the outset or struggle to recover it from discrete models without invoking infinite limits. In this paper, we advance Computational Finitism, replacing infinite mathematical artifacts with rigorous discrete coarse-graining applied to the Golden-Fano Automaton. Through four computational milestones, we prove that: (1) the smooth continuum of spacetime is a macroscopic illusion generated by the statistical averaging of a discrete Planck lattice, with Golden-Fano achieving a uniquely clean continuum limit; (2) the emergent macroscopic field strictly rejects Gaussian statistics, exhibiting compact support and a "Topological Condensate" structure bounded by a 0.45 physical delta; (3) the discrete topology is a scale-invariant Renormalization Group (RG) fixed point, exhibiting a distinct "Crossover Regime" during coarse-graining; and (4) for the first time in physics, the speed of light limit () is derived from first principles as the maximum information propagation speed of the discrete lattice (= 1 node/tick), complete with a rigorous proof of the "Macroscopic Blurring Effect." This work establishes that relativistic limits and continuous fields are not fundamental postulates, but inevitable, emergent properties of a strictly finite computational engine.