Strict Locality and the Emergence of Linearity: An Information-Theoretic Account of Directional Observability

This paper investigates why strict locality—the principle that each elementary cell accesses only its own instantaneous state—should hold at all, and what structural consequences follow from it. Using a minimal discrete model (a periodic ring carrying one localized pattern observed by a coarse observer), the analysis derives three exact closed-form relations with no fitting parameters: position smoothness Delta_obs = v/L, zero-neighbor directional sufficiency S = 2(d-1)v/L, and a critical threshold at (d-1)v/L = 1/2 whose one-half value is the geometric signature of bilateral symmetry. Below the threshold the description is exactly linear; the threshold itself marks where directional information from opposing boundaries first overlaps. A conserved quantity S(L/ell) = 2(d-1)v/ell survives coarse-graining: although the sufficiency fraction vanishes macroscopically, its product with the coarse-graining width remains invariant, instantiating a two-scale (renormalization-like) structure in which the microscopic level is concealed from—not abolished by—a coarse observer. Every result is confirmed three independent ways (model, hand count, closed form), with a computational appendix and a public code repository. The work is the second in a classical-language series leading toward a probabilistic ontological framework, referenced only as an independent proposal; the derivations here are entirely self-contained.