A Finite Dyadic Pregeometry over Spacetime: A Non-Continuum Ansatz for Unified Field Theory

This paper proposes a finite dyadic pregeometry over a spacetime manifold as an ansatz for unified field theory. A connected smooth four-dimensional manifold M supplies macroscopic localization, ordering, and refinement structure. Fundamental physical data at fixed resolution are finite: each observational cell carries a six-bit observer-system register G = F_2^3 + F_2^3, edges carry dyadic transition labels, and faces carry consistency defects F = dA. The 64-state local register is derived from first principles. Binary distinguishability gives F_2 variables; a record-bearing subregister requires a nondegenerate complementary dyad cycle; an n-bit subregister has 2^(n-1) complementary dyads; the minimum cyclic structure with adjacent and opposite dyad relations requires four dyads, hence n = 3; observer-system coupling gives 3 + 3 = 6 bits and 2^6 = 64 local states. Continuum fields, variational action functionals, gauge-redundant representations, and gravitational field equations are treated as reconstruction targets in suitable scaling regimes. At the foundational level, the unified object is a dyadic cochain system on finite covers of M. Matter-like persistence corresponds to stable vertex-register content, interaction-like change to edge cochains, curvature-like behavior to face defects and path dependence, and measurement to observer-mask quotients of system registers. Elementary propositions establish minimality, observer projections, path independence, Walsh diagonalization, finite information pseudometrics, and additional structural constraints.