The Sobolev-Ozok Lattice (SOL): A Minimal Axiomatic Framework for Emergent Spacetime, Propagation, and Curvature

This paper presents a minimal axiomatic formulation of the Sobolev-Ozok Lattice (SOL), a discrete theoretical framework in which spacetime, propagation, and curvature emerge from a single scalar coherence field defined on a Planck‑scale lattice. Instead of assuming a pre‑existing manifold, metric, or separate matter and geometry fields, SOL starts from seven primitive axioms: a discrete substrate, a scalar coherence field, bounded local updates, interface flux, global conservation, a Sobolev energy hierarchy, and a criterion of physical admissibility. Within this closed system, coherence is given a unified definition as the transmissibility of local state updates across lattice interfaces, showing that gradient, flux, and transmission‑based descriptions are equivalent projections of the same underlying structure. From this axiomatic core, the paper derives: the first‑order (k = 1) sector, describing propagation, invariant signal speed, action closure, coherence stiffness, and attenuation; the second‑order (k = 2) sector, where curvature, an emergent metric, and Einstein‑like geometric closure arise in the coarse‑grained limit. Propagation and geometry are thus shown to be successive Sobolev orders of one scalar field, rather than independent physical domains. The equivalence principle, causal structure, and gravitational coupling emerge as consequences of coherence conservation and bounded update rules, rather than as separate postulates. The purpose of this work is not to review all SOL applications, but to isolate a minimal, closed foundational core from which previously developed SOL results such as wave dynamics, Planck‑scale limits, curvature emergence, and gravitational coupling can be understood as consequences of a single axiomatic system. As such, the paper serves as a foundational entry point and organizational reference for the broader SOL research program. Declaration of Tools Used: This manuscript was prepared and typeset using LaTeX via Overleaf. Language refinement and stylistic polishing were assisted by the Overleaf AI Editor. All scientific content, mathematical derivations, conceptual development, and conclusions are original and authored by the undersigned. This paper is part of the Sobolev-Ozok Lattice (SOL) research program. Project webpage (papers, figures, updates): https://ozokozcasol.github.io/Sobolev-Ozok-Lattice/