The Sobolev-Ozok Lattice (SOL) Model: A Discrete Framework for Spacetime and Energy Dynamics

The Sobolev-Ozok Lattice (SOL) model is a foundational theoretical framework that explores how effective spacetime structure may emerge from discrete Planck-scale dynamics. Rather than assuming a preexisting smooth manifold or background metric, the SOL framework reformulates the underlying substrate of physical space as a lattice of discrete cells governed by coherence and resolution constraints. Mathematical structure is introduced through discrete Sobolev norms, which provide a controlled way to characterize stability, regularity, and scale-dependent behavior across the lattice. In this work, a scalar coherence field defined on lattice sites serves as the primary organizing quantity. Finite differences of this field generate discrete analogues of gradients and Laplacians, allowing propagative behavior and curvature-like responses to arise as coarse-grained consequences of local coherence redistribution. A resolution hierarchy indexed by Sobolev order is introduced, where first-order coherence gradients govern bounded propagation, and second-order coherence variations encode effective geometric response in the continuum limit. Time and metric structure are not postulated a priori, but instead emerge as effective descriptions once stable coherence regimes are established. The present paper focuses exclusively on the mathematical formulation and internal consistency of the SOL framework. No claim is made that the model constitutes a complete physical theory, nor are phenomenological parameters, experimental fits, or force specific dynamics introduced. Instead, the objective is to establish a minimal and well defined structural setting in which familiar continuum concepts such as propagation, differential operators, and curvature can arise from discrete coherence dynamics. This work serves as the foundational formulation of the SOL program. Subsequent papers address extensions of the framework to dynamical fields, coupling structures, curvature dynamics, and phenomenological implications. 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.