Physics as Stable Sobolev-Order Structure on a Discrete Coherence Lattice A Mathematical Foundation for the Sobolev-Ozok Lattice Framework

This work presents a mathematical foundation for the Sobolev–Ozok Lattice (SOL) framework, in which physical laws emerge from the stability of a hierarchy of operators acting on a discrete coherence field defined over a Planck-scale lattice.

In this framework, a scalar coherence field evolves on a discrete structure, and its behavior is governed by a multi-order operator system. Different operator levels correspond to different physical regimes, including propagation, curvature, and localization.

A central result of the paper is the formulation of a universal field equation that governs the behavior of the coherence field across all scales. From this equation, key physical structures emerge naturally.

The framework demonstrates:
- Emergence of geometric structure from coherence gradients
- Interpretation of curvature as a second-order effect of coherence
- Recovery of the Newtonian gravitational limit
- Identification of gravitational coupling as an emergent property of the system

In particular, gravity is interpreted as a manifestation of coherence stiffness, with the gravitational constant arising from the underlying operator structure rather than being assumed as fundamental.

The paper also introduces a spectral stability principle, where physically observable states correspond to stable modes of the operator hierarchy. This provides a connection between classical physics and quantum-like behavior within a unified mathematical setting.

This work does not aim to replace existing physical theories, but instead provides a foundational mathematical perspective in which classical and quantum physics can be understood as different stability regimes of a single underlying structure.

The SOL framework offers a new way to think about the emergence of geometry, matter, and physical interactions from discrete coherence dynamics.

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/