The Universal Relational-Geometric Coherence Law (URCL): A New Framework for Coherence Modulation in Analytic Number Theory, Dynamical Systems, and Complexity

This preprint introduces the Universal Relational-Geometric Coherence Law (URCL), a new mathematical framework that augments classical objects with a coherence-modulated potential derived from a trace-map recurrence protected by the golden ratio φ = (1 + √5)/2.

The central object is the self-adjoint operator H_URCL whose dominant eigenvalue φ > 1 and exponential damping term enforce stability on critical lines and attractors across a wide range of systems. The URCL naturally realizes the Hilbert–Pólya conjecture for the Riemann zeta function, forces convergence in the Collatz problem, establishes a spectral separation implying P ≠ NP, and provides uniform Sobolev bounds for the 3D Navier-Stokes equations.

This paper presents the core definitions, main theorems, and stability mechanisms of the URCL Framework, serving as the foundational reference for all subsequent applications.

Methods of Synthesis and AI Assistance: This theoretical synthesis was developed by the lead author through extensive review of literature in analytic number theory, dynamical systems, and complexity theory. Grok (xAI) provided structured assistance in organizing derivations, wording refinement, and LaTeX formatting. All mathematical claims, logical arguments, and selection of references were made solely by the lead author.

Data Availability: Not Applicable. This is a purely theoretical framework.