Coherent Field Ontology U: Structural Axioms and Dynamics

This work presents a speculative but structurally formalized framework for a continuous ontological field denoted by U, proposed as a foundational substrate underlying observable physical and mathematical structures. The document does not claim to provide a proof of the Riemann Hypothesis, a unified physical theory, or an experimentally validated ontology. Instead, it introduces a rigorously separated conceptual architecture intended to explore whether discrete structures may emerge naturally from a continuously vibrating, globally coupled field incapable of perfect equilibrium. The central premise is that absolute structural equalization in a universal vibrational continuum is dynamically impossible. As a consequence, local fluctuations arise spontaneously and may self-reinforce into regions of concentrated coherence (“prime-like structures”) or depleted coherence (“human zeros”). These structures are interpreted not as fundamental independent objects, but as emergent reorganizations of coherence inside a single indivisible field. The framework introduces: a continuous structural field U; local coherence amplitude A and structural phase Φ; an internal self-projection operator Π producing direct and mirrored modes; a non-local interaction kernel K governing redistribution and competition of coherence; emergence of discrete structures from intrinsic instability rather than imposed discreteness. A major emphasis of the work is epistemic discipline. The document explicitly separates: 1. Ontological hypotheses about reality; 2. Mathematical representations used instrumentally; 3. Existing formal mathematical objects such as the Riemann zeta function. The work rejects automatic identification between: mathematical primes and physical coherent structures; zeta zeros and coherence deficits in U; complex analytic constructions and physical ontology. Complex numbers and spectral mathematics are treated as representational tools rather than fundamental constituents of reality. The model is therefore intentionally non-dogmatic and remains exploratory. Its objective is not metaphysical closure, but the construction of a minimally consistent structural language capable of generating future testable dynamics. Open mathematical problems identified in the document include: stability of the proposed evolution equations; admissible forms of the non-local kernel K; spontaneous formation of coherent attractors; emergence of discrete spectra from continuous dynamics; possible mappings between spectral mathematics and structural projections of U. The framework should be understood as a research program in structural emergence, coherence dynamics, and continuous-to-discrete organization — not as a completed physical theory.