The Emergence of Formal Structure in Unified  Coherence Theory From Ontological Irreducibilities to Axiomatic  Foundations

Deriving the minimal axioms required by coherence as invariant ground and its complementary modalities.This paper continues the development of Unified Coherence Theory by deriving the first formal axiomatic layer from the ontological foundations established previously. In Paper 1 it was shown that becoming is ontologically prior to being and that coherence functions as the invariant ground binding symmetry and asymmetry across scales. The principal irreducibilities—coherence, symmetry, asymmetry, resonance, and the observer/consciousness polarity—were identified as primitive conditions that cannot be removed without explanatory loss. Here we demonstrate that these irreducibilities impose precise, non-arbitrary constraints on any adequate formal system. We articulate four minimal axiomatic requirements that follow directly from the ontology: relational continuity, complementary invariance and differentiation, mediation and transmissibility, and local reduction with reintegration. These requirements are not free stipulations but the first disciplined crystallizations of ontological necessity. The result is a principled starting point for formal development that prepares subsequent work in coherence mathematics, physical applications, and operator theory without anticipating their full technical content. Axioms in this framework are understood as stabilized expressions of generative ontological conditions rather than as arbitrary foundations. Keywords: Unified Coherence Theory; axiomatic foundations; ontological irreducibility; coherence; generative ontology; relational continuity; symmetry; asymmetry; resonance; reduction; reintegration