Coherence Mathematics: The Formal Realization of Ontological Irreducibilities

Beginning the algebraic and structural development of Unified Coherence Theory from the axiomatic base Paper advances Unified Coherence Theory by initiating the development of coherence mathematics—the formal structures that realize the ontological foundations of Papers 1 and 2. Paper 1 established that becoming is ontologically prior to being and that coherence functions as the invariant ground from which symmetry, asymmetry, resonance, and the observer/consciousness polarity emerge as irreducible conditions. Paper 2 derived from these irreducibilities four minimal axiomatic requirements: relational continuity, complementary invariance and differentiation, mediation and transmissibility, and local reduction with reintegration. The present paper demonstrates that these requirements are realizable in disciplined mathematical form. We introduce the basic objects and operations of coherence mathematics: a relational field equipped with a coherence functional, dual symmetry/asymmetry transformations, resonance mediators, and paired reduction/reintegration maps. These structures constitute the first layer of disclosed mathematical ontology. They are compelled by, rather than imposed upon, the generative conditions previously identified. The deeper question of the foundations of ontological mathematics—the generative ground from which mathematics itself emerges and through which it discloses reality—is acknowledged as fundamental and is reserved for dedicated later treatment. The achievement here is the demonstration that the axiomatic requirements of Paper 2 admit coherent algebraic and structural realization, thereby securing a non-arbitrary formal base for subsequent physical and operator-theoretic development. Keywords: Unified Coherence Theory; coherence mathematics; ontological irreducibility; relational field; coherence functional; symmetry; asymmetry; resonance; reduction; reintegration; generative ontology