The Coherential Engine: Mathematical Foundations and Computational Realization

This article introduces the foundational structure of Coherential Mathematics, a mathematical framework built upon three intrinsic regimes of form: the positive, the negative, and the transitional. These regimes are formalized through the Coherential Triplet (C+,C−,T)(\mathcal{C}^+, \mathcal{C}^-, \mathcal{T})(C+,C−,T), a categorical architecture that captures not only the existence of coherent structures but also their deformation, loss, and partial reconstruction. At the operational level, the theory develops three internal algebras—A+A^+A+, A−A^-A−, and AETA_{\mathrm{ET}}AET—which describe the dynamics of coherence, structural negativity, and transitional behavior. These components are linked by the Coherential Triangle of functors (F,G,K)(F, G, K)(F,G,K), establishing the core asymmetry between form, deformation, and reconstruction. The central result of the article is the Fundamental Theorem of Non-Commutativity, which shows that no real system can traverse the functorial cycle without irreversible structural loss. This irreversibility is quantified by the operator Δ\DeltaΔ, which measures the non-recoverable difference between ideal laminar form and actual structure. Classical mathematics emerges as the degenerate case Δ=0\Delta = 0Δ=0, in which negative and transitional regimes collapse and only perfect coherence remains. The theory naturally extends to proto-applications in physics, information theory, computation, complex systems, and emergence, offering a structural language for phenomena governed by asymmetry, dissipation, and irreversible reconfiguration. This work provides the mathematical foundation for the Coherential Engine—presented in a separate publication—and establishes the core formal principles of the broader Ontology of Coherence.