Matrix Formalization and Cyclicity of Coherence χ in Temporal Bifoliation Systems

We present a complete mathematical formalism for temporal bifoliation theory, a structural architecturethatresolvesthefundamentalincompatibilitybetweenmacroscopicreturnwithout restoring forces, persistence of microscopic memory, and global stability. We develop a matrix parametrization of coherence χ as a cyclic structural variable that modulates the coupling between operational (τ) and compensatory (ξ) temporal regimes. We introduce a non-circular redefinition of the structural interference magnitude ∥∆∥S as a composite functional that disaggregates geometric and informational costs through a dominance parameter λ ∈[0,1]. We derive fundamental equations for the evolution of χ, prove critical behavior theorems, and establish universality predictions based on λ. The implementation includes computational algorithms for calculating ∥∆∥S and simula- tions of the complete system. As paradigmatic examples, we analyze the perturbed harmonic oscillator and a nonlinear oscillatory model, showing how different λ values (0.2 for thermal perturbations, 0.8 for structural ones) reproduce distinctive physical behaviors, including multistability and self-sustained limit cycles. This formulation provides a unified operational framework that simultaneously explains phenomena currently treated as independent –quantum decoherence, BMS gravitational memory, non-Markovian processes– without introducing ad hoc mechanisms or unobserved restoring forces, establishing the foundations for an effective field theory of structural coher- ence.