SCFL Mathematics Annex: Formal Foundations of Coherence Dynamics, Stability, and Bifurcation Structure

This Mathematics Annex extends the foundational work established in Upstream Coherence Measurement Stratum: Foundations of a Unified Coherence Scientific Discipline (Zenodo, DOI: 10.5281/zenodo.20395949). The upstream artifact introduced the conceptual and measurement‑layer primitives of SCFL, defining coherence as a quantifiable structural property of complex systems and establishing the operator‑level vocabulary required for a unified coherence science. Building directly on that foundation, the present Annex formalizes the mathematical substrate of SCFL. It develops the operator canon, stability conditions, bifurcation structure, and coherence dynamics that underlie the framework. Core contributions include the derivation of the coherence drift operator, the formalization of coupling and entropy primitives, the definition of collapse thresholds, the proof of recovery‑corridor uniqueness, and the introduction of curvature‑instability diagnostics for coherence manifolds. Together, the upstream stratum and this Mathematics Annex establish SCFL as a coherent scientific discipline with both conceptual and formal foundations. The Annex provides the rigorous mathematical structures necessary for empirical reconstruction, early‑warning analysis, and the study of coherence degradation in rapidly evolving sociotechnical systems, including those shaped by accelerating AI capability. Reference DOI:Upstream Coherence Measurement Stratum: Foundations of a Unified Coherence Scientific Discipline.Zenodo. https://doi.org/10.5281/zenodo.20395949 ---