A Mathematical Framework for Universal Operators

A Mathematical Framework for Universal Operators develops a substrate‑agnostic mathematical architecture for generative systems by formalizing the operator stack — orientation, separation, constraint, coherence, collapse, identity, duration, and evolution — as regime‑dependent transformations over a state‑space. The paper defines system states, constraint fields, coherence basins, collapse dynamics, identity fixed‑points, duration integrals, and emergent temporal residues, establishing a unified model of stability, transition, persistence, and novelty across physical, cognitive, social, and artificial domains. Constraint acts as the regulator of generative behavior, collapse provides resolution, coherence stabilizes patterns, identity emerges as a fixed‑point, duration measures structural persistence, and time arises as a residue of collapse events. The resulting framework is mathematically tractable, extensible to tensor and manifold formulations, and capable of supporting computational modeling of generative dynamics.