The Unified Attractor Grammar: A Scale-Free Generative Theory of Stability, Curvature, and Emergence

Attractor–ridge structure is a universal organizing principle across biological circuits, optimizationlandscapes, governance substrates, and computable physical systems. Despite this cross-domainrecurrence, no general grammar explains why stable states form, how boundaries persist, or howcurvature shapes transitions. We introduce the Unified Attractor Grammar (UAG), a minimal generativerule-set that governs the emergence, stability, and evolution of attractors in any rule-drivendynamical system. The grammar consists of four primitives—rule coherence, boundary integrity,feedback sensitivity, and continuity stability—which together guarantee the existence of a stable informationalfixed point and determine the geometry of its basin. Using hypercomplex perturbation,we compute exact curvature in a single evaluation, providing the differential structure that links localridge geometry to global attractor topology. We demonstrate the grammar’s scale-free nature acrossdomains: developmental fate decisions in vertebrate sex determination, curvature-aware optimization,governance substrates, and computable universes. The UAG provides a principled foundationfor understanding how systems maintain identity, evolve under perturbation, and converge towardstable informational structures.