Coherence Vortices. A Mathematical Grammar for Post-Critical Emergence

Abstract

Complex systems approaching a critical transition are often studied through the mechanisms that constrain, destabilise, or collapse their previous organisation. This article examines the complementary question: how does a new collective regime become organised after the critical passage? Building on the Drain Vortex Principle, we propose Coherence Vortices as a transversal mathematical grammar for post-critical emergence. The grammar follows recurrent functional relations---nucleation, coupling, coherence, stabilisation, memory, and contextual compatibility---while allowing each domain to retain its own entities, mechanisms, scales, observables, and characteristic times. A continuous organisational restructuring provides the narrative thread, contrasted with physical, chemical, biological, neural, and medical systems. Landau-type landscapes, network coupling, the Coupling Factor, contextual coherence measures, Kuramoto dynamics, attractor stability, hysteresis, coarse-graining, and Turning Points provide complementary mathematical tools rather than a single universal formalism. Change (rho) and resilience (eta) are introduced as transversal dimensions describing how emerging organisations respond to perturbation and contextual transformation. The framework further distinguishes persistence from contextual compatibility, allowing coherent and resilient regimes to remain locally organised while becoming incompatible with the larger whole. Finally, backward and forward fertility are used to evaluate whether the grammar can illuminate known processes and generate new concepts, observables, questions, and research pathways.