The Systemic Coherence Function: Gradient-Flow Modeling of Coherence v4

Coherence-the emergence of stable, aligned, and integrative structure-appears across physical, biological, neural, cognitive, and social systems. Yet most disciplines treat coherence indirectly through domain-specific constructs such as order parameters, synchronization indices, patternforming instabilities, or consensus variables. This work introduces the Systemic Coherence Function (SCF) as a phenomenological gradient-flow framework that treats coherence itself as the modeled field. We define a scalar coherence field Φs(r, t) governed by a gradient flow on a coherence-tension functional F [Φs]. The resulting SCF equation is a driven nonlinear parabolic partial differential equation combining diffusion, nonlinear local relaxation, external drive, and structural forcing in a substrate-generalizable form. We provide operational definitions of Φs across domains, sketch a generic microscopic-to-macroscopic route to an SCF-type equation, and develop a concrete neural case study: starting from a spatially extended Kuramoto model, we apply an Ott-Antonsen reduction [18, 19], write z = Φse iΨ for the local complex order parameter, and obtain an approximate evolution equation for ∂tΦs in a near-threshold regime. This yields an SCF equation with a quartic potential whose quadratic coefficient α(K, ∆) recovers the classical Kuramoto synchrony threshold, with Kc = 2∆/J0 in SCF parameters. We present numerical simulations of one-dimensional driven SCF systems with double-well potentials. A well-mixed model exhibits multistability and hysteresis in Φs as a function of drive amplitude. A spatially extended model with localized Gaussian drive shows coherence-front propagation from a seeded region, and the numerically computed coherence-tension functional decreases monotonically in time under time-independent forcing, confirming its Lyapunov character. We outline how SCF predictions can be compared directly to spatial Kuramoto simulations by matching front speeds and coherence profiles. Throughout, SCF is presented as a mathematically conservative but conceptually unifying modeling program: a canonical parametrization of coherenceseeking flows that allows coherence phenomena across domains to be studied in a shared field-theoretic language.