Why Order Formation is Universal: Analyticity and Constraint Morphisms

Why do physically distinct systems—gravitational clustering, neural synchro-nization, bacterial swarming—admit similar mathematical descriptions? We providea rigorous answer grounded in the analyticity of physical law: morphisms betweenmodels of order formation exist because smooth (analytic) dynamics admit linearapproximation near fixed points.We formalize the Order Formation Category OFC, where objects are dy-namical systems with bifurcation structure and morphisms are limiting proceduresthat preserve constraint geometry. We prove that each morphism in the OrderEmergence Structure (OES)—connecting Newton →HMF →Kuramoto →Vic-sek →MIPS—arises from the analyticity of dynamics: overdamped limits exploitadiabatic slaving of fast variables, phase reduction exploits smooth phase responseto weak perturbations, and coarse-graining exploits Taylor expansion in gradients.Crucially, this analyticity near fixed points is distinct from global behavior: theorder parameter still exhibits nonlinear power-law scaling r∼(λ−λc)1/2 at phasetransitions. The morphisms preserve this critical behavior precisely because theyrespect the underlying constraint geometry.We derive explicit error bounds for each morphism using singular perturba-tion theory and validate predictions numerically. Extension to biological systems(honeybee learning, slime mold optimization) and cognitive systems (Free EnergyPrinciple) demonstrates substrate-independence of the constraint geometry.This framework supports causal eliminativism: order emerges not through cau-sation but through constraint satisfaction, made universal by the analyticity ofphysical law.