Why Order Formation is Universal: Analyticity and Constraint Morphisms

Why do physically distinct systems—gravitational clustering, neuralsynchronization, bacterial swarming—share the same mathematics at phasetransitions? We show that the answer lies in a single property: the analyticity(smoothness) of physical dynamics. Near fixed points, analytic functions admitTaylor expansion, and this expansion structure alone determines the universalityclass.We formalize this insight through the Order Formation Category (OFC), whereobjects are dynamical systems with bifurcation structure and morphisms arelimiting procedures—overdamped limits, phase reduction, coarse-graining—thatpreserve constraint geometry. The Order Emergence Structure (OES) mapsthe resulting web: Newton → HMF → Kuramoto → Vicsek → MIPS. Each morphismarises from analyticity yet preserves nonlinear scaling r ~ (λ − λc)1/2.Comprehensive numerical validation across five systems confirms the framework:Stuart-Landau bifurcation (exact theory agreement), Kuramoto critical coupling(Kc = 2Δ within 2.5%), HMF→Kuramoto overdamped morphism (Kc preserved forγ ≥ 5), motility-induced phase separation (three-regime structure identified), andShimizu critical slowing down (τ ~ α−1 confirmed). Extensions to honeybee sociallearning, slime mold optimization, and the Free Energy Principle demonstratesubstrate-independence. The mathematics of order formation is one—a web ofmorphisms woven by smoothness.