A Reduction Network for Order Formation: From Newton to MIPS

How are gravitational clustering, neural synchronization, and bacterial swarmingrelated? This paper presents a reduction networkan explicit diagram of limitingprocedures connecting models of order formation across three centuries of physics.We trace the paths: Newton →HMF (phenomenological modeling), HMF →Kuramoto (overdamped limit), Stuart-Landau →Kuramoto (phase reduction), Ku-ramoto →Vicsek (spatial extension), Vicsek →MIPS (coarse-graining). Each arrowspecies validity conditions and what is preserved or lost.While individual connections are well-established (Acebrón et al., Gupta et al.,Strogatz), the network structure itself has not been systematically pre-sented. We demonstrate its utility as a model selection guide: given a physicalsystem's properties (inertia, spatial structure, noise), one can identify the appropri-ate node and applicable theorems.The framework extends beyond physics. Shimizu's and Kuramoto's work hasbeen applied to organizational dynamics and economic systems, suggesting that themathematics of emergenceattraction, instability, symmetry breaking, saturationmay be substrate-independent.