Mutual Inductive Coherence Redefined: From Field Theory to Relational Emergence

Shimizu's concept of "mutual inductive coherence" (相互誘導合致, sōgo yūdōgatchi) describes how living systems achieve order through reciprocal influencewithout central control or predetermined plan. This paper reformulates the conceptwithin the relational emergence framework. We argue that mutual inductive coherenceis: (1) Kuramoto synchronization in relational space̶coupled oscilators reachingphase-locking without central clock; (2) the process by which β emerges frominteraction̶saturation parameters crystalizing through the synchronization processitself; (3) meaning co-creation through relational resonance̶significance arising atnodes where configurations mutualy amplify. The reformulation preserves Shimizu'skey insights̶emergence without design, order from mutual influence, the livingquality of coherent systems̶ while grounding them in the mathematical structure ofthe reduction network. This provides both formal precision and connection to thebroader framework of order formation across physical, biological, and social systems.