Adaptive systems governed by hierarchical precision regulation are commonly describedin terms of stability, error minimization, and resource-constrained optimization. Yet thestructural conditions under which such systems undergo qualitative reorganization remainunderexplored. This paper demonstrates that precision-regulated hierarchical systems ad-mit genuine structural phase transitions—critical reorganizations of inferential architecturearising from constrained precision redistribution.Under finite global precision capacity, redistribution eigenmodes generically determinestability, and their eigenvalue crossings constitute the precise spectral mechanism of struc-tural phase transition. We formalize a general class of precision flow equations on hierar-chical inference systems subject to finite capacity constraints. Analyzing the Jacobian ofthe constrained dynamics, we derive explicit stability conditions and identify bifurcationpoints at which redistribution eigenmodes lose stability. At these critical thresholds, smallshifts in precision allocation can induce large-scale reconfiguration of hierarchical dominance,corresponding to the emergence of new effective inferential modes.This analysis reframes adaptive regulation as a dynamical balance between stabilityand generativity. Stability is maintained within bounded regions of the precision manifold;beyond critical boundaries, the system reorganizes endogenously. By embedding precisionregulation within constrained dynamical systems and bifurcation theory, this frameworkdemonstrates that structural reorganization is not an exception to adaptive inference, butan intrinsic consequence of finite precision capacity.
Takashi Kubo (Sun,) studied this question.