Precision Conservation in Hierarchical Inference: A Geometric Redistribution Principle under Capacity Constraints

Adaptive theories of biological and artificial systems typically model hierarchical inferenceas minimizing prediction error or variational free energy, with precision (inverse prediction-error variance) treated as a modulatory gain. We reformulate this perspective by showing that,under finite resource constraints, hierarchical adaptation is governed by a geometric conservationstructure of precision allocation. When total allocable precision is bounded, confidence cannotbe uniformly amplified; it must be redistributed across hierarchical levels.Within a hierarchical variational framework, we derive coupled multi-time-scale dynamicsin which fast state updates attenuate precision-weighted prediction errors, while…