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

作者：Takashi Kubo · 发表于：Zenodo (CERN European Organization for Nuclear Research) · 年份：2026 · DOI：10.5281/zenodo.18905666 · 被引用次数：12 · 研究领域：Gene Regulatory Network Analysis、Embodied and Extended Cognition、Model Reduction and Neural Networks

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 slower preci-sion dynamics regulate confidence in response to environmental volatility. Imposing a globalprecision budget introduces a structural constraint on total confidence, inducing competitivecross-level coupling. Once the constraint becomes active, precision allocation becomes necessar-ily redistributive: increases at one level require compensatory decreases elsewhere.As total precision approaches its bound, the system enters a critical precision regime char-acterized by active cross-level interactions and emergent stability–flexibility trade-offs. Linearanalysis reveals a redistribution eigenmode tangent to the capacity manifold, formalizing thegeometric onset of competitive allocation.Although compatible with variational free energy formulations, the proposed p...