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 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 principle doesnot depend on a specific objective function. Treating precision as a bounded dynamical variableyields testable predictions of cross-level gain redistribution and establishes a general geometricredistribution principle for resource-constrained hierarchical inference in neural and artificialsystems.