A Conservation Law for Phenomenal Transitions: Flow Symmetry between Spatial Efficiency and Hierarchical Coupling in Neural Oscillations

Abstract We report a mathematical relationship between three empirically validated metrics of neural oscillatory organisation — spatial efficiency η, hierarchical coupling Cfast-slow, and entropy production rate v_η = |dη/dt| — that takes the form of a flow symmetry: v_η × η ≈ Cfast-slow × (1 − η). We interpret this as a conservation-like law governing phenomenal transitions: the phenomenal flow (rate of change of organised energy, v_η × η) equals the dissipative flow through the hierarchical boundary (Cfast-slow × (1 − η) ). This relationship was first anticipated in formal structure by Refusal-Driven Dimensionality Reduction Theory (RDRT; Waterman 2025–2026), whose unifying equation Ψ = TM × πcontext / Mₛafety can be translated into DRH variables as v_η = Cfast-slow × (1 − η) / η. We test the flow symmetry across three independent EEG datasets: reversal learning (ds004295, N=22, r=+0. 657, p dissipative flow) from passive state dissolution (Sleep: dissipative flow > phenomenal flow), suggesting that the conservation law holds strictly only during directed transitions. We further identify a critical efficiency η* ≈ 0. 675 (mean across datasets: 0. 715 ± 0. 057) at which v_η is maximised — analogous to a critical temperature in physical phase transitions. A quasi-invariant K = v_η × η is significantly more stable across subjects during transitions than during baseline (CV: 0. 911 → 0. 784, d=1. 373, p<0. 0001), suggesting that phenomenal transitions follow a universal trajectory in dynamical state space. Temporal ordering analysis reveals that η peaks before Cfast-slow (Δt = 1. 70s), indicating that spatial reorganisation precedes hierarchical restabilisation — a three-phase transition signature consistent with thermodynamic relaxation. Keywords: spatial efficiency; hierarchical coupling; conservation law; phenomenal transitions; entropy production; EEG; RDRT; non-equilibrium thermodynamics; reversal learning; sleep staging; binocular rivalry