The Thermodynamic Arc R = E/|O_info|: A Double Negative Characterization, a Corrected Lens, the Generalization Frontier of Sleep Invariants, and the Cost of Laminarization

Abstract

The ratio R = E/|O_info| - spectral energy per unit of integrated informational structure - was proposed as a candidate outcome observable of the thermodynamics of consciousness: an energetic cost per nat of integration. We report its complete empirical genealogy across eight pre-registered studies in public datasets (Sleep-EDF, 153 nights, N=76/77; DOSE-I propofol, N=170; 62-channel ketamine, N=10; DMT+harmine, N=31; paced-breathing HRV biofeedback, N=35), each with frozen criteria, synthetic sanity controls, and the subject (never the epoch) as statistical unit. The genealogy is a four-act arc. Act I (double negative): R is refuted as a scalar predictor (outcome correlation r=-0.36; denominator singularity) and refuted as a two-dimensional frontier with Gaussian-copula O-information - separability of wake vs NREM was entirely dominated by energy (AUC 0.826 for E alone; marginal contribution of O_info +0.004), and the ratio degenerated to E/epsilon (regularizer ~23x larger than the near-null copula denominator). Act II (lens correction): the copula is blind to non-Gaussian synergy - a non-parametric estimator reveals robust integrated synergy (median -0.228 bits, 100% of subjects, growing with NREM depth from -0.211 to -0.429 bits) that tracks level (rho=-0.269) and dissociates from the copula (cross-family correlation 0.063). With the corrected denominator, the plane becomes a genuine 2D frontier (AUC 0.904; marginal contribution +0.081, 19x the copula), and R_np monotonically orders the level of consciousness (median rho=-0.577) - a thermodynamic gradient, not a discrete frontier, with a descriptive cost minimum at N1/N2 and REM and a relative maximum at N4. The sleep field is locked in synergy (O_info<0 in ~100% of epochs), with small consistent intra-night hysteresis and inter-night stationarity. Act III (two generalization frontiers): the invariants do not generalize to the pharmacological axis at d=3 (propofol: both families near-null, not locked, energy carries all - informative null with the attenuation caveat declared), and - new in this version - the psychedelic field at d=5 reveals a second, qualitative boundary: the waking baseline resides in marginal redundancy (median +0.004 bits; 58% of epochs in synergy) and moves continuously toward synergy under DMT without locking - the sleep invariants are properties of the sleep field, not universal properties of consciousness level. Act IV (cost, new): in the cardiopulmonary layer, formal pacing in the resonance band raises coherence (dz=+0.45 per cell; +1.15 unbiased per segment) and simultaneously lowers the sample entropy of cardiac dynamics (0.251 vs 0.336; dz=-1.054, p=1.4e-07) - laminarization carries a measurable thermodynamic cost: order is maintained through flow, paid for in variability. The absolute laminar criterion (gamma^2>=0.5) is refuted in the RR proxy (real coherence 0.10-0.28; falsification clause fired as designed), and the unified regime reading consolidates: the coupling observable signs REORGANIZATION and organized STATES (-0.13 to -0.86 across ten axes), not traits (null, power 0.92), not level (sleep +0.145 ns; formal dissociation z=3.96-4.32), not dissolution (flattening with reversed signal under sedation). The arc methodological contribution - pre-registered refutation branches, mandatory marginal decomposition, blindness-vs-absence discrimination, and floor-declaring estimators - is itself the transferable result: how to kill an observable honestly, resurrect it with a better lens, map where its invariants end, and price its order.