Sleep as Attractor Basin Navigation Cost: A Unified Geometric Theory of Sleep Function, Topology, and Deep-Time Survival

This paper presents a unified geometric theory of sleep derived from the attractor landscape framework (S + N + G → R), in which the global attractor landscape G is physically implemented as the organism's epigenetic state, the coherence gradient field N is physically implemented as the circadian clock, social information, and light, and S is the structural substrate maintained by sleep-dependent growth hormone release and stem cell renewal. The framework makes three inseparable predictions about rest across all life: Prediction 1 (Quantity): The total cost of occupying a position in the attractor landscape must be discharged per cycle. Cost decomposes into three axes: between- basin switching cost (Axis 1), within- basin metabolic cost (Axis 2), and within-basin computational intensity (Axis 3), plus T6 transition cost and T7 immune-basin cost when applicable. Duration of discharge tracks total cost within a topology class. Basin depth predicts survival horizon. The oldest surviving lineages sleep the least. Prediction 2 (Topology): The architecture of the discharge process is a geometric prediction of the landscape position, not an independent biological variable. Eight topologies are derived: T1 (metabolic quiescence), T2 (distributed bilateral rest, ectothermic Navigators), T3 (consolidated bilateral deep sleep, endothermic organisms), T4 (permanent Basin Ω, unihemispheric alternation), T5 (fragmented shallow sleep with rebound diagnostic), T6 (transition-cost sleep, developmental basin crossings), T7 (immune-basin sleep, pathogen challenge), T8 (suspended state with blocked discharge, hibernation). Prediction 3 (Substrate): The cost of basin navigation is fundamentally oxidative. Reactive oxygen species (ROS) accumulate in proportion to navigational intensity. Sleep discharges synaptic cost (via slow-wave downscaling, Synaptic Homeostasis Hypothesis) and oxidative cost (via glymphatic pulsation) in the same window. The proximate death mechanism in sleep deprivation is ROS accumulation in the gut epithelium (Vaccaro et al., Cell, 2020). From these three predictions the paper derives: two deep-time survival strategies (Lock and Navigator) and an unstable zone; the dimensional invariant (oldest surviving lineages sleep least; minimum of sleep spectrum = maximum of survival spectrum); a complete three-channel sonar instrument for landscape position determination (duration, topology, behavioural flexibility); and the result that the four dominant sleep function theories in the current literature (SHY, energy conservation, glymphatic clearance, memory consolidation) are correct descriptions of mechanisms within T3 and are subsumed by the framework as special cases. The paper further identifies that the attractor geometry framework is internally consistent across independent derivation paths: the sleep geometry arrived at BMAL1 as the molecular N field governing cellular repair basin entry, which is geometrically identical to the structure independently derived in the Identity Axis Theorem (doi:10.5281/zenodo.18898788, 2026-03-07) from cancer molecular biology. Zero geometric incompatibilities were found across five contact points between the two frameworks. This convergence generates one novel prediction (P-CONVERGENCE-1): IAT drug efficacy will be inversely correlated with degree of G degradation at A gene loci, testable in existing H inhibitor trial datasets with matched epigenomic data. Eleven a-priori predictions (P-INVARIANT-1 through P-INVARIANT-11) and five closure predictions (P-CLOSURE-1 through P-CLOSURE-5) are stated. None are derivable from any existing sleep function theory. All were stated before literature confirmation was sought. This is a preprint intended for journal submission.