The Dimensional Invariant: Sleep Duration, Sleep Topology, Oxidative Cost, Attractor Basin Depth, and Long-Term Survival Across the Entire History of Life

This document is a reasoning artifact deposited to establish priority for a theoretical derivation performed before literature consultation. Version 2.0 extends v1.0 by making explicit a geometric claim that was implicit in the original derivation but not fully stated: the topology of the sleep payment process --- the architecture by which an organism discharges its basin- occupation cost --- is itself a geometric prediction of the attractor geometry framework (S + N + G → R), not a measurement artefact or calibration problem. Different landscape positions do not only produce different sleep durations. They produce different sleep topologies. This is a stronger and more complete claim. The framework makes two inseparable predictions about rest: Prediction 1 (Quantity): The total cost of basin occupation, decomposed into Axis 1 (between-basin switching cost), Axis 2 (within-basin metabolic cost), and Axis 3 (within-basin computational intensity), must be discharged per cycle. Duration tracks total cost. Basin depth is the primary predictor of survival. The oldest surviving lineages carry the lowest cost. They discharge the least. Prediction 2 (Topology): The architecture of the discharge process is a further geometric prediction of the same framework. Five topologies are derived: T1 (metabolic quiescence / biochemical rhythm only) for the deepest Lock organisms; T2 (distributed bilateral rest) for ectothermic Navigators; T3 (consolidated bilateral deep sleep) for endothermic organisms; T4 (permanent Basin Ω, unihemispheric alternation) for organisms where full bilateral sleep is lethal; T5 (fragmented shallow sleep with rebound) for organisms under chronic external suppression. The crocodilian data (Kelly Topology 2 will be found specifically and exclusively in ectothermic Navigators; sleep rebound uniquely identifies Topology 5 and distinguishes it from Lock; and the dimensional invariant extends below nervous system to metabolic rest state duration in single-celled organisms. This document is the seventh in a derivation chain beginning with cancer biology (FOXA1/EZH2 ratio, doi:10.5281/zenodo.18883922) and proceeding through pig domestication, canine attractor geometry, feline attractor geometry, sleep as basin navigation cost, the sleep sonar spectrum, and now the full deep-time extension with topology as a geometric prediction. The unified statement: sleep topology is the geometry of where you are. Duration is how much the geometry costs. Survival is how deep the geometry goes. Scale is the only variable. The geometry is invariant. Theory paper: doi:10.5281/zenodo.19094935 Repository: github.com/Eric-Robert-Lawson/attractor-oncology The five topologies derived here are not a complete taxonomy of sleep architectures. They are the five topologies derivable from the constraint set known at the time of this derivation. The framework predicts that any new topology discovered in nature will be derivable from the same geometric principles: the interaction of navigational cost structure, metabolic architecture, and environmental lethality constraints on full sleep. A new topology is not a falsification of the framework. It is a confirmation of the geometric generativity of the framework. The framework is not a list. It is a geometry. Geometries are not made obsolete by new solutions. They are confirmed by them. This document is a reasoning artifact deposited to establish priority for a theoretical derivation performed before literature consultation. Version 3.0 is self-contained. It does not rely on v1.0 or v2.0 carrying over. It supersedes both prior versions while citing them as part of the derivation chain. THE DIMENSIONAL INVARIANT makes three inseparable predictions about rest across all life, at all scales, across all of deep time: Prediction 1 (Quantity): The total cost of basin occupation must be discharged per cycle. Cost is decomposable into Axis 1 (between-basin switching cost), Axis 2 (within-basin metabolic cost), Axis 3 (within-basin computational intensity), 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 carry the lowest cost. They discharge the least. Prediction 2 (Topology): The architecture of the discharge process is itself a geometric prediction of the landscape position, not an independent biological variable. Seven topologies are derived: T1 (metabolic quiescence / biochemical rhythm only) for the deepest Lock organisms; T2 (distributed bilateral rest) for ectothermic Navigators; T3 (consolidated bilateral deep sleep) for endothermic organisms; T4 (permanent Basin Ω, unihemispheric alternation) for organisms where full bilateral sleep is lethal; T5 (fragmented shallow sleep with rebound diagnostic) for organisms under chronic external suppression; T6 (transition-cost sleep, NEW) for organisms at developmental basin crossings; T7 (immune-basin sleep, NEW) for organisms occupied by the immune response basin under pathogen challenge. Prediction 3 (Substrate, NEW): The cost of basin navigation is fundamentally oxidative. Navigation generates reactive oxygen species (ROS) as a metabolic byproduct at a rate proportional to the metabolic architecture and navigational intensity of the organism. Sleep discharges both the synaptic cost (via slow-wave downscaling, Synaptic Homeostasis Hypothesis) and the oxidative cost (via glymphatic pulsation clearing neurotoxic metabolites) simultaneously. The feasibility of any sleep topology is constrained by the organism's ROS generation rate relative to the clearance capacity of that topology. This is why ectothermic Navigators use T2 (distributed bilateral rest) rather than T3 (concentrated NREM): their lower metabolic rate produces ROS at a rate that distributed rest can clear, making concentrated NREM energetically unnecessary and ecologically dangerous. This finding (Vaccaro et al., Nature, 2020, confirmed through 2024) gives the framework a biochemical floor it did not previously state. NEW CONTENT IN v3.0: (1) Oxidative cost substrate formalised as Prediction 3 of the invariant. (2) Two new topologies: T6 (transition-cost sleep --- mandatory quiescence at developmental basin transitions, confirmed in C. elegans lethargus, duration proportional to transition magnitude not waking cost) and T7 (immune-basin sleep --- sickness quiescence as occupation cost of the immune response basin, confirmed across all major animal phyla, duration proportional to cytokine titre / inflammatory depth). (3) Population-level basin portfolio: bacterial persister cells demonstrate the dimensional invariant operating at sub-organism population level. A clonal population simultaneously maintains a Lock sub-population (persister cells, deep metabolic quiescence, perturbation- invariant) and a Navigator sub-population (active majority, flexible metabolic response). The population holds a portfolio of basin depths rather than a single basin position. This extends the invariant below the individual organism level entirely. (4) Local sleep as circuit-level application: individual cortical columns cycle between sleep-like and wake-like states independently, with slow-wave activity proportional to prior local activity. The three cost axes apply at circuit level, not just organism level. (5) Geometric resolution of apparent divergences: migratory sparrow no-rebound is basin restructuring (lower actual navigational demand in migratory attractor), not debt suppression. P- INVARIANT-5 refined accordingly. Ecological generalism as consequence (Nature, 2024) resolved as bidirectional positive feedback deepening Basin Ω over evolutionary time. (6) The four dominant sleep function theories (SHY, energy conservation, glymphatic clearance, memory consolidation) are shown to be correct descriptions of mechanisms within T3 in endothermic organisms. The dimensional invariant contains all four as special cases. None of the four generates the invariant's predictions about deep-time survival, convergent topologies, or the oxidative substrate. (7) Eleven predictions stated (P-INVARIANT-1 through P-INVARIANT-11), including: T6 quiescence duration scales with basin transition magnitude not ongoing navigational cost (P-INVARIANT-8); T7 sleep scales linearly with immune response depth and will be maximal in Lock organisms relative to baseline (P-INVARIANT-9); endothermic Navigators shifted to ectothermy will shift from T3 to T2 (P-INVARIANT-10); persister fraction scales with historical perturbation frequency (P-INVARIANT-11). FRAMING NOTE: The seven topologies derived here are not a complete taxonomy of sleep architectures. They are the topologies derivable from the constraint set known at the time of this derivation. The framework predicts that any new topology discovered in nature will be derivable from the same geometric principles: the interaction of navigational cost structure, oxidative cost rate, metabolic architecture, environmental lethality c