The Dimensional Invariant: Sleep Duration, Sleep Topology, 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.