The Dimensional Invariant: Sleep Duration, 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. The derivation, its claims, and the date of derivation are the priority-establishing record. Literature confirmations are recorded separately within the document. The attractor geometry framework (S + N + G → R), developed across a series of prior documents (theory paper: doi:10.5281/zenodo.19094935), predicts that sleep duration is a measurement of the total navigational cost of an organism's attractor basin position, decomposable into three axes: between- basin switching cost (Axis 1), within- basin metabolic cost (Axis 2), and within- basin computational intensity (Axis 3). This document extends the framework to its full temporal scope and proposes a dimensional invariant: across all life, at all scales, across all of deep time, sleep duration (or metabolic rest-state duration in pre-neural organisms) is a proxy for attractor basin depth, and basin depth is the primary predictor of long-term survival. The relationship between sleep and survival is U-shaped with a flat minimum. Two convergent survival strategies are identified and formalised: (1) The Lock — deep single basin, minimal sleep, perturbation-invariant, maximum survival horizon (300–600 Myr+), confirmed independently across at least seven animal phyla (Porifera, Cnidaria, Mollusca, Chelicerata, Chondrichthyes, Sarcopterygii, Lepidosauria); (2) The Navigator — shallow multiple basins, moderate sleep, ecological generalism as Basin Ω of ecological flexibility, long survival horizon (100–300 Myr), confirmed independently across at least six lineages. An unstable zone between the two strategies — the narrow specialist neither deep enough to be a Lock nor flexible enough to be a Navigator — is identified as the region where mass extinction concentrates its casualties. The primary pre-specified prediction (P-INVARIANT-1): a cross-phyla analysis controlling for body mass and metabolic rate will show a negative correlation between sleep duration and lineage age, detectable across all animal phyla simultaneously. This prediction is not generated by any existing sleep function theory. It requires the dimensional invariant framework to generate it. Five additional predictions are stated (P-INVARIANT-2 through P-INVARIANT-6), including: mass extinction survivor lineages will sleep less than closely related casualties; the unstable zone boundary is predictable from sleep duration and dietary breadth alone; Lock organisms are undomesticable; and the dimensional invariant is detectable in single-celled organisms via metabolic rest state duration, below the level of nervous system entirely. 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. The unified statement: the attractor geometry framework is a single geometric principle operating identically at every scale at which life organizes itself. Scale is the only variable. The geometry is invariant. Theory paper: doi:10.5281/zenodo.19094935 Repository: github.com/Eric-Robert-Lawson/attractor-oncology