Evolution as Active Geometry: The Geometric State Equation of the Tree of Life

Any process that generates information at a constant rate into a branching hierarchy must embed into hyperbolic space: exponentially growing lineages cannot pack into polynomial-growth Euclidean geometry. We derive a geometric state equation,{kappa} = (h ln 2/(n-1))2, relating the curvature{kappa} of the embedding manifold to the entropy rate h and dimension n, with zero adjustable parameters. Back-solving the dimension across every system tested--from decade-old viral outbreaks to 3.8-billion-year cellular lineages to domain-level species phylogenies--yields n = 2.00 {+/-} 0.05: evolution is two-dimensional. Curvature, by contrast, is scale-dependent. At the inter-domain scale, a neural encoder trained on 5,550 genomes with no phylogenetic supervision finds an optimal curvature range{kappa} {approx} 1.28-1.34, set by the Kolmogorov complexity profile of the biosphere; at the intra-domain scale, direct [H]2 embeddings of the complete GTDB bacterial (107,000 tips), archaeal (5,900 tips), and fungal (1,600 tips) species trees yield{kappa} = 3-16. Both regimes obey the state equation at the scale-appropriate entropy rate. Fifteen viral families trace the predicted curvature-entropy curve at Pearson r = 0.996; fifteen protein families confirm the predicted 3.1x curvature increase from a 4-letter to a 20-letter alphabet. The universal invariant is the dimension, not the curvature. The geometry of the tree of life is not a historical accident but a constraint imposed by the information capacity of the genetic code. Graphical Abstract O_FIG O_LINKSMALLFIG WIDTH=174 HEIGHT=200 SRC="FIGDIR/small/710612v2_ufig1.gif" ALT="Figure 1"> View larger version (40K): org.highwire.dtl.DTLVardef@148fbf6org.highwire.dtl.DTLVardef@95f2e9org.highwire.dtl.DTLVardef@7ea052org.highwire.dtl.DTLVardef@14f8999_HPS_FORMAT_FIGEXP M_FIG C_FIG The tree of life embeds into 2D hyperbolic space with curvature determined by the geometric state equation{kappa} = (h ln 2)2. Top: Voronoi tessellation of 5,550 genome embeddings in the Poincare disk, colored by domain (Bacteria, Archaea, Eukarya). LUCA occupies the center; cell boundaries are hyperbolic geodesics (circular arcs orthogonal to the disk boundary). Bottom: The state equation predicts curvature from entropy alone across a 13-fold range--from the compressed inter-domain hierarchy ({kappa} {approx} 1.3) through domain-level species trees ({kappa} = 3-16)--and across both DNA and protein alphabets (3.1 x increase), with zero adjustable parameters. Fifteen viral families confirm the curvature-entropy curve at r = 0.996.