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Abstract 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, κ = ( h ln 2 / ( n −1)) 2 , relating the curvature κ 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 κ ≈ 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 κ = 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 3.1× 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 The tree of life embeds into 2D hyperbolic space with curvature determined by the geometric state equation κ = ( h ln 2) 2 . Top: Voronoi tessellation of 5,550 genome embeddings in the Poincaré 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 ( κ ≈ 1.3) through domain-level species trees ( κ = 3–16)—and across both DNA and protein alphabets (3.1 × increase), with zero adjustable parameters. Fifteen viral families confirm the curvature-entropy curve at r = 0.996.