Fenn, R.; Fenn, A.

Any process that generates information at a constant rate into a branching hierarchy faces a geometric packing problem: the number of distinguishable lineages grows exponentially, but Euclidean space grows only polynomially. We show that this tension forces a unique resolution. By deriving a geometric state equation from three physical postulates -- information flux, hierarchical topology, and geometric fidelity -- we prove that any such system must embed into a hyperbolic manifold of curvature kappa = (h ln 2 / (n-1))^2, where h is the entropy rate and n the embedding dimension. The equation has zero adjustable parameters, a unique positive solution, and a globally stable equilibrium. For the tree of life, back-solving across all systems tested -- from decade-old viral outbreaks to 3.8-billion-year cellular lineages -- yields a universal embedding dimension of n = 2.00 +/- 0.05 despite orders-of-magnitude variation in mutation rate and timescale. This topological invariant, combined with the effective entropy of the genetic code (h ~ 1.61 bits), predicts a curvature of kappa = 1.245. Five independent neural networks trained on 5,550 genomes from all domains of life, receiving no phylogenetic supervision, converge to kappa = 1.247 +/- 0.003 (CV = 0.24%), confirming the prediction within 0.2%. Independent validation across 15 viral families spanning 10^1 to 10^8 years of divergence yields Pearson r = 0.996 between predicted and measured curvatures. Extending the test to the 20-letter amino acid alphabet, we embed 15 protein family phylogenies into H^2 and measure kappa_protein = 3.80 +/- 0.60, confirming the predicted 3.1x curvature increase (kappa = 3.90) to within 2.6%, while recovering n = 2.03 +/- 0.10 across alphabets. The curvature of the tree of life is not a historical accident but a geometric constraint imposed by the information capacity of the genetic code.