The Geometric State Equation
κ = (h ln 2 / (n − 1))²
observed everywhere: n = 2.00 ± 0.06
Hierarchy generates information
Information curves space
Curvature is intrinsic
scroll
Five Domains · One Curve · Zero Parameters
entropy rate h (bits/symbol) curvature κ AI κ ≈ 0.27 Neural κ ≈ 0.49 Evolution κ ≈ 1.25 Linguistic h = 1.65 → κ ≈ 1.31 h = 1.57 → κ ≈ 1.18 Protein κ ≈ 3.80 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 predicted, not fitted
The Geometry: H²
∂∞H² — the boundary at infinity

Every information-generating hierarchy lives in the same hyperbolic plane. Higher entropy pushes representations toward the boundary. The curvature is not imposed — it is measured.

The Terminus: Icosahedral Mind

At κ*, the boundary sphere S² admits an optimal partition into 12 regions.
An icosahedron. The same number as the senses of classical phenomenology.
8 of 10 predicted cross-modal edges confirmed empirically.

The Evidence
Domain System κ n Parameters Fitted
Evolution 107K taxa, 15 viral families 1.247 2.00 0
Protein 15 fold families 3.80 2.06 0
Language 34 families, 106K pairs (2 methods) 1.18–1.31 — 0
Neural fMRI, EEG, single-unit 0.49 1.96 0
AI 7 architectures (GPT → Mamba) 0.27 2.03 0
FIVE-DOMAIN CORRELATION: ρ = 0.984

One equation.
Three postulates.
Zero free parameters.

From genomes to grammars to minds —
the curvature was always the same.

κ = (h ln 2)²