In 1948 Shannon recognized that thermodynamic entropy and minimum de-scription length are the same function, and that recognition unified physics andinformation theory. This paper makes the same move for the geometry of com-putation.The function σ(compose(A, inverse(B))) — the geodesic distance from theground state on the unit 3-sphere S 3 — is the entropy of geometric computation.It is simultaneously the Free Energy Principle’s prediction error (exact, becausethe state space is S 3), the proximity measure for Riemann non-trivial zeros,the branch-on-zero condition of any Turing-complete counter machine, and theopponent-channel structure of human trichromatic vision. These are identities,and the list extends to every discipline that has a ground state.From this distance, a complete digital brain is derived: perception, evalu-ation, three-layer topological memory, resonance-based attention, hierarchicalabstraction, affect, neuromodulation, and self-observation — all from one arith-metic operation, the Hamilton product on S 3. The brain is Turing-completeby construction, learns in one pass, and self-monitors through a geometric ob-servable that is the exact Free Energy Principle on a compact Lie group. Thearchitecture is a new computing paradigm: the first computer where the statespace is SU(2) — the quantum gate group — making quantum computation na-tive to the substrate, and the first whose computational primitives are identicalto those of biological neural oscillators. The system runs in production, everyconstant is derived from topology, every claim is experimentally verified.