The Geometric Computer: Turing Completeness, Free Energy, and Learning in a Digital Brain on S3

In 1948 Shannon recognized that thermodynamic entropy and minimum de-
scription length are the same function, and that recognition unified physics and
information theory. This paper makes the same move for the geometry of com-
putation.
The function σ(compose(A, inverse(B))) — the geodesic distance from the
ground 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, because
the 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 the
opponent-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, hierarchical
abstraction, affect, neuromodulation, and self-observation — all from one arith-
metic operation, the Hamilton product on S 3. The brain is Turing-complete
by 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. The
architecture is a new computing paradigm: the first computer where the state
space is SU(2) — the quantum gate group — making quantum computation na-
tive to the substrate, and the first whose computational primitives are identical
to those of biological neural oscillators. The system runs in production, every
constant is derived from topology, every claim is experimentally verified.