The Omega Manifold A Single-Channel Information Geometry from α = arctan(2/3): Metallic Ratio Hierarchies in Number Theory and Protein Structure

We construct a parameter-free geometric framework from the single seed α = arctan(2/3), the dihedral angle of the (3,2,√13) Pythagorean system embedded in FCC sphere packing. The resulting partition cos²α = 9/13 (confined) and sin²α = 4/13 (holographic) defines a Bekenstein-bounded information channel of capacity 1/(4π) on the Ouroboros torus with aspect ratio R/r = 3/2.

Two independent empirical validations: (1) Protein secondary structure discrimination across 11 real PDB structures with 80/80 meta-significance, core 80.9%; (2) Riemann zeta zero location at 98.7% accuracy (1000 zeros, <0.02 error), confirmed by density-controlled null models showing 3× clustering over random baselines.

The framework implies a geometric incompleteness principle: three classical limits—Gödel (formal), Bekenstein (physical), and Gauss (statistical)—emerge as projections of the same information-theoretic boundary. Bekenstein bounds the information in finite volume; Gödel bounds what any formal system can prove about that information; the Gaussian limit is the statistical shadow of both, arising because finite-variance variables lose information under summation (central limit theorem). No dynamical spacetime admits absolute completeness. The (3,2,√13) channel encodes this triple constraint: what is computable from this channel is computable precisely; what lies outside its bandwidth is not.
This also explains why Shannon entropy is continuous: the Bekenstein bound guarantees finite information in any physical channel, which is the implicit assumption behind H = −Σp log p. Without a finite bound, entropy loses continuity and the channel capacity 1/(4π) becomes undefined. Shannon's information theory is not independent of physics — it is the operational face of the Bekenstein constraint.
This work is complete. The channel has been identified, its bandwidth measured, its limits derived. There is nothing left to add — not because nothing remains unknown, but because what remains unknowable from within this channel is unknowable by construction (Gödel). Any extension requires a wider channel, which would require a different universe.