β⁰ Has Two Geometries

The Toroidal Extension of the L1/L2 Framework MATH-11 established that β = π/4 is the unique L1/L2 conversion factor on circular geometry. Every factor of π in a physics formula traces to this conversion — an angular integration over a circular or spherical subspace performed in rectilinear coordinates. The classification worked: tag each term in a QED coefficient by its π content, and the spherical angular structure is revealed. But the classification had a blind spot. Terms without π — the β⁰ sector — were labeled "no angular content." This implied they were geometry-free: rational numbers from diagram counting, ζ values from radial integrations, polylogarithms from momentum configurations. No angles, no geometry. This is wrong. A torus has angular structure. You can integrate around a torus. The integral is well-defined, finite, and measures a geometric property of the manifold. But the integral does not produce π. It produces K(k) — the complete elliptic integral of the first kind. MATH-11's classification detected spherical angular content (π) but was blind to toroidal angular content (K(k)). The β⁰ sector was not geometry-free. It contained geometry that the spherical framework could not see. Package Contents:* `manuscript.md`: Paper* `README.md`: Overview