The β⁰ Frontier

MATH-11 established that β = π/4 is the unique conversion factor between L1 (taxicab) and L2 (Euclidean) metrics 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. This gives a classification tool for QED coefficients. Each term in the Schwinger series aₑ = Σ Aₙ (α/π) ⁿ can be tagged by its π content: β⁰: no π. The term comes from topology and number theory — Feynman diagram combinatorics (rational coefficients), nested radial integrations (ζ values), or specific momentum configurations (polylogarithms). No angular integration contributed. β²: contains π². One angular integration over one circular subspace of loop momentum. π² = 16β² — one L1/L2 conversion squared, or equivalently two L1/L2 conversions (one per angular coordinate on the 2-sphere). β⁴: contains π⁴. Two independent angular integrations. Each contributes π² = 16β². The classification counts how many times the computation converted from rectangular to spherical coordinates. More β powers means more spherical geometry in the diagram.