S³ geometry · 2026

The Projection Angle: α as a 4D→3D shadow of the S³ knot, and its running as the turning ring

R. G. Measey

Abstract

The fine-structure constant enters the confined-photon framework as a projection: α = tan β, where β = arctanα = 0.418° is the angle at which the S³ knot is viewed when its 4D geometry is cast into our 3-space. We make three claims and bound each honestly. First, a single power of α is a 4D→3D projection — it is structure, and its exponent is forced to be an integer; many powers of α are scale ratios numerically disguised as α-powers (conflating them is a category error). Second, the running of α is the projection angle opening: dβ/d lnE = (2/3π)sin²β ∝ α² — a self-coupling loop matching 1-loop QED. Third, the lone Mach-closure α is one such projection, yielding Hubble constant H0 = 68.97 ± 0.002 km/s/Mpc. α's magnitude is treated as an input.

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