Alpha-Cyclic Scale Theory: A Universal (g,q)-Sphere of Radius R = α⁻¹

We present comprehensive evidence for a universal relation between mass and characteristic size spanning 32 orders of magnitude — from neutrinos (0.001 eV) to supermassive black holes (6.5×10⁹ M☉). In logarithmic coordinates (g,q) defined with respect to the Planck scale, all physical objects lie approximately on a circle of radius R = α⁻¹ = 137.036, where α is the fine-structure constant. Ordinary least-squares fit of 129 objects yields R = 140.30 ± 4.39 (Δ = 2.38%). Block bootstrap (4000 resamples) gives a 95% CI [132.47, 149.10] containing α⁻¹ (z = 0.74σ). Cross-validation (3000 iterations, 80/20 split): train R = 140.91 (IQR: 138.71–142.59), test R = 137.02 (IQR: 132.44–157.92). Bootstrap (10000 resamples): median R = 140.09, 95% CI [131.61, 162.79]. Leave-category-out: removing any category leaves R within 5% except neutrinos (Δ = 104%). Independent verification on 11 new objects shows 7/11 lie within 3% of the sphere. Random data reproduces R = α⁻¹ in only 6.1% of trials. Parameterising the circle by an angle θ reveals a cyclic structure — the alpha-cycle — connecting particles, nuclei, neutron stars, and black holes in distinct angular sectors. We propose the Alpha-Cycle Scale Law: any physical object with mass m and characteristic size R satisfies (g − g₀)² + (q − q₀)² = (α⁻¹)², where g = ln(R/L_P) − ln(m/M_P) − ln 2 = ln(R/R_S), q = ln(R/L_P) + ln(m/M_P) = ln(R/λ_C), and (g₀, q₀) = (128.947, 134.448). The fine-structure constant thus acquires a new meaning — a universal scale parameter determining the radius of the (g,q)-circle encompassing all physics. The circle is parameterised by an angle θ that cycles through different physical regimes — the alpha-cycle. Each sector (particles, nuclei, neutron stars, black holes, galaxies) occupies a distinct angular range governed by a different fundamental interaction, yet all share radius R = α⁻¹. If confirmed by further data, the alpha-cycle represents a new law of scale: a universal connection between mass, size, and fundamental constants that emerges from the geometry of Planck-normalised space. We invite independent replication on larger and more diverse datasets.