The Adelic Universe: Dark Matter, the Fine-Structure Constant, and the Arithmetic of {2, 3, 7}

We observe that the extended Randall-Sundrum framework with D = 11 dimensions, N = 7 spokes, and kL = 37 warp depth is arithmetically closed under the prime set {2, 3, 7}. Every rational quantity in the framework — coupling ratios, mixing angles, dark matter fraction, nuclear binding coefficients, and the fine-structure constant itself — factors over these three primes. No other prime appears in any multiplicative structure. The fine-structure constant decomposes exactly as 1/α = 2N + 3² + 1/(2² × 7), and this decomposition uniquely selects kL = 37 as the warp depth at which the exponents refer back to the geometry’s own structural parameters. The dark matter fraction (observed 84.3±0.3%) is computed from the Euler factor at p = 7 in the Dedekind zeta function of Q(ζ7): (6/7) × (1 − 7−2) = 288/343 = 25 × 3²/7³ = 83.97% (0.3% error). The (1+α) correction appearing four times across 120 orders of magnitude is identified as the leading contribution of the 7-adic propagator, dominated by the p = 2 Euler factor at s = N. The SEMF coefficients governing nuclear binding (denominators 9, 8, 6, 63), iron-56 = 2³ × 7, and the magic numbers 28 = 2² × 7 and 126 = 2 × 3² × 7 inherit the spoke geometry’s adelic structure. (Four of the seven nuclear magic numbers factor over {2, 3, 7}; the remaining three — 20, 50, 82 — do not and are identified as an open question in Section XIII.) The error hierarchy — percent-level for topological observables, sub-percent for spectral, sub-ppm for couplings — matches the Euler product structure with exponent s increasing from s ≈ 2 (topological) to s ≈ kL (coupling). The framework’s 19 Standard Model parameters reduce to three structural numbers built from two seed primes. The arithmetic of the universe is the arithmetic of {2, 3}.