We demonstrate that the [proposed] Decomposition, grounded in the single algebraic identity phi^2 = phi + 1, extends to atomic physics with sub-percent precision across three independent domains. Spectral: alpha_em^{-1} = N x W = 137.337 (0.22% vs. CODATA), where N = 294 and W = 0.467134 are derived from the topological invariants of a 233-site Aubry-Andre-Harper lattice at critical coupling. The bracket count N is independently derivable as N = F(13) + F(10) + F(5) + F(2) = 233 + 55 + 5 + 1, a spectral topology invariant generalizable to any Fibonacci lattice. This cascades through QED to predict the Bohr radius (0.22%), Rydberg energy (0.44%), and proton charge radius (0.14%). Spatial: Anchoring the hydrogen 1s peak at the Cantor shell center maps 42% of the probability into the sigma_2-sigma_4 wall zone; the H_2 bond length matches sigma_4 outer to 0.5%. Information-theoretic: The bipartite entanglement entropy at the Cantor sigma_4 outer wall reaches S = 0.6908 nats, 99.66% of the theoretical maximum ln 2, with zero free parameters. The Cantor outer wall lies 5.3% beyond the equal-partition point (p = 0.535), preserving 99.66% of the theoretical maximum entropy ln 2. Three additional exact identities are proved from phi^2 = phi + 1 alone: the unity partition 1/phi + 1/phi^3 + 1/phi^4 = 1, the boundary law 2/phi^4 + 3/phi^3 = 1, and the pi-phi bridge pi = 4*arctan(1/phi) + 4*arctan(1/phi^3). Systematic comparison with all metallic means and quadratic irrationals shows phi is unique: it alone produces a critical eigenstate with minimal fractal dimension D_2 ~ 0.538, a nested five-sector Cantor spectrum, stable baryon fraction Omega_b ~ 0.048, and dark-energy-dominated expansion. The fine-structure constant is the cumulative wall fraction of the Cantor vacuum: alpha_em = 1/(N x W). The hydrogen atom is the same self-consistent structure at bracket 117.