Background: The Aubry—André—Harper (AAH) Hamiltonian, a one-dimensional quasiperiodic tight-binding model with irrational modulation frequency, exhibits a Cantor-set energy spectrum at its self-dual critical point. This spectrum has been experimentally realized in ultracold atoms, superconducting qubits, and photonic waveguides, yet its structural features have not previously been applied to the prediction of atomic properties. Methods: We constructed the AAH Hamiltonian on a lattice of D = 233 sites (a Fibonacci number) with modulation frequency α = 1/φ, where φ denotes the golden ratio, and potential amplitude V = 2J (the self-dual critical point). Numerical diagonalization yielded 233 eigenvalues whose gap structure was analyzed to extract five spectral constants. These constants were combined into a closed-form algebraic formula for the ratio of van der Waals radius to covalent radius, r(vdW)/r(cov), using six prediction modes parameterized solely by each element's electron configuration. No constants were fitted to experimental atomic data. Results: The formula was evaluated for 54 elements (Z = 3—56). Of these, 42 (78%) yielded predicted ratios within 10% of observed values, 53 of 54 (98%) within 20%, with a mean absolute error of 6.7%. Only one element (boron) exceeded a 20% error. The best-predicted element, cesium, showed agreement to 0.2%. The residual deviations correlated significantly with independently measured material properties: hardness (ρ = +0.66 for p-block elements), melting point (ρ = −0.61), and electrical conductivity (r = −0.74 for metals). For the lanthanide series (Z = 57—71), the four-gate architecture generated three confirmed predictions: (a) the van der Waals radii should be approximately constant across the series, consistent with Alvarez's crystallographic finding of a 232 ± 9 pm mean; (b) covalent radii should contract monotonically, matching the observed lanthanide contraction from 207 to 175 pm; and (c) the worst conductor should occur at f7 half-filling and the best at f14, confirmed by gadolinium (0.74 MS/m) and ytterbium (3.51 MS/m). Conclusions: These findings indicate that the Cantor-set band structure of the AAH spectrum encodes quantitative information about atomic radius ratios across the periodic table. The formula achieves accuracy comparable to semi-empirical screening models while employing zero adjustable parameters. Its residuals constitute systematic indices of material properties rather than random noise.