Fibonacci Band Structure of the Aubry–André–Harper Spectrum and Its Correspondence with Atomic Shell Degeneracies and Radius Ratios

Background: The Aubry–André–Harper (AAH) Hamiltonian at its self-dual critical point produces a Cantor-set energy spectrum whose hierarchical structure is governed by the Fibonacci sequence. Whether this spectral architecture has any quantitative relationship with the atomic shell structure has not been systematically investigated. Methods: We constructed an AAH Hamiltonian on Fibonacci-length lattices (D = 13 to 377 sites) with a modulation frequency α = 1/φ and critical coupling V = 2J. For each lattice, the eigenvalue spectrum was computed by exact diagonalization and decomposed into five principal bands via gap analysis. Band state counts, subband decompositions, and interband ratios were tabulated. Independently, a closed-form formula for the ratio r(vdW)/r(cov) was constructed from five spectral constants extracted at D = 233 using seven prediction modes parameterized solely by the electron configuration. The formula was evaluated against experimental data for 54 elements (Z = 3–56). Residual deviations were correlated with independently measured material properties. Results: At even-index Fibonacci lattice sizes, all five band state counts were Fibonacci numbers, with outer-to-inner ratios converging to the golden ratio φ. Within the center band, 89% (8/9) of the subband sizes were Fibonacci numbers; the single exception was consistently adjacent to an isolated singleton eigenvalue near E ≈ 0, and the pair summed to a Fibonacci number. The ratios of the successive atomic shell capacities (6/2 = 3, 10/6 = 5/3, 14/10 = 1.4) matched the Fibonacci convergentsexactly for s→p and p→d and matched the principal spectral ratio BASE = 1.408 to 0.6% for d→f. The radius ratio formula achieved a 6.2% mean error across 54 elements with zero free parameters (44/54 within 10%). The formula residuals correlated with the Mohs hardness at ρ = +0.73 (N = 20, p < 0.001). Conclusions: The AAH Cantor spectrum exhibits a Fibonacci band hierarchy that corresponds numerically to atomic shell degeneracies. The spectral constants predict atomic radius ratios with accuracy comparable to that of semiempirical methods when no adjustable parameters are used. These correspondences invite further investigation into whether quasiperiodic spectral organization underlies the atomic structure.