Anomalous Power-Law Behavior in Atomic Form Factors: Evidence for Fractional Schrödinger Dynamics

The fractional Schrödinger equation with α = 6/5 predicts bound-state wavefunctions with power-law tails ψ(r) ~ r^{-(d+α)}, yielding atomic form factors F(Q) = 1 - c|Q|^{6/5} that exhibit a cusp at Q = 0 rather than the smooth parabola of standard quantum mechanics. Tested against Waasmaier-Kirfel analytical form factors for 11 elements (Z = 1 to 79). Free-exponent fit yields β = 1.411 ± 0.144, with 10/11 atoms favoring the cusp model by BIC. Null test confirms this is not a fitting artifact (p < 10^{-6}). Zero free parameters.