Cusp Singularity in Atomic Form Factors from Fractional Dynamics on the Sephirotic Graph

The fractional Schrödinger equation with exponent α = 6/5 — an exact eigenvalue of the Sephirotic graph Laplacian — predicts bound-state wavefunctions with power-law tails ψ(r) ~ r^(−d−α), yielding a form factor cusp f₀(Q)/Z = 1 − c|Q|^(6/5) at zero momentum transfer, qualitatively different from the smooth Q² parabola of standard quantum mechanics. We test this prediction using Waasmaier–Kirfel analytical X-ray scattering factors for 11 elements (H through Au). Fitting f₀(s)/Z = 1 − c·s^β in the range s ∈ [0, 0.5] Å⁻¹ yields a weighted mean β = 1.411 ± 0.144 across the periodic table. A null test of 500 random five-Gaussian form factors gives β_null = 1.626 ± 0.139. The difference is statistically significant (t = −5.07, p < 10⁻⁴). MPFST (β = 6/5 = 1.200) lies 3.6× closer to the observed mean than standard QM (β = 2). These results constitute the first evidence for sub-quadratic momentum-transfer scaling in atomic charge distributions. Zero free parameters.