An Inverse-Squared-Logarithm Fine-Structure History: Conditional Tests with Atomic Clocks and Quasar Spectra

Motivated by our numerical investigation of a non-autonomous spectral construction, we test a fine-structure history $\Delta\alpha/\alpha_0=\Gamma\{\ln^{-2}(t/t_*)-\ln^{-2}(t_0/t_*)\}$. Identifying an iteration index with cosmic time and coupling the structural parameter to electromagnetism are additional physical hypotheses. We fix the logarithmic reference scale to the conventional Planck time and condition the redshift--age conversion on a flat matter--$\Lambda$ background. A reproducible analysis of 293 published quasar measurements, using the authors' selection and a declared fixed excess-scatter budget, gives $\Gamma=-1.700\pm0.969$ and improves $\chi^2$ by $\(3.078\)$ over constant $\alpha$. Telescope subsets prefer opposite signs; allowing separate offsets reduces the improvement to $\(1.392\)$. Combining the quasar summaries with a verified optical-clock drift of $(1.8\pm2.5)\times10^{-19}\,\yr^{-1}$ gives $\Gamma=-0.00347\pm0.00476$, with $\Delta\mathrm{AIC}=+1.469$ relative to constant $\alpha$. The same clock constraint predicts $\Delta\alpha/\alpha_0=-0.00558\pm0.00775$ ppm at $\(z=4\)$ under this history. Nearby inverse-log exponents have nearly identical shapes after normalization to the present drift. A separate ESPRESSO measurement is consistent with the clock-constrained history. We also assess 2026 molecular and nuclear-clock results while retaining their additional coupling assumptions. The analysis constrains a specified homogeneous time dependence; it neither detects varying $\alpha$ nor establishes a physical identification of Riemann spectral dynamics.