Geometric Structure in Strong Lensing Redshift Ratios: A Phenomenological Analysis

This technical note investigates the statistical distribution of a simple, dimensionless interaction ratio R = \frac{z_s}{z_s - z_l}, constructed from observed source and lens redshifts in strong gravitational lensing systems. Using a sample of 40 lenses with spectroscopically confirmed redshifts, we analyze whether observed configurations occupy random geometric regimes or preferentially cluster within a restricted interaction range. The empirical mean interaction ratio is found to be $\bar{R} = 1.49 \pm 0.07$, with a 95% confidence interval of $[1.36, 1.62]$. This interval approaches, but does not center on, the theoretical reference value $\sqrt{e} \approx 1.649$, which arises as the unique positive fixed point of a minimal exponential duality map. The reference value is not treated as a fundamental constant, but as a benchmark against which observed distributions can be compared. Selection effects inherent to strong lensing catalogs—including alignment bias, magnification bias, and redshift completeness—are explicitly discussed. The analysis is structured to be falsifiable: future samples with larger size and better-characterized selection functions should either reproduce the observed clustering through null-model simulations or exhibit systematic convergence toward the reference scale. Independent of the duality hypothesis, the interaction ratio $R$ provides a compact, model-independent diagnostic for classifying strong lensing configurations by geometric regime.