Power-law Coupling and Mock Validation of a Surface-density Anisotropy Diagnostic for 85 Rich Galaxy Clusters

We assess a surface-density anisotropy diagnostic, $A_{\mathrm{a}\mathrm{n}\mathrm{i}}$, and its association with the slope $\alpha$ of a power-law galaxy profile. We reprocess 20-annulus profiles of 85 optically rich clusters on a canonical grid, $0\le A_{\mathrm{a}\mathrm{n}\mathrm{i}}\le 1.92$, and an extended grid reaching 1.99. Twenty-four canonical fits and 12 extended fits reach their respective upper boundaries; 69 of 85 upper $\mathrm{\Delta }{\chi }^2=1$ bounds remain open, and only four clusters have two-sided intervals. Among 40 preselected power-law fits, the observed $\alpha$-$A_{\mathrm{a}\mathrm{n}\mathrm{i}}$ Spearman coefficient is 0.616. In a model-native forward-recovery experiment with 11 injected values, all 935 noiseless profiles recover the input within 0.02. Across 46,750 Poisson catalogs matched to observed count and background conditions, recovery instead has root-mean-square error (RMSE) 0.992 and Spearman $\rho =0.093$; 40.0% of fits reach the canonical upper boundary and 81.4% have an open upper interval. Fixing both background and radial scale improves RMSE to 0.564 in a 9,350-catalog sensitivity test. As a separate structural control, 4,250 Poisson catalogs generated from fitted power laws reproduce the observed slope association: the median mock coefficient for the 40-cluster subset is 0.608 (95% range 0.453-0.731). Thus the diagnostic is numerically recoverable under its defining model without noise, but the available photometry does not reliably identify it for individual clusters. The power-law correlation can arise from shared projected profile shape and is not independent evidence for orbital anisotropy.