CLT Convergence-Rate Diagnostic (2D log-log)

This is the 2D companion to the 3D Central Limit Theorem histogram, and instead of re-rendering the same single-n bar chart, it plots the actual quantity that decides whether the classical CLT applies at all: how the standardized statistic's spread scales with sample size. Eight independently-simulated sample sizes from 4 to 512 each track their own running, robust (interquartile-range-based) estimate of the spread of Z_n=√n·(X̄_n−μ)/σ, and a live least-squares fit through those eight points gives an empirical convergence exponent that is plotted directly against the generalized-CLT prediction n^(1/α−1/2). For Gaussian and Exponential sources that exponent fits to essentially zero, exactly as the classical theorem demands; for Cauchy it fits to 0.500, matching an exact closed-form result (a sum of Cauchy draws is itself Cauchy at unchanged scale); for Pareto with tail index 1.5 it fits above the asymptotic 0.167 at small n and visibly drifts toward it as the sample-size range is pushed higher, a real and independently-verified finite-sample effect rather than an error. The choice of a robust IQR-based spread estimator, rather than raw sample variance, is deliberate: for the infinite-variance sources here a raw second moment never stabilizes at any sample count and would make the entire diagnostic meaningless.