A 1/R Law for Kurtosis Contrast in Balanced Mixtures

Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection with effective width Reff (participation ratio), the population excess kurtosis obeys |kappa(y)|=O(kappamax/Reff), yielding the order-tight O(cbkappamax/R) under balance (typically cb=O(log R)). As an impossibility screen, under standard finite-moment conditions for sample kurtosis estimation, surpassing the O(1/√T) estimation scale requires Rlesssim kappamax√T. We also show that purification -- selecting m!ll!R sign-consistent sources -- restores R-independent contrast Omega(1/m), with a simple data-driven heuristic. Synthetic experiments validate the predicted decay, the √T crossover, and contrast recovery.