Kurtosis-based Independent Component Analysis (ICA) weakens in wide, balanced mixtures. We prove a sharp redundancy law: for a standardized projection of effective width $R_{\text{eff}}$, the population excess kurtosis obeys $|\kappa (y)|=O(\kappa _{\max }/R_{\text{eff}})$, order-tight as $O(c_{b}\kappa _{\max }/R)$ under balance. As an impossibility screen, surpassing the $O(1/\sqrt{T})$ estimation scale requires $R\lesssim \kappa _{\max }\sqrt{T}$. We further show that purification, the selection of $m\!\ll \!R$ sign-consistent sources, restores $R$-independent contrast $\Omega (1/m)$. Synthetic experiments validate the decay, the $\sqrt{T}$ crossover, and contrast recovery.
A 1/R Law for Kurtosis Contrast in Balanced Mixtures
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