Truncation, Sampling, and Huber Geometry for Supremum Characteristic-Function Independence Testing

We study a fixed-window supremum characteristic-function statistic for mutual independence. The frequency window is structural rather than merely computational: for generic even empirical samples with scalar blocks and absolutely continuous marginal laws, unrestricted maximization reaches the sharp population ceiling independently of the data-generating dependence, whereas every fixed window has an explicit \(n-1/2\) uniform deviation bound. In the two-block problem we combine an exact correlated-Rademacher product \(χ^2\) identity with the sharp Huber contaminated-null envelope. The envelope has a boundary-to-interior contact bifurcation at contamination \(1/2\), but this is not the clean-law identifiability threshold. Using the classical total-variation criterion for intersections of Huber neighbourhoods, we construct clean-law overlap obstructions and identify the exact full-range attainment threshold \(1-1/√2\): at and above this level a clean law with dependence one can be observationally indistinguishable from an independent clean law. For fixed support and frequency geometry we give a robust Huber minimax separation rate \(Θ(ep+n-1/2)\) near \((ep,n-1/2)=(0,0)\). In the separate complete-row replacement model, we establish the rate \(Θ(η+n-1/2)\), with explicit sensitivity and clean-overlap lower bounds. The exact finite-contamination clean-overlap profile remains open, but we prove its sharp small-contamination first-order coefficient \(3√3\); the binary lower curve of Theorem [thm:cleanoverlap] is not optimal.