Least-Atom Extremizers for Resampling Noise

A single least-mass atom determines the extremal behavior of resampling noise on a finite probability space. A fixed-moment comparison gives the sharp generalized Samorodnitsky inequality for every $1<q<2$ through one scalar curve, and identifies all nonconstant scalar extremizers. The same principle determines every $L^p$-to-$L^q$ resampling norm for $1<p<q<\infty$, resolving a conjecture of Cao, Fan, Han, Qiu, and Wang. The sharp hypercontractive parameter increases strictly with the least atom. The uniform-binary phase diagram is explicit, and product equality cases are classified for every nonuniform reference law and for the uniform binary law.