Report Resolution in Federated Multiple Testing: An Inverse-Square Law for Power Loss

Federated machine-learning evaluations and other multi-site studies may test one hypothesis family without pooling records, so each site sends only a finite-valued report per hypothesis. We compare the supremal full-alternative average power under strong family-wise error control with the centralized-oracle value from the full $p$-value array. Under an independent two-group model with fixed $K$ and $S\ge2$, suppose the known local alternative densities satisfy $g_s\in C^1[0,1]$ and $g_s'(u)<0$ on $[0,1)$. For $K\ge2$, also suppose the dual integrand has an active kink at an oracle dual minimizer for the full-alternative objective. Then the infimal deficit over all deterministic, one-shot, componentwise maps shared across hypotheses with at most $m$ symbols per site is $\Theta(m^{-2})$. Equivalently, $b$ bits per site per hypothesis incur a deficit of order $4^{-b}$. Equal-width reports attain this rate; even nonuniform, disconnected measurable cells cannot improve its exponent. Duality and conditional-Jensen gaps link the two optima. In a $K=S=2$ $\mathrm{Beta}(1,2)$ model, one-, two-, and three-bit equal-width reports retain $71.9$%, $97.1$%, and $99.2$% of numerical centralized-oracle power; a one-bit threshold candidate with a classwise value certificate retains $91.5$%. Thus, three bits come within one percent of oracle power.