arXiv:2603.13407v2 Announce Type: replace Abstract: Le Cam deficiency is the largest increase in Bayes risk caused by retaining a statistic, over all priors and decision problems with losses in $[0,1]$. We compute its exact leading term for finite hypothesis families in growing shuffle models. For a bounded correlated message bundle in an iid categorical background with an observed finite tag, the coefficient is a convex program over exposed faces with Gaussian-polynomial scores; its order is set by the first differing conditional factorial-moment degree. Tag-preserving kernels can be suboptimal, and an exact example has family coefficient $4/3$ times the largest pairwise coefficient. For a critical rare-event channel along $m+1$ consecutive compositions, retaining a distinguished-symbol count has loss $n^{-1/2}\kappa\varphi(0)\mathcal V_m(\mu)+o(n^{-1/2})$, where $\mu=\theta\lambda$, $\kappa^2=\chi^2(q\|p)/(1-\theta)>0$, and $\varphi$ is the standard normal density. We evaluate $\mathcal V_2$ for all $\mu>0$. For every $m\ge2$, the full-chain value $e^{-\mu}(1-1/\sum_{k=0}^m\mu^k)$ is exact if and only if $\sum_{k=2}^{m+1}\mu^k\le2$; an explicit large-mean regime is attained by the endpoint pair. Exact-arithmetic computer-assisted certificates give the complete four-composition phase diagram. At $\mu=\sqrt2$, the ratio to the largest pairwise coefficient is strictly below two for every finite window and tends to two as $m\to\infty$. A universal upper bound of two and Hypothesis P, which asserts that admissible vertex paths attain every window coefficient, remain open outside the proved regimes. The comparison principle is classical; the contributions are the model-specific scores, exact coefficients, and certificates.