On Local Population-Risk Certificates

We develop finite-sample certificates for local population-risk increments $P{\delta }_v=R({\theta }_0+v)-R({\theta }_0)$, $v\in \mathcal{D}$. The primitive object is an expected-valid upper endpoint ${\widehat{\mathsf{U}}}_{\mathcal{D}}$ satisfying $\mathbb{E}{\sup }_{v\in \mathcal{D}}\{P{\delta }_v-{\widehat{\mathsf{U}}}_{\mathcal{D}}(v)\}\le 0$. This uniform criterion certifies any measurable update selected from the same sample and allows penalties to depend on empirical geometry. The main construction is a cross-fitted ridge calibration for linear feature classes. A pilot fold learns the ridge metric, the complementary fold calibrates the squared mean error in that metric, and complete split averaging recovers the full empirical covariance in the directional quadratic form ${\widehat{q}}_{X,\lambda }$. The optimized diagnostic scale is $\{{\widehat{q}}_{X,\lambda }(h){\widehat{r}}_{X,n_{\mathrm{p}},\lambda }^{\mathrm{c}\mathrm{f}}\mathrm{/}n{\}}^{1\mathrm{/}2}$, and the calibrated trace factor ${\widehat{r}}_{X,n_{\mathrm{p}},\lambda }^{\mathrm{c}\mathrm{f}}$ is compared with the ordinary ridge effective dimension ${\widehat{r}}_{X,\lambda }$. For nonsmooth losses, an exact fixed-mask decomposition ${\delta }_v=J_v^0+R_v^{\circ }+C_v$ separates frozen Taylor fluctuations, good-path remainders, and interface crossings. Applying the linear and composite certificates componentwise yields endpoints for same-sample expected local search and concentrated release rules.