Four-Certificate Minimax Rates over Public Class Pairs for Bounded Partially Linear Models

We study estimation of a bounded partially linear coefficient from two public nuisance classes with separate approximation and stochastic complexity certificates. In the strict envelope interior, the worst-public-pair expected absolute risk for $(a,s,b,t)\in [0,1{]}^4$ has rate $n^{-1/2}+(a+s^2)(b+t^2)+(s\wedge t{)}^2+t(a\wedge t)+s(b\wedge s)$. Matching directional lower bounds close the gap between the Gu–Yin–Cai–Fan lower envelope and TAME's calibration upper. We also construct one rule attaining this rate without any of the four budget inputs. Double calibration removes approximation-dependent orientation, and empirical local Rademacher fixed points calibrate the stochastic radii. The general rate persists over full finite dictionary ${\ell }_1$ balls and finite unions of rank-one balls. Full interval boxes instead have sharp rate $n^{-1/2}+(a+s^2)(b+t^2)$, and full affine Hilbert balls have rate $n^{-1/2}+ab$; both are attained without any budget input. The model requires only positive average residual variance and bounded envelopes. Results include all sample sizes and zero budgets; a uniform upper connects to saturation. The guarantees are statistical; effective optimization requires additional interfaces.