Weighted Krylov Obstructions and a Shifted-Wishart Posterior

Two concrete pieces of the finite-accuracy transition for unrestricted adaptive matrix–vector queries are isolated, without closing the transition itself. First, a weighted Chebyshev interpolation argument proves matching bounds, for all finite parameters, for a precisely defined deterministic-width, full-block span-query class, whose query complexity is $\Theta(\min\{n,\frac{k}{\sqrt\varepsilon}\log\frac{en}{k}\})$. The lower bound uses integer spectral multiplicities, an elementary reciprocal-Gaussian-Gram moment, and a rotational argument that permits arbitrary output subspaces. It does not permit arbitrary fresh query directions or selective-column updates. Second, an exact adaptive posterior is derived for a singular Wishart matrix whose positive eigenvalues have a deterministic additive shift, and a conditional first-exit formula is given. An elementary Gaussian resolvent identity then shows that this shifted law admits a rank-one averaged upper bound at the smaller scale $\varepsilon^{-1/2}\log(1+n\sqrt\varepsilon)$, so it cannot witness a linear lower bound in the missing transition.