An all-spectrum upper bound for the isospectral Nyström minimax problem

For the real isospectral Nyström minimax problem, an oversampled limiting-profile estimate is proved and used to obtain an upper bound for every positive finite spectrum: $R_{n,k}\le\min\{k+1,n-k,C\sqrt{k}\}$ for $2\le k\le n-2$, with the explicit universal constant $C=2^{34}e^{2560}$. The argument completes the conditional upper-bound reduction in the author's preceding research package. Its main ingredients are a clipping inequality, rectangular Gaussian inverse-trace small-ball estimates, a common orthogonal rotation for all subsets, and an exact-spectrum grouping construction with additional observations granted to the selector. Every construction chooses the orientation before the subset. Singular limiting precision blocks are assigned infinite inverse-trace cost. No matching dimension-growing lower bound for $R_{n,k}$ is established, and the sharp joint dependence on $n$ and $k$ is not determined. The very large constant is retained to make the probability estimates explicit, not for numerical use.