Sharper bounds for fixed-partition HODLR approximation

For the fixed-partition HODLR approximation problem with $n=2^Lk$, this note
develops the bounds
$c\min\{n,kL/\varepsilon+k/\varepsilon^2\}\le q_*(n,k,\varepsilon)\le C\min\{n,kL^2/\varepsilon+kL/\varepsilon^2\}$.
The upper bound is obtained by a row-isotropic perturbation inequality for
truncated SVD, a noisy Gaussian Nyström calculation, a constant-accuracy
perforated pilot, and a globally shared final regression. The new lower term
$kL/\varepsilon$ uses weighted hierarchical Stiefel frames, large fixed
rank-constrained anchors, and a transcript-divergence argument valid for
adaptive queries on both sides. The earlier $k/\varepsilon^2$ lower-bound
argument is included in an appendix, so the combined argument can be read
without an external unpublished dependency. These bounds still do not match
in general. They retain the characterization $q_*=\Theta(n)$ for
$\varepsilon\le\sqrt{k/n}$ and rule out the previously plausible rate
$kL+k/\varepsilon^2$ as a uniform answer.