My second paper asked the smallest perturbation that rotates a matrix’s dominant right singular subspace by a given angle, and answered it exactly. This one narrows to a sharper, harder version of the same question. Fix the matrix. Prescribe only the right subspace — leave the left frame free — and demand that the perturbed matrix keep that subspace at the top of its spectrum. Then ask the exact spectral-norm backward error: the size of the smallest perturbation that makes it so.
You might expect the answer to be the worst single direction — the hardest two-plane you have to bend, and no more. It isn’t. The central phenomenon of the paper is multidirectionality: the exact spectral-norm cost can be strictly larger than the supremum of every single two-plane compression bound. The perturbation has to satisfy several rotations at once, and they obstruct one another — a cost that no one-direction estimate can see, because it lives in how the directions interfere.
The gap is not a fluke of one matrix. It is proved on a nonempty open set — a full-dimensional family, not a measure-zero curiosity — so the obstruction is generic, part of the geometry rather than an artifact of a special example.
And it is pinned, not plotted. The base instance is certified exactly: an exact-rational semidefinite dual on one side and interval arithmetic on the other, closing the strict inequality by hand. The claim is that the cost exceeds the single-direction bound, and the certificate proves the strictness rather than showing a suggestive picture.
The result is extracted from the same backward-error program as the companion paper and the subspace-boundary result that started the thread. As everywhere on this site, the objects carry descriptive names, not mine. It is a preprint, under review at BIT Numerical Mathematics, not yet refereed; the load-bearing theorems are proved analytically and the base instance is interval-certified; the computation that ships with it is non-load-bearing reproduction and adversarial stress testing, fixed seeds and a pinned environment, self-verifying against a SHA-256 manifest with sha256sum -c. You don’t trust the author; you trust the trail.
→ The paper + reproduction packet (Zenodo, CC BY) · the exact backward-error law it sharpens · where the thread began
Method note: language-model tools assisted with prose and analysis-code generation under author-defined statements, checks, and reporting constraints; I reviewed and take responsibility for every analysis and claim.