Perturbation theory usually answers a forward question: push a matrix by some error E and its dominant singular subspace turns by at most some angle. This paper answers the exact inverse. Fix a rectangular matrix with a singular gap g, and ask the smallest perturbation — measured in the spectral norm — that rotates the top-k right singular subspace by a principal angle θ. There is an exact answer, not a bound.
The answer is (g/2)·sin 2θ. Not “at most,” not “on the order of” — the exact infimum of ‖E‖₂ for that specific matrix, attained by an explicit block perturbation for every angle below π/4, with its smallest value g/2 reached at π/4. The double-angle shape is the classical one; what is new is the per-instance exact inverse, in the original gap, for the rectangular, right-subspace, spectral-norm case.
The proof turns on one compression lemma. Every principal pair between the true subspace and a perturbed one reduces to a two-column problem — so the whole k-dimensional question collapses to something you can solve by hand in 2×2. That reduction is the load-bearing idea; the exact law falls straight out of it.
Bolt the exact backward error onto the exact projection-metric distance to the edge of the totally positive Grassmannian — the boundary from the previous paper — and you get a sharp universal stability radius: a single perturbation size, below which the positive structure survives for every matrix sitting a given distance δ from the boundary, wrapped in a clean √(1−δ²) envelope. A semidefinite specialization and an oriented-matroid (chirotope) generalization follow as corollaries.
Being straight about what it is: a preprint, not yet peer-reviewed. The double-angle constant is classical Davis–Kahan, and I cede it; Wedin’s sin Θ theorem is cited only through Dopico’s inspected modern restatement — I cite what I read. Novelty is claimed for the rectangular, right-only, spectral-norm, per-instance exact law and the positive-Grassmannian exit envelope, and nothing more.
Same discipline as the rest of this site, pointed at pure mathematics. The three load-bearing results are proved analytically; the computations that ship with the paper are non-load-bearing reproduction and adversarial stress tests — fixed seeds, 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 subspace-boundary result it continues
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.