Exact Backward Error for Singular Subspaces and the Sharp Universal Stability Radius of the Positive Grassmannian

This record archives the paper "Exact Backward Error for Singular Subspaces and the Sharp Universal Stability Radius of the Positive Grassmannian" (B. M. Lindsey) together with its self-verifying reproduction packet. The paper settles the exact inverse (backward-error) question for right singular subspaces in the spectral norm: for a fixed rectangular matrix with singular gap g, the exact infimum of ||E||_2 required to rotate the top-k right singular subspace by a principal angle theta <= pi/4 is (g/2) sin 2theta -- per-instance exact, attained on [0, pi/4) by an explicit block perturbation, with infimum g/2 at pi/4. The engine is a principal-pair compression lemma that reduces every k to a two-column problem; combined with the exact projection-metric distance to the boundary of the totally positive Grassmannian it yields a sharp universal stability radius with envelope sqrt(1 - delta^2). A semidefinite specialization and an oriented-matroid (chirotope) generalization follow as corollaries. The three load-bearing results are proved analytically; the packet's computations are non-load-bearing reproduction and adversarial stress tests (fixed seeds, pinned environment, self-verifying via sha256sum -c MANIFEST.sha256). The manuscript contains an AI-use disclosure in accordance with SIAM policy.