Some matrices carry a hidden kind of positivity. Not that every entry is positive — something subtler: the top-k directions the matrix actually cares about, its dominant singular subspace, sit inside the “positive Grassmannian.” It is a fragile property, and I wanted to know exactly how fragile.
The answer is an exact number. Measured the way perturbation theory really measures a subspace — the projection metric — the distance from that subspace to the edge where positivity breaks is exactly a quantity I call δ₊: the smallest, over the coordinate row-blocks of an orthonormal basis, of the least singular value. Geometrically it is the distance to the nearest coordinate Schubert divisor. From it falls a computable perturbation radius, g·δ₊/(1+δ₊): hit the matrix with anything smaller and the positive structure is guaranteed to survive.
There is no hidden dependence on how big the matrix is. The radius is set by the geometry inside δ₊, not by the ambient dimension or by k. And it is sharp — a clean projector family shows the radius is optimal to first order as the subspace approaches the boundary, so the constant is not an artifact of a loose bound.
Where it earns its keep is optimal transport. The Gaussian Sinkhorn couplings you get from entropic OT are totally positive, but that entrywise total positivity is brittle: a small nonseparable tilt of the cost breaks it quickly. The subspace-level positivity is far more robust — it outlasts the whole-matrix property well past the point where total positivity is already gone. This paper makes that gap quantitative.
Being straight about what it is: a preprint, not yet peer-reviewed. I claim novelty only for the synthesis. The Grassmann-condition-number and singular-subspace-perturbation literatures are cited and ceded; those measure how far a subspace is displaced, whereas the new quantity here measures the loss of positive-Grassmannian membership. The classical pieces are cited as classical.
Same discipline as the rest of this site, pointed at pure mathematics. The result was checked symbolically, in double precision across thousands of randomized and adversarial trials, and again at 50 digits; two independent expert reviewers went through it; and it ships with a reproduction packet that self-verifies against a SHA-256 manifest. You don’t trust the author; you trust the trail.
→ The paper + reproduction packet (Zenodo, CC BY) · the methodology it inherits
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.