My last paper fixed a matrix and asked the smallest perturbation that turns its dominant singular subspace by a given angle. This one asks a different question about the same geometry: not how far a subspace moves, but where it is allowed to go. The sign pattern of a subspace’s Plücker coordinates — its chirotope — cuts the Grassmannian into chambers. Fix a subspace in one chamber and a prescribed target sign pattern, and ask the exact distance it must travel to reach that pattern.
The distance is set less by how deep the target wall sits than by whether the sign change is realizable at all. The transition metric is the projection-metric distance from the subspace to the closure of the target chamber, and it resolves into a competition among the walls that have to be crossed together. In the smallest nontrivial case — rank-2 subspaces in four dimensions — it collapses to a clean two-branch law, min{𝒟, √(1−𝒟²)}: an exact distance for every single sign flip, proved by an explicit pair of nearest points.
One dimension up the simplicity ends. In the rank-3 case the sign-compatibility constraint forces a strict metric premium — reaching certain patterns costs strictly more than the nearest wall alone would suggest, because you cannot cross one wall without respecting the others. That contrast is the structural heart of the paper: an exact two-branch law where the geometry permits it, and a certified strict premium where it does not. The premium is not asserted from a picture; it is pinned with explicit interval enclosures.
Bolt the subspace distance onto the exact backward-error law from the companion paper and it lifts to matrices: a universal lower bound on the perturbation needed to drive a matrix’s singular subspace into a prescribed sign pattern, wrapped in the exit factor t·√(1−t²), sharp as an infimum. Two dualities run through the whole thing — the chirotope’s discrete sign data reads out as a continuous distance to the chamber walls, and forward perturbation bounds turn into exact backward, inverse costs.
As everywhere on this site, the objects carry descriptive names — chirotope boundary margin, chirotope exit radius, the universal exit factor — not mine; if the community finds them useful it can attach a name. And it is a preprint, not yet peer-reviewed. The load-bearing theorems are proved analytically; the rank-3 premium is interval-certified; the computation that ships with the paper is non-load-bearing reproduction and adversarial stress testing — 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 exact backward-error law it builds on · the subspace-boundary result that started the thread
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.