My third paper found the exact distance to reach a prescribed sign pattern in the smallest cases — a single flip in four dimensions, and a first look one dimension up. This one takes the rank-two case all the way. It is the general transition geometry of realizable rank-two chirotopes: not one move, but the whole family of single-step moves, in any ambient dimension.
The main result is an all-n single-mutation law. For a rank-two subspace in any number of dimensions, it gives the exact geometry of a single sign mutation — crossing one chirotope wall — in closed form, uniformly in the dimension. Where my earlier paper answered the four-dimensional case, this answers every case at once.
And it locates where the simple story ends. In Gr(2,5) there is a certified strict realizability premium: reaching certain patterns costs strictly more than the nearest wall alone would suggest, because the sign change is not independently realizable — you cannot cross the wall you want without paying for the ones tied to it. That premium is the structural payoff of taking the general case seriously.
Underneath, it is a meeting of two languages. The discrete data of an oriented matroid — which sign patterns can exist at all — reads out as a continuous distance to the chamber walls. The “transition geometry” is the exact dictionary between them for rank two, and it is where the chirotope-distance paper stops and this one begins.
The load-bearing theorems are proved analytically and the Gr(2,5) premium is certified; a firewall held the whole way — no theorem statement, hypothesis, quantifier, or numbered equation changed between the frozen result and the deposited version. It is a preprint, under review at Discrete & Computational Geometry; the arXiv version of record (math.CO) follows once endorsed. The objects carry descriptive names, not mine; the reproduction packet is non-load-bearing and 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 chirotope-distance paper it continues · the sign-vector certificate margins
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.