General Transition Geometry of Rank-Two Chirotopes

Abstract

Let W ∈ Gr(2,n) and let M be a realizable rank-two oriented matroid. Building on the prescribed-target transition metric τ_M(W) = d_π(W, cl C_M), we develop a rank-two geometry for moving from a fixed source plane to a prescribed realizable chirotope stratum, organized by an obstruction ladder: coordinate incidence gives a sign-free benchmark; each target cocircuit restricts admissible witness directions to an oriented cone; several cocircuit requirements must be jointly compatible inside one two-plane; and the resulting constrained plane determines the full target distance. Three results make the theory sharp. First, for every n ≥ 3, every uniform W ∈ Gr(2,n), and every realizable single mutation χ^(I), we prove the exact universal law τ_{χ^(I)}(W) = min{ 𝒹_I(W), √(1 − 𝒹_I(W)²) }, so a single mutation has no within-facet compatibility premium in rank two. Second, we give an exact Gr(2,5) multi-sign transition in which the full target chamber is strictly farther than a specified unsigned joint-wall benchmark, and which also strictly separates a certified single-cocircuit floor from the full target-compatible two-cone face. Third, arbitrary rank-two targets admit two equivalent finite descriptions — multi-cocircuit transversality and ordered projective slope charts — reducing the metric core to a two-column generalized-eigenvalue problem; a counterexample shows that the resulting fixed-sector reductions do not imply a global polynomial bound on the balanced-response combinatorics. A reproduction package accompanies this deposit; all decisive quantities are exact (rational/algebraic), with floating-point used only as numerical corroboration.