The Metric Geometry of Realizable Chirotope Transitions

This record archives the paper "The Metric Geometry of Realizable Chirotope Transitions" (B. M. Lindsey) together with its supplement and a self-verifying reproducibility archive. It is the third paper of a three-stage program: the first gives the exact projection-metric distance from a positive k-plane to the positive-Grassmannian boundary (a dimension-free first-exit stability radius); the second solves the inverse singular-subspace backward-error problem exactly, sharpening that radius to its universal envelope; the present work asks the distance to a PRESCRIBED target chirotope chamber. The signs of the Plucker coordinates of a k-plane W in R^n form its chirotope and are locally constant away from the coordinate walls. For a realizable target chirotope chi', the transition distance tau(W) is the projection-metric distance from W to the closed target chamber. Its geometry rests on the wall depth d_I = sigma_min(U_I) and an exact witness-subspace formula for the sign-free cost of approaching several walls at once. In Gr(2,4) every realizable single mutation satisfies tau = min{d_I, sqrt(1 - d_I^2)}. In Gr(3,6) this simplicity fails: an interval-certified single mutation has tau > d_I, so realizability can impose a strict metric premium. Finally, for a gapped matrix A with dominant right singular subspace W_k(A), the spectral-norm backward error obeys r(A) >= g(A)*Psi(tau), sharp as an infimum on shifted projectors. The results identify an intrinsic metric geometry of realizable chirotope transitions on the Grassmannian and transfer it sharply to singular-subspace backward error. The load-bearing results are proved analytically (main article plus supplement SM1-SM7); the archive's computations are non-load-bearing corroboration and interval-certified verification (fixed environment; self-verifying via sha256sum -c SHA256SUMS). The manuscript contains an AI-use disclosure in accordance with SIAM policy.