From Exponential Phase Ambiguity to Long-Range Rigidity: Matroid Fibers, Determinantal Interference, and Boundary Instability in Phaseless Plücker Geometry

What, precisely, is lost when complex Plücker coordinates are replaced by their complete labeled squared magnitudes? We establish a support-sensitive answer that links complex Grassmannian measurement geometry, matroid connectivity, graph holonomy, perfect-matching interference, and boundary conditioning. For a cactus fundamental bipartite support, the full phaseless fiber modulo diagonal-unitary gauge has exactly 2^{b_} elements, where b_ counts cycles with nonreal holonomy. Consequently, even a connected representable matroid can retain arbitrarily many independent binary phase orientations. A rank-three example exhibits two such orientations and recovers them using two oriented Bargmann triangle invariants. The contrasting mechanism is long-range synchronization: for a nonzero complex tridiagonal ladder of arbitrary size, higher-order minors couple all nonreal cycle orientations into one globally conjugate pair, even across arbitrarily long intervals of real-phase cycles. A continuant endpoint determinant identity proves this all-strata result, whereas deletion of one linking edge yields four orbits and an explicit family with no uniform inverse stability near the boundary. The contribution is an exact bridge between combinatorial support, algebraic interference, and identifiability versus conditioning, beyond generic principal-minor rigidity. This bridge supplies mathematical criteria for distinguishing persistent phase ambiguity from recoverable relative orientation and suggests how support geometry and nonvanishing links may inform future phaseless sensing, determinantal inference, and stability-aware reconstruction.