Geometric Constraint on Residue Phases: Resolving the N(2190) Anomaly and Diagnosing Exotic States

We derive a parameter-free geometric constraint on residue phases dictated by the pole-threshold angle. Using the N(2190) anomaly as a test case, this constraint reveals a sign ambiguity in prior data; correcting it yields a phase of $-{28}^{\circ }\pm {10}^{\circ }$, matching our prediction. This consistency validates the method as a model-independent diagnostic for distinguishing compact from molecular states, offering a rigorous tool for exotic spectroscopy.