Nonlinear Boundary Degeneracy and the Limits of Scalar Diagnostics

Scalar order parameters are widely used to describe critical transitions in nonlinear systems, yet their structural validity is rarely stated explicitly. This paper identifies a sharp obstruction to scalar diagnostics that arises when regime boundaries are generated by interacting degeneracies. We show that if the reduced boundary ideal is non-principal, then the boundary-induced comparability structure necessarily contains incomparabilities, and no scalar index can be faithful (even up to monotone transformation) to the underlying boundary constraints. Our analysis yields a boundary-first classification. In the codimension-one principal regime, the boundary is a hypersurface with a canonical generator Δred (unique up to units and admissible reparametrizations), and scalar crisis diagnostics are structurally pinned down to a single channel. Beyond this regime, scalarization fails intrinsically: the failure is ideal-theoretic rather than a calibration artifact. We illustrate the mechanism by a minimal two-fold construction producing a non-principal boundary, and introduce the positive-degree Betti Profile Index BPI∗ as a minimal cohomological gatekeeper that becomes informative exactly when the scalar paradigm is no longer defensible.