A Redshift-Shape Consistency Test for Single-Canonical-Scalar Birefringent Quintessence

This short technical note derives a necessary redshift-shape consistency test for single-canonical-scalar birefringent quintessence. If one canonical scalar field both sources dynamical dark energy and produces cosmic birefringence through a photon coupling of the form (\theta F_{\mu\nu}\widetilde F^{\mu\nu}), then the scalar rolling history that determines (w_\theta(z)) also determines the birefringence history (\beta(z)). The amplitude depends on the coupling ratio (c_\gamma/f), but the normalized redshift shape (\beta(z)/\beta(z_{\rm ref})) is coupling-independent once the scalar history is fixed. The note emphasizes that this relation is a necessary null test, not a sufficient proof of the one-field hypothesis. It also discusses the strict all-dark-energy case, the subcomponent case, the no-phantom limitation of canonical scalar dynamics, benchmark coupling scales, and the role of anisotropic birefringence as a perturbation-level consistency check.