Induced Couplings and Causal Bounds from Nondegenerate Dirac Lagrangians
The standard Dirac Lagrangian is linear in the field derivatives and therefore has a vanishing Hessian. We identify the minimal null deformations of this Lagrangian that make the covariant Legendre map locally invertible. Imposing global phase invariance, reality, proper Poincaré covariance, and absence of external background tensors leaves the two-parameter spinorial bivector $\mathcal E^{μν}=\ellσ^{μν} +\ell_5{}^\starσ^{μν}$, where the star denotes the Hodge dual and $\ell^2+\ell_5^2\neq0$. This extends the analysis of \cite{struckmeier2024pauli} by a parity-odd term. After minimal $U(1)$ gauging, this free null term is no longer variationally trivial and induces magnetic- and electric-dipole Pauli operators, together with an identically conserved dipole current. These dipole terms make the species-dependent regularization lengths directly constrainable by precision moment measurements. Metric-affine gauging in a Lorentz-spinor prescription then produces spin-curvature, torsion, and nonmetricity corrections to the Dirac operator. Applying the Velo-Zwanziger criterion, we exclude all nonzero pure axial, pure trace, and mixed axial-trace torsion backgrounds, as well as nonzero pure Weyl and second-trace nonmetricity backgrounds. The combined trace-vector nonmetricity sector is not excluded only when the effective trace vector vanishes. Tensor torsion and general tracefree nonmetricity remain unclassified without further algebraic assumptions, while the Levi-Civita limit preserves the metric light cone and leaves only lower-order curvature-dependent mass terms. Thus, gauging a Legendre-regular Dirac representative turns a free variational ambiguity into observable couplings, while non-Riemannian sectors are sharply constrained by causality bounds.