A theoretical study has constructed complete bases of parity-violating monomials for generally covariant scalar-tensor gravity (GST) and spatially covariant gravity (SCG) up to derivative grade d=4. Including all derivative partitions, each description contains one linearly independent monomial at d=3 and thirteen at d=4. At the Lagrangian level, six integration-by-parts relations reduce polynomial Lagrangians to combinations of one d=3 and seven d=4 monomials with general coefficient functions.

The work also establishes the correspondence between the two bases, retaining general coefficients and total derivatives, and proves their independence under integrations by parts. Among the seven d=4 SCG monomials, five contain neither the lapse velocity nor the time derivative of the extrinsic curvature. For the two remaining monomials, the authors performed a Hamiltonian constraint analysis with a standard Einstein-Hilbert kinetic term and a lapse-dependent potential.

On the regular branches considered, the lapse-velocity term propagates four degrees of freedom, while the extrinsic-curvature-velocity term propagates six degrees of freedom, both alone and when combined with the lapse-velocity term. These results provide a complete basis for constructing parity-violating scalar-tensor theories with three degrees of freedom up to d=4 within this setting, with the parity-violating Lagrangian spanned by six GST polynomials: one at d=3 and five at d=4.

Counting degrees of freedom is key to distinguishing healthy theories from those plagued by instabilities or ghost modes. By fixing which operator combinations are genuinely independent and how many propagation modes each produces, the work avoids duplicates and provides a reusable catalogue for future model building with parity violation, relevant in cosmology and in the search for signatures of physics beyond general relativity.