This paper builds a systematic catalogue of parity‑violating interactions in a class of gravity theories called spatially covariant gravity (SCG). “Parity‑violating” means the laws treat left‑ and right‑handed circular polarizations of gravitational waves differently. The authors extend a previous construction to include all polynomial operators that carry five total derivatives. Their goal is to show what kinds of parity‑odd terms can appear at this derivative order and how those terms would affect the simplest gravitational‑wave signals.

To make the list manageable, the authors count time and space derivatives separately (d = d_t + d_s). They then write all candidate monomials and reduce them using standard manipulations: integrations by parts, index symmetries, three‑dimensional curvature identities, the Schouten identity, and the Cayley–Hamilton relation. After this algebraic reduction they find a 59‑element basis of independent parity‑odd SCG monomials at total derivative order d = 5. These split into three sectors by derivative content: 2 monomials with (d_t,d_s) = (0,5), 40 with (2,3), and 17 with (4,1).

The authors further sort the 59 operators by whether they involve certain time‑derivative structures. Thirty‑five monomials do not contain either the lapse velocity (denoted L_u ln N, the time change of the lapse function) nor L_u K_{ij} (a higher normal derivative of the spatial metric). Eleven monomials do contain the lapse velocity, and 13 contain the L_u K_{ij} type pieces. The last two groups carry potentially dangerous extra time derivatives and therefore need a separate degeneracy and constraint analysis to decide whether they introduce unwanted degrees of freedom. The present paper highlights this need rather than completing that stability analysis.