Scalar-polarization limits from LIGO, Virgo and KAGRA bound the transverse-positivity branch of postquantum-classical gravity

In the two-coupling form of postquantum-classical gravity studied by Oppenheim and Sajjad, gravitational-wave data bound the diffusion of the Newtonian potential only if positivity is imposed on all ten metric components. If only the transverse components are required to be positive, the scalar diffusion constant D2,scalar is bounded only by LISA Pathfinder, at 1e-55, which leaves about 10.9 orders of magnitude open above the full-positivity ceiling. We point out that the dynamical scalar of the theory, psi, obeys the same stochastic wave equation as the tensor modes, and that on shell the psi-dependent part of the constrained potential is Phi = -psi. The resulting tidal field is a transverse breathing wave of strain h_b = 2 psi, so its stochastic background is constrained by the scalar-polarization searches of the LIGO, Virgo and KAGRA collaborations. In the overlap-function normalisation used for those searches, Omega_S/Omega_T = (1+2x)/(12x) with x = (alpha - 3 beta)/alpha. Hence D2,scalar <= 4 (1+8x)/(1+2x) Omega_S^max / A for every x > 0, where A is Omega_T per unit 1/alpha at 25 Hz. With the published O1 to O4c1 limits and log-uniform amplitude priors, the ceiling on the transverse-positivity branch is D2,scalar <~ 1.0e-65. An estimated uniform-prior equivalent gives 4e-65. Either way, 9.4 to 10.0 of the 10.9 open orders are closed, and the transverse branch remains about 1.5 orders of magnitude wider than the full-positivity window computed from the same data. The result assumes that psi is damped by Hubble friction in the same way as the tensor modes, an extension of the treatment Oppenheim and Sajjad apply to the tensor sector, and it inherits their linearised, Markovian approximation.