Spin-Inversion Degeneracies in Restricted Inspiral Waveforms for LISA

Spin inversions appear in several settings. Analytical work predicts a single spin flip during unequal-mass supermassive binary inspiral, while numerical relativity and post-Newtonian calculations show repeated flip-flop motion in comparable-mass binaries. Secular spin evolution also predicts additional cases driven by spin-induced mass quadrupoles. Whether these effects can be distinguished in gravitational-wave data is still unclear. We combine the secular spin angle equations with a quasi-circular second post-Newtonian frequency evolution and build a restricted waveform weighted by the sky-averaged LISA sensitivity. We study five near-equal-mass injections with detector-frame total mass M=2×105M⊙, including Kerr flip-flops and one quadrupole-induced case. Each injection is compared with physically evolving waveforms constrained to have no orbital-plane crossings. We search all four no-inversion sectors, vary the masses, spin magnitudes, and initial spin angles, and maximize over time, phase, and overall amplitude. Large spin motion does not by itself lead to a clearly different waveform in this restricted model. We compare configurations with different spin magnitudes, relative spin orientations, and angular excursions at eleven observer inclinations between 0° and 90°. With each injection normalized to signal SNR ρ⋆=100, four of the five cases leave a residual SNR below unity for the 0° viewing direction along the total angular momentum. At nonzero inclinations, the residual associated with the largest match found reaches 7.07 in the two-active-spin case at 55°. These upper bounds on the minimum residual within the allowed template family suggest potential distinguishability of spin inversions and motivate studies with more complete waveforms and the full LISA response.