Detectability of a binary system containing a 1.4 M_⊙ MSP with LISA, shown as a function of companion mass and gravitational-wave frequency (bottom x-axis) or orbital period (top x-axis). Grey contours indicate LISA’s 4-year signal-to-noise threshold (ρ_4 yr = 7) for sources located at 5 (dark grey), 8 (grey), and 11 kpc (light grey). Dashed horizontal lines mark approximate boundaries between different classes of MSP binaries: black widows (low-mass companions, M ≲ 0.06 M_⊙), redbacks and He WDs (intermediate-mass companions, 0.06 ≲ M ≲ 0.5 M_⊙), and more massive donors such as CO/ONeMg WDs or He stars (M ≳ 0.5 M_⊙). The grey solid curve marks the Roche-lobe overflow (RLOF) boundary for WD donors, indicating the onset of stable mass transfer. Observed MSP binaries are shown as coloured symbols, following the same classification as in Fig. 1. UCXBs lying near the WD RLOF line are shown as light blue pentagons. The two-segment grey arrow illustrates the expected evolutionary track of close binaries.