% Table 4: Full OAT results — includes low-mass boundary rows for ground-based detectors
\begin{table*}[ht]
\centering
\caption{Fisher $1\sigma$ uncertainties on $H_0$ [$\kms$] from GW memory, scanning baseline
parameter variations one-at-a-time (OAT). The $N\!=\!100^\star$ column gives the direct
Fisher result for $d_L\!\leq\!700$\,Mpc; $N_{\rm lo}$/$N_{\rm hi}$ are full-volume
projections at the low/high merger-rate bounds. The mass range column shows
$[M_{\rm min}, M_{\rm max}]$\,[$M_\odot$]; a dash indicates the detector default is used
(no lower cut applied for the baseline; low-mass boundary rows show explicit lower limits).
Each cell: mean\,$\pm$\,scatter (MC error).}
\label{tab:sigma_H0_combined}
\setlength{\tabcolsep}{4pt}
{\small
\begin{tabular}{l l r r r c c c c}
\toprule
Det. & Approx. & $T_{\rm obs}$\,[yr] & $f_{\rm min}$\,[Hz] & $N_{\rm samp}$ & $M$\,range\,[$M_\odot$] &
  $N\!=\!100^\star$ & $N_{\rm lo}$ & $N_{\rm hi}$ \\
\midrule
%% -----------------------------------------------
%% aLIGO/Virgo
%% -----------------------------------------------
HLV & MWM & 8 & 5 & -- & $[{-},{100}]$ & $398.8\pm32.4(10.2)^\star$ & $96.1\pm33.4(10.6)$ & $58.8\pm20.4(6.5)$ \\
 & MWM & 4 & 5 & -- & $[{-},{100}]$ & -- & $96.5\pm34.0(10.7)$ & $59.1\pm20.8(6.6)$ \\
 & MWM & 16 & 5 & -- & $[{-},{100}]$ & -- & $97.2\pm32.7(10.3)$ & $59.5\pm20.0(6.3)$ \\
 & MWM & 8 & 20 & -- & $[{-},{100}]$ & -- & $96.2\pm33.4(10.6)$ & $58.9\pm20.4(6.5)$ \\
 & MWM & 8 & 40 & -- & $[{-},{100}]$ & -- & $109.0\pm28.0(8.9)$ & $66.7\pm17.1(5.4)$ \\
 & MWM & 8 & 60 & -- & $[{-},{100}]$ & -- & $136.5\pm32.9(10.4)$ & $83.6\pm20.1(6.4)$ \\
 & MWM & 8 & 5 & -- & $[{-},{80}]$ & -- & $96.2\pm33.4(10.6)$ & $58.9\pm20.4(6.5)$ \\
 & MWM$^\dagger$ & 8 & 5 & -- & $[{-},{100}]$ & -- & $95.2\pm33.2(10.5)$ & $58.3\pm20.3(6.4)$ \\
 & NRSur & 8 & 5 & -- & $[{-},{100}]$ & -- & $91.8\pm21.6(6.8)$ & $56.2\pm13.3(4.2)$ \\
\midrule
%% -----------------------------------------------
%% ET
%% -----------------------------------------------
ET & MWM & 8 & 2 & -- & $[{-},{100}]$ & $26.1\pm1.6(0.5)^\star$ & $1.70\pm0.33(0.10)$ & $0.984\pm0.191(0.061)$ \\
 & MWM & 4 & 2 & -- & $[{-},{100}]$ & -- & $1.68\pm0.36(0.11)$ & $0.971\pm0.205(0.065)$ \\
 & MWM & 16 & 2 & -- & $[{-},{100}]$ & -- & $1.76\pm0.31(0.10)$ & $1.01\pm0.18(0.06)$ \\
 & MWM & 8 & 5 & -- & $[{-},{100}]$ & -- & $1.73\pm0.33(0.11)$ & $0.998\pm0.193(0.061)$ \\
 & MWM & 8 & 20 & -- & $[{-},{100}]$ & -- & $2.86\pm0.54(0.17)$ & $1.65\pm0.31(0.10)$ \\
 & MWM & 8 & 40 & -- & $[{-},{100}]$ & -- & $3.32\pm0.63(0.20)$ & $1.92\pm0.36(0.12)$ \\
 & MWM & 8 & 2 & -- & $[{-},{80}]$ & -- & $1.71\pm0.33(0.10)$ & $0.987\pm0.190(0.060)$ \\
 & MWM$^\dagger$ & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.70\pm0.32(0.10)$ & $0.979\pm0.187(0.059)$ \\
 & NRSur & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.61\pm0.27(0.09)$ & $0.931\pm0.158(0.050)$ \\
\midrule
%% -----------------------------------------------
%% CE
%% -----------------------------------------------
CE & MWM & 8 & 2 & -- & $[{-},{100}]$ & $13.1\pm1.3(0.4)^\star$ & $1.06\pm0.30(0.09)$ & $0.611\pm0.172(0.054)$ \\
 & MWM & 4 & 2 & -- & $[{-},{100}]$ & -- & $1.04\pm0.26(0.08)$ & $0.598\pm0.152(0.048)$ \\
 & MWM & 16 & 2 & -- & $[{-},{100}]$ & -- & $1.06\pm0.30(0.09)$ & $0.612\pm0.171(0.054)$ \\
 & MWM & 8 & 5 & -- & $[{-},{100}]$ & -- & $1.06\pm0.30(0.09)$ & $0.611\pm0.172(0.054)$ \\
 & MWM & 8 & 20 & -- & $[{-},{100}]$ & -- & $1.33\pm0.34(0.11)$ & $0.770\pm0.196(0.062)$ \\
 & MWM & 8 & 40 & -- & $[{-},{100}]$ & -- & $2.17\pm0.65(0.20)$ & $1.25\pm0.37(0.12)$ \\
 & MWM & 8 & 2 & -- & $[{-},{80}]$ & -- & $1.06\pm0.30(0.09)$ & $0.612\pm0.171(0.054)$ \\
 & MWM$^\dagger$ & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.05\pm0.29(0.09)$ & $0.607\pm0.170(0.054)$ \\
 & NRSur & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.00\pm0.26(0.08)$ & $0.578\pm0.151(0.048)$ \\
\midrule
%% -----------------------------------------------
%% ET+CE
%% -----------------------------------------------
ET$+$CE & MWM & 8 & 2 & -- & $[{-},{100}]$ & $11.7\pm1.0(0.3)^\star$ & $1.11\pm0.28(0.09)$ & $0.638\pm0.163(0.051)$ \\
 & MWM & 4 & 2 & -- & $[{-},{100}]$ & -- & $1.11\pm0.28(0.09)$ & $0.638\pm0.163(0.051)$ \\
 & MWM & 16 & 2 & -- & $[{-},{100}]$ & -- & $1.11\pm0.27(0.09)$ & $0.643\pm0.157(0.050)$ \\
 & MWM & 8 & 5 & -- & $[{-},{100}]$ & -- & $1.11\pm0.28(0.09)$ & $0.640\pm0.164(0.052)$ \\
 & MWM & 8 & 20 & -- & $[{-},{100}]$ & -- & $1.52\pm0.38(0.12)$ & $0.876\pm0.220(0.069)$ \\
 & MWM & 8 & 40 & -- & $[{-},{100}]$ & -- & $2.29\pm0.56(0.18)$ & $1.32\pm0.32(0.10)$ \\
 & MWM & 8 & 2 & -- & $[{-},{80}]$ & -- & $1.11\pm0.28(0.09)$ & $0.639\pm0.163(0.051)$ \\
 & MWM$^\dagger$ & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.10\pm0.28(0.09)$ & $0.634\pm0.161(0.051)$ \\
 & NRSur & 8 & 2 & -- & $[{-},{100}]$ & -- & $1.06\pm0.24(0.08)$ & $0.611\pm0.141(0.045)$ \\
\midrule
%% -----------------------------------------------
%% LISA (light seeds)
%% -----------------------------------------------
LISA-L & MWM & -- & -- & 16384 & $[{10^{4}},{10^{6}}]$ & -- & $341.9\pm118.5(37.5)$ & $108.1\pm37.5(11.8)$ \\
 & MWM & -- & -- & 4096 & $[{10^{4}},{10^{6}}]$ & -- & $335.9\pm111.6(35.3)$ & $106.2\pm35.3(11.2)$ \\
 & MWM & -- & -- & 8192 & $[{10^{4}},{10^{6}}]$ & -- & $338.4\pm115.3(36.5)$ & $107.0\pm36.5(11.5)$ \\
 & MWM$^\dagger$ & -- & -- & 16384 & $[{10^{4}},{10^{6}}]$ & -- & $304.2\pm87.5(27.7)$ & $96.2\pm27.7(8.7)$ \\
 & NRSur & -- & -- & 16384 & $[{10^{4}},{10^{6}}]$ & -- & $274.2\pm80.5(25.5)$ & $86.7\pm25.5(8.0)$ \\
\midrule
%% -----------------------------------------------
%% LISA (heavy seeds)
%% -----------------------------------------------
LISA-H & MWM & -- & -- & 16384 & $[{10^{6}},{10^{8}}]$ & -- & $510.9\pm389.3(123.1)$ & $255.4\pm194.6(61.6)$ \\
 & MWM & -- & -- & 4096 & $[{10^{6}},{10^{8}}]$ & -- & $517.3\pm306.8(97.0)$ & $258.7\pm153.4(48.5)$ \\
 & MWM & -- & -- & 8192 & $[{10^{6}},{10^{8}}]$ & -- & $518.4\pm370.9(117.3)$ & $259.2\pm185.5(58.7)$ \\
 & MWM$^\dagger$ & -- & -- & 16384 & $[{10^{6}},{10^{8}}]$ & -- & $355.3\pm204.1(64.6)$ & $177.6\pm102.1(32.3)$ \\
 & NRSur & -- & -- & 16384 & $[{10^{6}},{10^{8}}]$ & -- & $453.0\pm305.4(96.6)$ & $226.5\pm152.7(48.3)$ \\
\bottomrule
\end{tabular}
}% end \small
\begin{minipage}{\linewidth}
\smallskip\footnotesize
$^\dagger$~MWM with NRSur7dq2 prior bounds ($q\!\geq\!0.5$, $|\chi|\!\leq\!0.8$).
$^\star$~Direct Fisher result at $d_L\!\leq\!700$\,Mpc ($N\!=\!100$); no $\sqrt{N}$ extrapolation.
$^\ddagger$~LISA uses SFR-like prior ($\alpha\!=\!2.7$); true SMBHB rate evolution poorly constrained.
$N_{\rm lo}\!=\!5$, $N_{\rm hi}\!=\!50$ total LISA events over the mission.
A dash (--) in the $M$ range lower bound indicates the ground-based default with no lower cut applied.
\end{minipage}
\end{table*}