\begin{table}[t]
\centering\scriptsize
\begin{threeparttable}
\caption{Boundary study in the exponential model (Theorem~\ref{thm:exponential}(v)): mean of $\sqrt N(\theta^{(q)}_r - \widehat\theta_N)$ when $K \approx cN^{\alpha(r,q)}$, against the limit $-\theta_0(\beta_r c_{\mathrm{eff}}^r)^{2^q}$ ($q\ge1$) or $\beta_r\theta_0 c_{\mathrm{eff}}^r$ ($q=0$), with $c_{\mathrm{eff}} = K/N^{\alpha}$ after rounding $m$ to a multiple of the finest split.}\label{tab:boundary}
\begin{tabular}{@{}ccrrrrrrrr@{}}
\toprule
$(r,q)$ & $c$ & $N$ & $K$ & $m$ & reps & mean gap (s.e.) & finite-$m$ prediction & limit at $c_{\mathrm{eff}}$ & gap/limit \\
\midrule
(1,1) & 0.5 & $2^{16}$ & 2,048 & 32 & 2000 & -0.266 (0.00039) & -0.2667 & -0.25 & 1.06 \\
(1,1) & 0.5 & $2^{18}$ & 4,096 & 64 & 2000 & -0.1289 (0.00013) & -0.1291 & -0.125 & 1.03 \\
(1,1) & 0.5 & $2^{20}$ & 16,384 & 64 & 2000 & -0.2582 (0.00013) & -0.258 & -0.25 & 1.03 \\
(1,1) & 0.5 & $2^{22}$ & 32,768 & 128 & 1000 & -0.127 (6.4e-05) & -0.127 & -0.125 & 1.02 \\
(1,1) & 0.5 & $2^{24}$ & 131,072 & 128 & 400 & -0.2541 (0.0001) & -0.254 & -0.25 & 1.02 \\
(1,1) & 1 & $2^{16}$ & 4,096 & 16 & 2000 & -1.138 (0.0012) & -1.138 & -1 & 1.14 \\
(1,1) & 1 & $2^{18}$ & 8,192 & 32 & 2000 & -0.5328 (0.00039) & -0.5329 & -0.5 & 1.07 \\
(1,1) & 1 & $2^{20}$ & 32,768 & 32 & 2000 & -1.065 (0.00038) & -1.066 & -1 & 1.07 \\
(1,1) & 1 & $2^{22}$ & 65,536 & 64 & 1000 & -0.5158 (0.00019) & -0.516 & -0.5 & 1.03 \\
(1,1) & 1 & $2^{24}$ & 262,144 & 64 & 400 & -1.032 (0.00029) & -1.032 & -1 & 1.03 \\
(1,1) & 2 & $2^{16}$ & 8,192 & 8 & 2000 & -5.227 (0.0043) & -5.226 & -4 & 1.31 \\
(1,1) & 2 & $2^{18}$ & 16,384 & 16 & 2000 & -2.274 (0.0012) & -2.276 & -2 & 1.14 \\
(1,1) & 2 & $2^{20}$ & 65,536 & 16 & 2000 & -4.55 (0.0012) & -4.551 & -4 & 1.14 \\
(1,1) & 2 & $2^{22}$ & 131,072 & 32 & 1000 & -2.13 (0.00052) & -2.131 & -2 & 1.07 \\
(1,1) & 2 & $2^{24}$ & 524,288 & 32 & 400 & -4.262 (0.00088) & -4.262 & -4 & 1.07 \\
(1,2) & 0.5 & $2^{16}$ & 8,192 & 8 & 2000 & -0.1068 (0.00018) & -0.1068 & -0.0625 & 1.71 \\
(1,2) & 0.5 & $2^{18}$ & 16,384 & 16 & 2000 & -0.01011 (1.1e-05) & -0.01012 & -0.007812 & 1.29 \\
(1,2) & 0.5 & $2^{20}$ & 65,536 & 16 & 2000 & -0.02023 (1.1e-05) & -0.02023 & -0.01562 & 1.29 \\
(1,2) & 0.5 & $2^{22}$ & 262,144 & 16 & 1000 & -0.04046 (1.6e-05) & -0.04046 & -0.03125 & 1.29 \\
(1,2) & 0.5 & $2^{24}$ & 1,048,576 & 16 & 400 & -0.08092 (2.3e-05) & -0.08091 & -0.0625 & 1.29 \\
(1,2) & 1 & $2^{16}$ & 16,384 & 4 & 2000 & -3.163 (0.0045) & -3.165 & -1 & 3.16 \\
(1,2) & 1 & $2^{18}$ & 32,768 & 8 & 2000 & -0.2131 (0.00017) & -0.2134 & -0.125 & 1.71 \\
(1,2) & 1 & $2^{20}$ & 131,072 & 8 & 2000 & -0.4264 (0.00017) & -0.4265 & -0.25 & 1.71 \\
(1,2) & 1 & $2^{22}$ & 524,288 & 8 & 1000 & -0.8533 (0.00025) & -0.853 & -0.5 & 1.71 \\
(1,2) & 1 & $2^{24}$ & 2,097,152 & 8 & 400 & -1.705 (0.00038) & -1.706 & -1 & 1.71 \\
(1,2) & 2 & $2^{16}$ & 32,768 & 2 & 2000 & -257.5 (0.92) & -- & -16 & 16.09 \\
(1,2) & 2 & $2^{18}$ & 131,072 & 2 & 2000 & -513.1 (0.74) & -- & -32 & 16.03 \\
(1,2) & 2 & $2^{20}$ & 262,144 & 4 & 2000 & -12.64 (0.0044) & -12.64 & -4 & 3.16 \\
(1,2) & 2 & $2^{22}$ & 1,048,576 & 4 & 1000 & -25.29 (0.006) & -25.28 & -8 & 3.16 \\
(1,2) & 2 & $2^{24}$ & 4,194,304 & 4 & 400 & -50.55 (0.01) & -50.57 & -16 & 3.16 \\
(2,0) & 0.5 & $2^{16}$ & 2,048 & 32 & 2000 & -0.5522 (0.0085) & -0.5505 & -0.5 & 1.10 \\
(2,0) & 0.5 & $2^{18}$ & 4,096 & 64 & 2000 & -0.2589 (0.0058) & -0.2622 & -0.25 & 1.04 \\
(2,0) & 0.5 & $2^{20}$ & 16,384 & 64 & 2000 & -0.5192 (0.0058) & -0.5243 & -0.5 & 1.04 \\
(2,0) & 0.5 & $2^{22}$ & 32,768 & 128 & 1000 & -0.2469 (0.0056) & -0.256 & -0.25 & 0.99 \\
(2,0) & 0.5 & $2^{24}$ & 131,072 & 128 & 400 & -0.5122 (0.0089) & -0.5119 & -0.5 & 1.02 \\
(2,0) & 1 & $2^{16}$ & 4,096 & 16 & 2000 & -2.438 (0.014) & -2.438 & -2 & 1.22 \\
(2,0) & 1 & $2^{18}$ & 8,192 & 32 & 2000 & -1.102 (0.0086) & -1.101 & -1 & 1.10 \\
(2,0) & 1 & $2^{20}$ & 32,768 & 32 & 2000 & -2.208 (0.0088) & -2.202 & -2 & 1.10 \\
(2,0) & 1 & $2^{22}$ & 65,536 & 64 & 1000 & -1.036 (0.0085) & -1.049 & -1 & 1.04 \\
(2,0) & 1 & $2^{24}$ & 262,144 & 64 & 400 & -2.104 (0.014) & -2.097 & -2 & 1.05 \\
(2,0) & 2 & $2^{16}$ & 8,192 & 8 & 2000 & -12.19 (0.028) & -12.19 & -8 & 1.52 \\
(2,0) & 2 & $2^{18}$ & 16,384 & 16 & 2000 & -4.863 (0.014) & -4.876 & -4 & 1.22 \\
(2,0) & 2 & $2^{20}$ & 65,536 & 16 & 2000 & -9.737 (0.014) & -9.752 & -8 & 1.22 \\
(2,0) & 2 & $2^{22}$ & 131,072 & 32 & 1000 & -4.391 (0.013) & -4.404 & -4 & 1.10 \\
(2,0) & 2 & $2^{24}$ & 524,288 & 32 & 400 & -8.8 (0.018) & -8.809 & -8 & 1.10 \\
(2,1) & 0.5 & $2^{16}$ & 8,192 & 8 & 2000 & -0.5876 (0.0026) & -0.5863 & -0.25 & 2.35 \\
(2,1) & 0.5 & $2^{18}$ & 16,384 & 16 & 2000 & -0.04726 (0.00026) & -0.0472 & -0.03125 & 1.51 \\
(2,1) & 0.5 & $2^{20}$ & 65,536 & 16 & 2000 & -0.09306 (0.00027) & -0.09326 & -0.0625 & 1.49 \\
(2,1) & 0.5 & $2^{22}$ & 262,144 & 16 & 1000 & -0.1866 (0.00038) & -0.1859 & -0.125 & 1.49 \\
(2,1) & 0.5 & $2^{24}$ & 1,048,576 & 16 & 400 & -0.3725 (0.0006) & -0.3716 & -0.25 & 1.49 \\
(2,1) & 1 & $2^{16}$ & 16,384 & 4 & 2000 & -28.8 (0.17) & -- & -4 & 7.20 \\
(2,1) & 1 & $2^{18}$ & 65,536 & 4 & 2000 & -57.3 (0.2) & -- & -8 & 7.16 \\
(2,1) & 1 & $2^{20}$ & 131,072 & 8 & 2000 & -2.323 (0.0027) & -2.323 & -1 & 2.32 \\
(2,1) & 1 & $2^{22}$ & 524,288 & 8 & 1000 & -4.637 (0.0036) & -4.645 & -2 & 2.32 \\
(2,1) & 1 & $2^{24}$ & 2,097,152 & 8 & 400 & -9.287 (0.0057) & -9.288 & -4 & 2.32 \\
(2,1) & 2 & $2^{16}$ & 16,384 & 4 & 2000 & -28.72 (0.14) & -- & -4 & 7.18 \\
(2,1) & 2 & $2^{18}$ & 65,536 & 4 & 2000 & -57.07 (0.12) & -- & -8 & 7.13 \\
(2,1) & 2 & $2^{20}$ & 262,144 & 4 & 2000 & -113.8 (0.12) & -- & -16 & 7.11 \\
(2,1) & 2 & $2^{22}$ & 1,048,576 & 4 & 1000 & -227.7 (0.18) & -- & -32 & 7.12 \\
(2,1) & 2 & $2^{24}$ & 4,194,304 & 4 & 400 & -455.1 (0.33) & -- & -64 & 7.11 \\
(3,1) & 0.5 & $2^{16}$ & 8,192 & 8 & 2000 & -0.7203 (0.1) & -- & -0.0625 & 11.53 \\
(3,1) & 0.5 & $2^{18}$ & 32,768 & 8 & 2000 & -1.178 (0.0066) & -- & -0.125 & 9.42 \\
(3,1) & 0.5 & $2^{20}$ & 131,072 & 8 & 2000 & -2.339 (0.0077) & -- & -0.25 & 9.36 \\
(3,1) & 0.5 & $2^{22}$ & 524,288 & 8 & 1000 & -4.668 (0.012) & -- & -0.5 & 9.34 \\
(3,1) & 0.5 & $2^{24}$ & 2,097,152 & 8 & 400 & -9.308 (0.016) & -- & -1 & 9.31 \\
(3,1) & 1 & $2^{16}$ & 8,192 & 8 & 2000 & -0.6026 (0.0082) & -- & -0.0625 & 9.64 \\
(3,1) & 1 & $2^{18}$ & 32,768 & 8 & 2000 & -1.178 (0.0065) & -- & -0.125 & 9.42 \\
(3,1) & 1 & $2^{20}$ & 131,072 & 8 & 2000 & -2.328 (0.0084) & -- & -0.25 & 9.31 \\
(3,1) & 1 & $2^{22}$ & 524,288 & 8 & 1000 & -4.641 (0.01) & -- & -0.5 & 9.28 \\
(3,1) & 1 & $2^{24}$ & 2,097,152 & 8 & 400 & -9.299 (0.015) & -- & -1 & 9.30 \\
(3,1) & 2 & $2^{16}$ & 8,192 & 8 & 2000 & -0.6075 (0.0075) & -- & -0.0625 & 9.72 \\
(3,1) & 2 & $2^{18}$ & 32,768 & 8 & 2000 & -1.19 (0.011) & -- & -0.125 & 9.52 \\
(3,1) & 2 & $2^{20}$ & 131,072 & 8 & 2000 & -2.332 (0.0067) & -- & -0.25 & 9.33 \\
(3,1) & 2 & $2^{22}$ & 524,288 & 8 & 1000 & -4.65 (0.01) & -- & -0.5 & 9.30 \\
(3,1) & 2 & $2^{24}$ & 2,097,152 & 8 & 400 & -9.289 (0.016) & -- & -1 & 9.29 \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item Exact-Gamma simulation of block sufficient statistics; seed 20261028 plus a cell hash. The ratio converges to one only as $m\to\infty$; with $m \asymp N^{1/(2^{q+1}r)}$ the local sample sizes in the table are small and the finite-$m$ bias $\theta_0\sum_j w_js_j/(m-s_j)$ exceeds its leading term. Source: \texttt{results/processed/boundary\_summary.csv}.
\end{tablenotes}
\end{threeparttable}
\end{table}