\section{Monte Carlo evidence}\label{sec:mc}

\subsection{Design}
All experiments are paired: every estimator in a cell is computed on the same simulated sample and the same contiguous partition (observations are i.i.d., so contiguous and random partitions are equivalent in law). The designs are (i) the exponential model, $N \in \{2^{14}, 2^{16}, 2^{18}, 2^{20}, 2^{22}\}$, $m \in \{8, 16, \dots, 1024\}$, \numMCrepsExp{} replications per cell up to $N = 2^{20}$ and $500$ at $N = 2^{22}$ (where the $m = 8$ cell was dropped for cost); (ii) logistic regression with $p \in \{3, 8\}$ (intercept and standard normal covariates, coefficients $0.3$ and $\pm0.5/(1+0.15j)$), $N \in \{2^{14},\dots,2^{20}\}$, $m \in \{32, \dots, 1024\}$ with $m \ge 8p$, \numMCrepsLogit{} replications, plus a stress design with $p = 20$ and $m \ge 6p$ (500 replications); (iii) Poisson regression with the same structure and half the coefficients; (iv) robustness designs: a misspecified logistic model with an omitted quadratic term $0.5(x_1^2 - 1)$ in the index, an ill-conditioned logistic design (covariate covariance with condition number $100$), an exponential sample contaminated by one percent of observations drawn with ten times the scale, and logistic designs with very small blocks. Every cell records, for each estimator, the scaled gap to the centralized MLE $\sqrt N\norm{\tilde\theta - \thetaN}$, the scaled error $\sqrt N\norm{\tilde\theta - \theta_0}$, Wald intervals from the observed information at the estimate, the status of every local fit, the condition number of the aggregated Hessian and the Newton residuals. Replications in which local fits failed are kept, and the tables report both the root-mean-square gap over all replications and its median, together with the fraction of workers whose sub-block fits all converged; rows labelled ``valid workers only'' drop the workers with a failed fit from the pilot. In the misspecified designs the target of the coverage statistics is the pseudo-true value, estimated by the average of the centralized pseudo-MLE over replications. Monte Carlo standard errors accompany every summary in the processed files and are at most a few percent of the reported gaps. The cells were run on the author's laptop and on 18 nodes of the MacLustr cluster (\cref{app:replication}); the simulations are in memory and measure no communication.

\subsection{Rates, the frontier, and the random remainder}
\Cref{fig:rmsgap-exp,fig:rmsgap-logit} plot the root-mean-square scaled gap against $m$ at the largest $N$, and \cref{tab:mc_rates} fits log-log slopes over the failure-free cells. The ordering predicted by \cref{tab:exponents} holds in every design: the plain average decays like $m^{-1}$ (slopes between $-1.0$ and $-1.1$), one step from it like $m^{-2}$ ($-2.0$ to $-2.1$), two steps like $m^{-4}$ ($-4.1$ to $-4.2$). The half-block pilot has slope $-2$ in the exponential model but only $-1.2$ to $-1.4$ in the regressions, and the one-step estimators from the corrected pilots have slopes between $-2.3$ and $-2.9$ rather than $-4$; the analytic correction with one step has slope $-1$ everywhere. These shallower slopes are not a failure of the bias cancellation but the signature of the centered random remainder of \cref{lem:average}: once the bias of order $m^{-r}$ has been removed, the gap of the one-step estimator is governed by the square of the remainder, $(\sqrt K/N)^2$, whose scaled size $\sqrt N\,K/N^2 = (m\sqrt N)^{-1}$ decays only like $m^{-1}$ at fixed $N$. The term vanishes asymptotically, as \cref{thm:frontier} requires, but at finite $N$ it is the visible floor for every estimator whose bias has been cancelled, and it is the reason the proofs keep it. \Cref{fig:collapse} shows the gaps against $K/N^{\alphafr{r}{q}}$ for all $N$: the curves of different $N$ collapse for each $(r,q)$ wherever the bias term dominates.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig_rms_gap_vs_m_exp_p1.pdf}\hfill
\includegraphics[width=0.49\textwidth]{fig_rms_gap_vs_m_logit_p8.pdf}
\caption{Root-mean-square of $\sqrt N\,\lVert\tilde\theta - \widehat\theta_N\rVert$ against the local sample size $m$ at the largest $N$ of the design, exponential model (left, $N = \NexpMax$) and logistic regression with $p = 8$ (right, $N = 2^{20}$). Estimators are labelled by $(r,q)$; the analytic Cox--Snell correction uses one local fit; CSL is the full surrogate maximization of \citet{jordan2019communication} with one round. Logarithmic axes; the values at the smallest $m$ of the logistic design include replications with failed sub-block fits (\cref{tab:mc_logit_p8}).}\label{fig:rmsgap-exp}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig_rms_gap_vs_m_pois_p8.pdf}\hfill
\includegraphics[width=0.49\textwidth]{fig_rms_gap_vs_m_logit_p3.pdf}
\caption{As \cref{fig:rmsgap-exp}, Poisson regression with $p = 8$ (left) and logistic regression with $p = 3$ (right).}\label{fig:rmsgap-logit}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig_gap_collapse_logit_p8.pdf}
\caption{Root-mean-square scaled gap against $K/N^{\alpha(r,q)}$ for several $N$ (logistic regression, $p = 8$). Under \cref{thm:frontier} the bias part of the gap is a function of this ratio, so the curves for different $N$ coincide where the bias dominates and separate where the $(m\sqrt N)^{-1}$ floor takes over.}\label{fig:collapse}
\end{figure}

\subsection{Bias, variance, and the cost of higher order}
\Cref{tab:mc_exp_p1,tab:mc_logit_p3,tab:mc_logit_p8,tab:mc_pois_p3,tab:mc_pois_p8} report gaps and coverage, and \cref{fig:rmse-order} the RMSE against $\theta_0$ of the three pilots. Three facts qualify the exponents. First, higher order is not monotone in finite samples. Across all cells of all designs the quarter-block pilot has a smaller gap than the half-block pilot in \fracThreeBeatsTwoSmallM\% of the cells with $m = 16$ and in \fracThreeBeatsTwoLargeM\% of the cells with $m \ge 256$, where its bias is already negligible and the remainder amplification of the weights $(8/3,-2,1/3)$ dominates. In logistic regression with $p = 8$ at $m = 256$ the one-step estimator from the quarter-block pilot has gap \gapLogitEightThreeOneMTwoFiftySix{} against \gapLogitEightTwoOneMTwoFiftySix{} for the half-block pilot, because the quarter-blocks contain $64$ observations for $8$ parameters; in Poisson regression with $p = 8$ the ranking is reversed at $m = 128$ (\gapPoisEightThreeOneMOneTwentyEight{} against \gapPoisEightTwoOneMOneTwentyEight) and restored at $m = 256$ (\gapPoisEightThreeOneMTwoFiftySix{} against \gapPoisEightTwoOneMTwoFiftySix). Second, two Newton steps from the plain average, $(1,2)$, which share the exponent $7/8$ with $(2,1)$, have the smaller gap in \fracOneTwoBeatsTwoOne\% of the cells (\gapLogitEightOneTwoMTwoFiftySix{} against \gapLogitEightTwoOneMTwoFiftySix{} in the logistic example): the plain pilot's error is dominated by its deterministic bias, which two squarings remove, whereas the corrected pilot's error contains the random remainder, which one squaring turns into the $(m\sqrt N)^{-1}$ floor. Third, the analytic first-order correction of \citet{cordeiro1991bias}, which costs one local fit and no sub-blocks, beats the half-block jackknife with one step in \fracCSBeatsTwoOneLogit\% of the logistic cells and in every exponential and Poisson cell (\gapLogitEightCSMTwoFiftySix{} at $m = 256$ in the logistic example), but in none of the contaminated exponential cells, where the bias formula of the assumed model is wrong: the jackknife needs no formula and no third derivatives, which is its advantage under misspecification and outside the canonical GLM class. The stale-curvature variant of $(2,1)$ sits between the one-step and the pilot (\gapLogitEightStaleMTwoFiftySix{} at $m = 256$), in the order of \cref{thm:inexact}.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig_rmse_theta0_vs_m_exp_p1.pdf}\hfill
\includegraphics[width=0.49\textwidth]{fig_rmse_theta0_vs_m_logit_p8.pdf}
\caption{Root-mean-square of $\sqrt N\,\lVert\tilde\theta - \theta_0\rVert$ (error against the truth, not against $\widehat\theta_N$) for the pilots of order $1$, $2$, $3$, the one-step estimators from the plain and half-block pilots, and the analytic correction; exponential (left) and logistic $p = 8$ (right). The centralized MLE has RMSE close to $\sqrt{\mathrm{tr}\,\Sigma_\psi}$; a pilot whose RMSE exceeds it pays for its bias (order 1) or for its remainder variance and failed sub-block fits (order 3).}\label{fig:rmse-order}
\end{figure}

\subsection{Failures at small blocks}
When the finest sub-blocks are small relative to $p$, local MLEs fail to exist (separation in logistic regression) or converge to extreme values, and the construction inherits the failure: the tables report the fraction of workers whose sub-block fits all converged. In logistic regression with $p = 3$ and $m = 32$, \validLogitThreeRtwoMThirtyTwo\% of the workers have valid half-blocks but only \validLogitThreeRthreeMThirtyTwo\% have valid quarter-blocks; with $p = 8$ the quarter-block fits are valid for \validLogitEightRthreeMSixtyFour\% of the workers at $m = 64$ and for \validLogitEightRthreeMOneTwentyEight\% at $m = 128$; with $p = 20$ the half-block fits are valid for \validPtwentyRtwoMOneTwentyEight\% of the workers at $m = 128$ and the quarter-block fits for \validPtwentyRthreeMTwoFiftySix\% at $m = 256$. In those cells the root-mean-square gaps are dominated by a few replications in which an invalid local estimate enters the average and the Newton iteration starts far outside the basin of \cref{thm:exponential}(iv); the medians are orders of magnitude smaller, and dropping the invalid workers from the pilot (rows ``valid workers only'') restores the behaviour of the failure-free cells at the price of a small loss of data. The safeguard is a necessity, not an option: the theory assumes that local MLEs exist, and \cref{ass:lin} fails on exactly the sub-blocks that the higher-order patterns create.

\subsection{Coverage}
\Cref{fig:coverage} and the coverage columns of the tables compare empirical and nominal 95\% coverage of intervals built from the observed information at each estimate. Inside the frontier the coverage of $(2,1)$, $(1,2)$, $(3,1)$ and of the analytic correction with one step is indistinguishable from that of the centralized MLE. The plain average under-covers at every $m$ of the grids, since its bias $\theta_0/m$ exceeds the standard error $\theta_0/\sqrt N$ whenever $m < \sqrt N$; the one-step estimator $(1,1)$ and the half-block pilot $(2,0)$, which share the exponent $3/4$, under-cover for $m \lesssim N^{1/4}$ (in the exponential model at $N = 2^{20}$, coverage \covExpOneOneMThirtyTwo{} and \covExpTwoZeroMThirtyTwo{} at $m = 32$), and $(2,1)$ is nominal from $m = 16$ on (\covExpTwoOneMSixteen). Under misspecification the relevant target is the pseudo-true value: with it, all estimators inside the frontier have nominal coverage in the misspecified logistic design with either the observed information or the sandwich, because the information identity nearly holds for a Bernoulli outcome with the fitted mean; in the contaminated exponential design the two differ, the observed-information intervals of every estimator including the centralized pseudo-MLE cover the pseudo-true rate with probability \covContamFisherMSixtyFour{} at $m = 64$ while the sandwich intervals cover it with probability \covContamSandwichMSixtyFour{} (\cref{tab:mc_exp_contam_p1}), as \cref{cor:inference} and \cref{prop:robust}(ii) state.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{fig_coverage95_exp_p1.pdf}\hfill
\includegraphics[width=0.49\textwidth]{fig_coverage95_logit_p8.pdf}
\caption{Empirical coverage of nominal 95\% Wald intervals (observed information at the estimate) against $m$, exponential (left) and logistic $p = 8$ (right); Monte Carlo standard error about $0.005$ (exponential) and $0.007$ (logistic).}\label{fig:coverage}
\end{figure}

\subsection{The boundary}
\Cref{tab:boundary} and \cref{fig:boundary} report the exact-model boundary study: $K$ is set to $cN^{\alphafr{r}{q}}$ (rounded so that $m$ is a multiple of the finest split) and the mean of $\sqrt N(\iter{r}{q} - \thetaN)$ is compared with the limit of \cref{thm:exponential}(v) at the realized $c_{\mathrm{eff}} = K/N^{\alpha}$ and with a finite-$m$ prediction, $-\sqrt N\,\E[(\pilot{r} - \thetaN)^{2}]/\theta_0$ for $q = 1$, computed from the exact bias and the variance of the local remainder (\cref{app:exp}). The finite-$m$ prediction matches the Monte Carlo means to three digits in every cell. The ratio of the mean to the asymptotic limit decreases towards one as $m$ grows: for $(1,1)$ and $(2,0)$, where $m \asymp N^{1/4}$, it is within \boundaryRatioOneOneMax\% of one at $N = 2^{24}$; for $(2,1)$, where $m \asymp N^{1/8}$, the local sample sizes at $N \le 2^{24}$ do not exceed \boundaryMaxM{} and the ratio is still between \boundaryRatioTwoOneMin{} and \boundaryRatioTwoOneMax. The boundary constants are confirmed as the limit of a regime that is reached only at very large $N$; at any practical $N$ the finite-$m$ formula is the relevant prediction.

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig_boundary_exponential.pdf}
\caption{Boundary study in the exponential model: ratio of the Monte Carlo mean of $\sqrt N(\theta^{(q)}_r - \widehat\theta_N)$ to the asymptotic limit of \cref{thm:exponential}(v) (filled markers, $\pm1.96$ standard errors) and of the finite-$m$ prediction to the same limit (hollow markers), against the local sample size $m$ that $K = cN^{\alpha(r,q)}$ implies at each $N$.}\label{fig:boundary}
\end{figure}

\subsection{Ablations: patterns, rounds, curvature and quantization}
\Cref{tab:abl_exponential,tab:abl_logistic} vary the split pattern, the number of rounds $q \in \{0,\dots,3\}$, the curvature used by the coordinator and the number of mantissa bits of the transmitted score and Hessian, on paired samples. Three findings. (a) At equal order, the pattern $(1,2,3)$ of \citet{dhaene2015split} and the doubling pattern $(1,2,4)$ are close when all sub-block fits are valid; when the quarter-blocks are too small for the model, $(1,2,3)$, whose finest block is larger, and even the two-level pattern $(1,3)$ are far better, and the order-4 patterns are worse than order 3 at every $m$ of the grids. (b) With stale curvature the first round removes most of the gap but each further round gains far less than with exact curvature, and the single-machine Hessian is worse again, in the order of \cref{thm:inexact}; in the exponential model single-machine curvature is exact and coincides with the exact variant. (c) In the logistic design with $N = \ablNlogit$, $m = \ablMlogit$ ($K = \ablKlogit$, so $K = N^{\ablAlphaLogit}$), quantizing the Hessian to \bitsHalfBlockHessian{} mantissa bits leaves the one-step gap of the half-block pilot within five percent of its exact value, and the same number of bits suffices when the score is quantized as well; fewer bits degrade the gap smoothly. The requirement \eqref{eq:bits} is asymptotic and gives a smaller number at this $\alpha$; the ablation shows where the finite-$N$ constants put the threshold.

\subsection{Unequal blocks and contamination}
\Cref{tab:skew} reports the unequal-size experiment of \cref{prop:robust}: with size ratios $4$ and $16$ the size-weighted pilots behave as in the equal-size case, the equal-weight pilots have a gap of order one that does not vanish for $q = 0$, and one Newton step removes it. In the logistic design with $K = 1024$ and ratio $16$ the smallest blocks contain a few dozen observations for $p = 3$, the half-blocks separate, and both the equal-weight and the size-weighted jackknife pilots fail while the raw pilots survive: the same small-block failure as above, now caused by the size imbalance. The contaminated exponential design (\cref{tab:mc_exp_contam_p1}) is a misspecified exponential model whose pseudo-true rate differs from $\theta_0$; all estimators remain first-order equivalent to the centralized pseudo-MLE, as \cref{prop:robust}(ii) states.

\input{tables/mc_exp_p1}
\input{tables/mc_logit_p3}
\input{tables/mc_logit_p8}
\input{tables/mc_pois_p3}
\input{tables/mc_pois_p8}
\input{tables/mc_logit_p20_p20}
\input{tables/mc_logit_small_m_p8}
\input{tables/mc_logit_illcond_p8}
\input{tables/mc_rates}
\input{tables/boundary}
\input{tables/abl_exponential}
\input{tables/abl_logistic}
\input{tables/skew}
\input{tables/mc_logit_misspec_p3}
\input{tables/mc_exp_contam_p1}
\FloatBarrier