1\section{Results: historical polities and ethnographic societies}2\label{sec:hist}34All estimates in this section are \EMP\ and associational. Coefficients are per standard deviation of the standardized regressor unless noted; stars denote $p<0.10$ ($^{*}$), $0.05$ ($^{**}$), $0.01$ ($^{***}$) with the clustering stated in each table.56\subsection{H1: capacity and scale}7\label{sec:hist-h1}89Capacity and scale move together in every dataset (Tables~\ref{tab:h1-seshat} and~\ref{tab:h1-ea}). In Seshat, a one-unit increase in the capacity share (from no indicator present to all present) is associated with 2.36 units of the scale component in the primary specification with region and era effects (SE 0.17, $n=317$), which is 0.75 standard deviations of $D$ per standard deviation of $K$; with the warfare, production-infrastructure and military covariates the coefficient falls to 1.21 (SE 0.43, $n=191$), still clearly positive, and it is 2.18 (SE 0.13) with a region random intercept instead of fixed effects. With the latent capacity score the association is 0.37 scale units per IRT standard deviation (SE 0.09). Entering the four blocks separately, communication infrastructure carries most of the association (1.09, SE 0.35) while administration, rules and money, and records are individually indistinguishable from zero given the others---which, with inter-block correlations near 0.8, is a statement about collinearity rather than about mechanisms. In the Ethnographic Atlas, each additional level of jurisdictional hierarchy beyond the local community is associated with 0.70 log units of mean community size ($n=579$; Conley standard errors 0.09 and 0.10 at 500 and 1{,}000 km; 0.12 clustered by language family) and with 0.59 log units of population size ($n=799$); the ordered-logit specification in the other direction gives 0.49 log-odds of one more level per log unit of community size (SE 0.12). In the modern panel the within-country association of log population with capacity is 0.15 (SE 0.06) and the fifty-year-difference association 0.13 (SE 0.05), while the pooled cross-sectional coefficient is positive but imprecise (0.19, SE 0.24). The sign is stable across datasets, specifications and the two worlds; the first pre-registered primary test is passed. Nothing here identifies a direction.1011\begin{table}[H]12\centering\footnotesize13\caption{H1 in Seshat: scale regressed on capacity. OLS, standard errors clustered by Seshat region (42 clusters); region random intercept in the last row. $K$ is a share in $[0,1]$; the last column rescales the coefficient per standard deviation of $K$.}14\label{tab:h1-seshat}15\begin{tabular}{L{7.2cm} l r r r r}16\toprule17Specification & Term & Coef. & SE & $n$ & $R^2$ \\18\midrule19\input tables/tab_h1_seshat 20\bottomrule21\end{tabular}22\end{table}2324\begin{table}[H]25\centering\footnotesize26\caption{H1 in the Ethnographic Atlas (top) and in the modern panel (bottom).}27\label{tab:h1-ea}28\begin{tabular}{L{8.4cm} l r r r r}29\toprule30Specification & Term & Coef. & SE & $n$ & $R^2$ \\31\midrule32\input tables/tab_h1_ea 33\midrule34\input tables/tab_h1_modern 35\bottomrule36\end{tabular}37\end{table}3839\subsection{H2: crisis severity}40\label{sec:hist-h2}4142Table~\ref{tab:h2-seshat} and Figure~\ref{fig:coefplot} report the pre-registered primary test. In the primary specification, a one-standard-deviation higher pre-crisis mismatch is associated with $0.20$ higher log-odds that a given adverse consequence is recorded for the crisis (SE 0.06, $n=127$ crisis records in 117 polities and 31 Seshat regions with a defined pre-crisis index); with the rival covariates the estimate is $0.20$ (SE 0.09, $n=87$), and it is unchanged when the overall complexity component is added. The coefficient is positive and precise, survives the Holm correction over the three primary tests, and is stable under every pre-registered perturbation of the measurement model (Table~\ref{tab:h2-seshat-robust}): latent capacity ($0.15$, SE 0.09), population only ($0.27$, SE 0.11), lower and upper range bounds, inferred codes dropped, within-era and within-region standardization, ranks, quasi-polities dropped ($0.15$, SE 0.11), crises starting at least 25 years after the polity's start ($0.25$, SE 0.09), records with at least twelve coded items, no era effects, no region effects, no effects at all, and clustering by natural geographic area or by polity. Held-out macro-regions and eras (Table~\ref{tab:h5-seshat}, Figure~\ref{fig:forest}) give pooled coefficients between $0.18$ and $0.33$, every one with a confidence interval excluding zero except the block that drops the first millennium BCE ($0.20$, SE 0.17), and the within-block estimates are positive wherever they can be computed, if imprecise. By the standards of a conventional regression table, the mismatch hypothesis would be reported as confirmed.4344\begin{table}[H]45\centering\footnotesize46\caption{H2 in Seshat: crisis severity $Y_1$ (fractional logit; share of fourteen instability consequences coded present) with macro-region and era effects; standard errors clustered by Seshat region. The binary outcome $Y_{1b}$ uses region effects only (era cells are all-severe) and robust errors.}47\label{tab:h2-seshat}48\begin{tabular}{L{8.0cm} l r r r}49\toprule50Specification & Term & Coef. & SE & $n$ \\51\midrule52\input tables/tab_h2_seshat 53\bottomrule54\end{tabular}55\end{table}5657The pre-registration asked a different question, and the rest of Table~\ref{tab:h2-seshat} answers it. Scale alone is associated with severity at $0.26$ per standard deviation (SE 0.14); capacity alone at $-0.03$ (SE 0.20); the Turchin-style complexity component at $-0.01$ (SE 0.24); the persistence indicator at $0.35$ (SE 0.24). When scale and capacity are entered separately, scale carries $0.34$ (SE 0.16) and capacity $-0.21$ (SE 0.19): capacity has the sign the hypothesis predicts, but it is not distinguishable from zero. The placebo index, which replaces capacity by the share of production-infrastructure indicators that were excluded from $K$ precisely because they are not coordination capacity, is associated with severity at $0.26$ (SE 0.14)---the same as scale alone, and within sampling error of $P$. Out of sample (Table~\ref{tab:h2-seshat-cv}), the model with $P$ has a higher log-likelihood than the fixed-effects-only, capacity-only and complexity-only models in 79--85\% of the fifty cross-validation folds, but beats the scale-only model in 75\%, the persistence model in 69\% and the scale-plus-capacity model in 75\%, below the 80\% rule; with macro-regions held out it loses to scale alone in seven of ten folds and to scale plus capacity in eight. Because the principal-component loadings of $D$ and the standardization of $P$ were estimated on the full polity sample (without any outcome) before cross-validation, we repeated the leave-one-region-out comparison with loadings and standardization re-estimated inside each training fold: $P$ then beats scale alone in two of ten folds (\texttt{h2\_seshat\_cv\_noleakage.csv}), so the verdict does not depend on that pre-processing. For the binary outcome, which was declared underpowered in advance, the coefficient is negative and imprecise ($-0.18$, SE 0.20 with region effects; $-0.07$, SE 0.18 without), and no model discriminates severe from non-severe crises out of sample (areas under the curve 0.62--0.64 for every model, including the baselines). Reform consequences are unrelated to $P$. The conclusion we are obliged to draw is that what the mismatch index predicts about crisis severity is what polity scale predicts, and that nothing in the data distinguishes a capacity shortfall from a large population.5859\begin{table}[H]60\centering\footnotesize61\caption{Pre-registered robustness set for the Seshat primary coefficient (fractional logit of $Y_1$ on the alternative index, rival covariates, macro-region and era effects; region-clustered standard errors unless stated).}62\label{tab:h2-seshat-robust}63\begin{tabular}{L{8.0cm} l r r r}64\toprule65Variant & Term & Coef. & SE & $n$ \\66\midrule67\input tables/tab_h2_seshat_robust 68\bottomrule69\end{tabular}70\end{table}7172\begin{table}[H]73\centering\footnotesize74\caption{Out-of-sample comparison in Seshat. Top: share of folds in which the model with $P$ (rival covariates included) has a higher out-of-sample log-likelihood than the baseline, and the mean difference per observation; the pre-registered rule requires 0.80. Bottom: out-of-sample metrics for the binary outcome $Y_{1b}$ (ten repetitions of five-fold cross-validation).}75\label{tab:h2-seshat-cv}76\begin{tabular}{l l r r r}77\toprule78Baseline & Scheme & Folds & Win share & Mean $\Delta$ log-lik. \\79\midrule80\input tables/tab_h2_seshat_cv 81\bottomrule82\end{tabular}\\[3mm]83\begin{tabular}{l r r r}84\toprule85Model ($Y_{1b}$) & OOS log-lik. & AUC & Brier \\86\midrule87\input tables/tab_h2_seshat_cv_binary 88\bottomrule89\end{tabular}90\end{table}9192\begin{figure}[H]93\centering94\includegraphics[width=\textwidth]{figures/fig_h2_coefplot.pdf}95\caption{Coefficients (per standard deviation) with 95\% confidence intervals for the pre-registered specifications, baselines, placebos and negative controls. Left: Seshat crisis severity. Right: modern instability events.}96\label{fig:coefplot}97\end{figure}9899\begin{table}[H]100\centering\footnotesize101\caption{H5 in Seshat: $\beta_P$ with one macro-region or era held out (pooled column), the number of crisis records in the held-out block, and the coefficient estimated within the held-out block when it has at least twelve records.}102\label{tab:h5-seshat}103\begin{tabular}{L{5.6cm} r r r r l}104\toprule105Held-out block & Pooled $\beta_P$ & SE & $n$ & Held-out $n$ & Within-block (SE) \\106\midrule107\input tables/tab_h5_seshat_blocks 108\bottomrule109\end{tabular}110\end{table}111112\subsection{H4: no threshold}113\label{sec:hist-h4}114115A hinge in $P$ with its knot chosen on the training fold, and a three-degree-of-freedom spline, improve on the linear model out of sample in only 10\% and 8\% of the fifty folds (Table~\ref{tab:h4}); the in-sample hinge places its knot at $0.92$ standard deviations with a negative post-knot slope, the opposite of a tipping point. The smooth form is retained for Seshat.116117\subsection{Polity duration and succession}118\label{sec:hist-y2}119120The mismatch index is unrelated to the hazard of a polity's end (Table~\ref{tab:y2y3}): hazard ratios of 1.03 (SE 0.06) and 1.00 (SE 0.07, Schoenfeld test $p=0.98$) with 302 and 201 events. Capacity, entered with scale, is associated with a \emph{higher} hazard (ratio 1.75, SE 0.20 on the log scale), which we read as periodization rather than fragility: well-documented complex polities are divided into more, shorter Seshat polities than poorly documented simple ones. Discontinuous succession, coded for 207 polities with 37 discontinuities, is unrelated to $P$ ($0.02$ and $0.24$ log-odds, SE 0.39 and 0.33).121122\begin{table}[H]123\centering\footnotesize124\caption{Secondary historical outcomes. Cox proportional hazards for polity duration, stratified by macro-region with region-clustered robust errors (coefficients on the log-hazard scale, hazard ratio in the last column), and logit for discontinuous succession with region and era effects.}125\label{tab:y2y3}126\begin{tabular}{L{6.4cm} l r r r r r}127\toprule128Specification & Term & Coef. & SE & $n$ & HR & Events \\129\midrule130\input tables/tab_h2_seshat_y2y3 131\bottomrule132\end{tabular}133\end{table}134