1\section{Results: modern states, 1820--2025}2\label{sec:modern}34The modern panel is the form of the test that the mandate's event-risk model has in mind: a country-year panel with lagged predictors, fixed effects, forward-chaining validation and held-out regions. All estimates are \EMP\ and associational.56\subsection{H2: instability events in the next five years}7\label{sec:modern-h2}89In the primary specification (Table~\ref{tab:h2-modern}), a one-standard-deviation higher mismatch is associated with $0.39$ higher log-odds of at least one civil-war onset, coup or irregular regime end in the following five years (SE 0.09; 14,637 country-years, of which 2,692 have an event in their five-year window; the windows overlap, and the sample contains 901 distinct event-years out of 1,676 in the full panel), conditional on development, electoral democracy, time since the last event (with an indicator for countries whose spell is left-censored at panel entry), region and year. The coefficient is $0.72$ without covariates, $1.33$ in a conditional logit with country and decade effects (which uses only countries that experience at least one event window), and $0.11$ in probability units in a linear model with country and year effects. It is positive for every horizon (0.32 at one year, 0.38 at three, 0.45 at ten), largest for civil-war onsets (1.02, SE 0.19), small and imprecise for coups (0.19, SE 0.14) and intermediate for irregular regime ends (0.24, SE 0.09); it does not depend on the capacity aggregation, on standardizing within years, on adding land area to demand, on dropping the nineteenth century, on removing the democracy control or on adding the persistence control (Table~\ref{tab:h2-modern-robust}). Every held-out region and period gives a pooled coefficient of similar size, and every within-block estimate is positive (Table~\ref{tab:h5-modern}, Figure~\ref{fig:forest}). The second H2 primary test also passes the conventional bar.1011\begin{table}[H]12\centering\footnotesize13\caption{H2 in the modern panel: any instability event in $(t,t+5]$. Logit with region and year effects and country-clustered standard errors unless stated; $X$ = log GDP per capita, electoral democracy, cubic in years since the last event. ``Events'' counts country-years whose five-year window contains an event.}14\label{tab:h2-modern}15\begin{tabular}{L{8.4cm} l r r r r}16\toprule17Specification & Term & Coef. & SE & $n$ & Events \\18\midrule19\input tables/tab_h2_modern 20\bottomrule21\end{tabular}22\end{table}2324The baselines, placebo and negative control tell the rest (Table~\ref{tab:h2-modern}, Figure~\ref{fig:coefplot}). Population alone is weakly associated with instability ($0.12$, SE 0.07); capacity alone is strongly and negatively associated ($-0.63$, SE 0.13), as is development ($-0.58$ per log unit of GDP per capita). With population and capacity entered separately, population carries $0.17$ (SE 0.07) and capacity $-0.70$ (SE 0.14), and the restriction $\beta_D + \beta_K = 0$ that the mismatch index imposes is rejected (sum $-0.54$, SE 0.15, Wald $p<0.001$): in the modern panel $P$ is a misspecified summary of two components of unequal weight, and the row that adds $K$ to $P$ (which is algebraically the same model) shows the mismatch adding $0.23$ (SE 0.09) once capacity is held fixed---that is, the population component. The signal in $P$ is, here, overwhelmingly the capacity component with the sign reversed. The placebo that replaces capacity with electoral democracy is associated at $0.20$ (SE 0.06) against $0.45$ (SE 0.08) for $P$ with the same controls, so the capacity items do carry information beyond regime type, which is reassuring for the measurement model but not for the hypothesis. The negative control, read by the letter of the pre-registration, passes: past events in the five years before $t$ are associated with $P_t$ at $0.44$ (SE 0.10), against $0.48$ (SE 0.10) for future events with the same controls, so the forward association is not weaker than the backward one. But the margin is within one standard error, and a predictor that is nearly as strongly related to what has already happened as to what will happen behaves like a marker of chronically unstable countries rather than a leading indicator; we report both the literal verdict and this reading.2526\begin{table}[H]27\centering\footnotesize28\caption{Pre-registered robustness set for the modern primary coefficient (same specification as the primary row of Table~\ref{tab:h2-modern} unless stated).}29\label{tab:h2-modern-robust}30\begin{tabular}{L{6.8cm} l r r r r}31\toprule32Variant & Term & Coef. & SE & $n$ & Events \\33\midrule34\input tables/tab_h2_modern_robust 35\bottomrule36\end{tabular}37\end{table}3839Out of sample (Table~\ref{tab:h2-modern-cv}, Figure~\ref{fig:calibration}) the verdict is the same as in Seshat with the roles of the components exchanged. Under forward chaining (three temporal splits, the first of which trains on 1820--1945 and tests on 1946--1975), the model with $P$ has the highest mean area under the curve (0.732 against 0.722 for the fixed-effects model, 0.724 for population or capacity alone and 0.726 for both), but it beats the capacity-only and the scale-plus-capacity baselines in only one of the three splits on either metric; with regions held out it beats capacity alone in one of six folds on log-likelihood and three of six on discrimination. All models are reasonably calibrated under forward chaining (slopes 0.77--0.79) and all are badly calibrated when a whole region is held out (slopes about 0.40), which is itself a finding: the level of instability risk is regional, and no index built from scale and capacity transports it. Under the pre-registered 80\% rule, $P$ fails against the capacity-only and the scale-plus-capacity baselines in both schemes (the latter nests the former, so this is the expected outcome whenever the restriction is wrong), and the second H2 primary test is therefore not passed.4041\begin{table}[H]42\centering\footnotesize43\caption{Out-of-sample validation in the modern panel. Top: share of folds in which the model with $P$ beats the baseline on the metric; the rule requires 0.80. Bottom: mean metrics by scheme and model (forward chaining: three temporal splits; LORO: six held-out regions).}44\label{tab:h2-modern-cv}45\begin{tabular}{l l l r r r}46\toprule47Baseline & Scheme & Metric & Folds & Win share & Mean $\Delta$ \\48\midrule49\input tables/tab_h2_modern_cv 50\bottomrule51\end{tabular}\\[3mm]52\begin{tabular}{l l r r r r}53\toprule54Scheme & Model & OOS log-lik. & AUC & Brier & Calibration slope \\55\midrule56\input tables/tab_h2_modern_cv_summary 57\bottomrule58\end{tabular}59\end{table}6061\begin{figure}[H]62\centering63\includegraphics[width=\textwidth]{figures/fig_calibration_modern.pdf}64\caption{Out-of-sample calibration in the modern panel (decile means of predicted probability against observed event shares) for the model with $P$, the capacity-only baseline and the fixed-effects baseline, under forward chaining (left) and leave-one-region-out (right).}65\label{fig:calibration}66\end{figure}6768\begin{table}[H]69\centering\footnotesize70\caption{H5 in the modern panel: $\beta_P$ with one region or period held out (pooled), and estimated within the held-out block.}71\label{tab:h5-modern}72\begin{tabular}{L{5.0cm} r r r r l}73\toprule74Held-out block & Pooled $\beta_P$ & SE & $n$ & Within $n$ & Within-block (SE) \\75\midrule76\input tables/tab_h5_modern_blocks 77\bottomrule78\end{tabular}79\end{table}8081\begin{figure}[H]82\centering83\includegraphics[width=\textwidth]{figures/fig_h5_forest.pdf}84\caption{H5: the mismatch coefficient with each macro-region or era (Seshat, left) and each region or period (modern panel, right) held out, and the coefficient estimated inside the held-out block where possible.}85\label{fig:forest}86\end{figure}8788\subsection{Sensitivity to unobservables}89\label{sec:modern-sens}9091Table~\ref{tab:sensitivity} applies Oster's and Cinelli--Hazlett's diagnostics to the linear-probability version of the primary model. Adding the controls moves the coefficient from $0.082$ to $0.037$ while raising $R^2$ from 0.127 to 0.161; the bias-adjusted coefficient at $R_{\max}=1.3R^2$ and $\delta=1$ is $-0.030$, on the other side of zero. The partial $R^2$ of $P$ is 0.005 and the robustness value is 0.07: an unobserved confounder that explained seven percent of the residual variance of both the mismatch and the outcome would suffice to remove the association. Two caveats apply. Proportional selection ($\delta=1$) is a strong assumption when one of the controls, electoral democracy, correlates 0.75 with a component of the regressor, so the Oster adjustment overstates the plausible bias; and the robustness value, while twelve times the partial $R^2$ of $P$, has no benchmark covariate against which to judge it. The coefficient is fragile under these conventions, not shown to be spurious.9293\begin{table}[H]94\centering\footnotesize95\caption{Sensitivity of the modern primary coefficient to unobservables (linear probability model with the primary controls).}96\label{tab:sensitivity}97\begin{tabular}{L{10cm} r}98\toprule99Quantity & Value \\100\midrule101\input tables/tab_sensitivity 102\bottomrule103\end{tabular}104\end{table}105106\subsection{H4: a nonlinearity of negligible size}107\label{sec:modern-h4}108109With country-blocked folds, the hinge improves on the linear form in 11 of 15 folds and the spline in 12 (Table~\ref{tab:h4}); the spline meets the pre-registered 80\% rule exactly and the hinge does not. By the letter of the pre-registration, H4 is therefore not rejected in the modern panel. The improvement is 0.003 nats per observation; the in-sample curve (Figure~\ref{fig:h4curve}) is a mild convexity around the median rather than a step, and the hinge knot is at $-0.08$ standard deviations, i.e.\ at the centre of the distribution. We report this as a nonlinearity that survives the rule but is small, consistent with the smooth convexity of Theorem~\ref{thm:backlog} rather than with a tipping point; the pre-registration's falsifier for H4 was a hinge or step model that \emph{outperforms} the smooth one, and the form that passes is the smooth spline.110111\begin{table}[H]112\centering\footnotesize113\caption{H4: linear against hinge (knot chosen on the training fold) and natural cubic spline, out of sample.}114\label{tab:h4}115\begin{tabular}{L{5.4cm} r r r r r r}116\toprule117Panel & Folds & Hinge wins & Spline wins & Log-lik.\ linear & hinge & spline \\118\midrule119\input tables/tab_h4 120\bottomrule121\end{tabular}122\end{table}123124\begin{figure}[H]125\centering126\includegraphics[width=0.6\textwidth]{figures/fig_h4_curve.pdf}127\caption{Shape of the relation between $P$ and the five-year event probability in the modern panel, at covariate means, linear and spline fits.}128\label{fig:h4curve}129\end{figure}130