1% ============================================================================2% Section: formal framework (generated from research/framework.md; proofs kept in full)3% ============================================================================4\section{A formal framework: institutions as codes}5\label{sec:theory}67This section states the theory behind the Social Compression Hypothesis in a form that separates what is8\emph{proved} from what is \emph{conjectured}. Definitions are conventions; theorems are statements proved below9or standard results whose proofs are reproduced; conjectures are statements we believe worth testing, each with its10falsifier. Nothing in this section is an empirical finding. The results concern an idealized communication system;11Section~\ref{sec:theory-notclaimed} states what is not claimed and which pre-registered test, if any, can speak to12each statement.1314\subsection{Primitives and definitions}15\label{sec:theory-primitives}1617\begin{definition}[agents, tasks, plans]18\label{def:primitives}19A society has $N$ agents. In each period, coordination tasks arrive at rate $\lambda$ per agent. A task requires20that $k \ge 2$ participants settle on a common element of a finite \emph{plan set} $S$, $|S| = s$, the required plan21being drawn independently across tasks from a distribution $p$ on $S$ with Shannon entropy22$H(p) = -\sum_{x \in S} p(x)\log_2 p(x) \le \log_2 s$ \citep{shannon1948,cover2006}. Communication is counted in23\emph{messages} (one transmission between two agents) and in \emph{bits} (the length of what is transmitted).24Verification and record-keeping use a second channel whose cost is counted separately as \emph{administrative25overhead} (Definition~\ref{def:costs}).26\end{definition}2728\begin{definition}[coordination demand]29\label{def:demand}30The coordination demand of a period is the task volume that must be settled: $D = N \lambda k$ participant-slots31per period. In the worst case in which every pair of agents must be reconciled, $D$ is bounded by the number of32pairs, $N(N-1)/2$, i.e.\ $D = \Theta(N^2)$; under bounded $\lambda$ and $k$, $D = \Theta(N)$. Either way $D$ is33increasing in $N$.34\end{definition}3536\begin{definition}[coordination capacity]37\label{def:capacity}38Fix a tolerated error rate $\varepsilon \in (0,1)$. The coordination capacity $K$ of an institutional system is the39maximal number of tasks it can settle per period such that the fraction of tasks settled on a wrong or inconsistent40plan is at most $\varepsilon$. $K$ depends on the available communication budget (bits per period), the code in use41(Section~\ref{sec:theory-coding}), the routing structure (Section~\ref{sec:theory-hierarchy}) and the overhead42(Section~\ref{sec:theory-breakeven}).43\end{definition}4445\begin{definition}[utilization]46\label{def:utilization}47$\rho = D/K$. When $D$ and $K$ are measured in the same units (tasks per period), $\log D - \log K = \log \rho$48exactly.49\end{definition}5051\paragraph{Limitation (not a theorem).}52The pre-registered empirical index $P = z(\log D) - z(K)$ uses standardized proxies (population, territory,53settlement hierarchy for $D$; shares of administrative, legal/monetary, communication and record-keeping indicators54for $K$). It is a monotone proxy for $\log\rho$ only under Conjecture~\ref{conj:c1}; it is not a measurement of55$\rho$, and the Seshat and V-Dem indicators are not Shannon quantities.5657\subsection{Coding gain of a shared code}58\label{sec:theory-coding}5960\begin{definition}[code]61\label{def:code}62A \emph{shared code} is an injective map $c: S \to \{0,1\}^*$ known to all participants, with lengths63$\ell(x) = |c(x)|$. It is \emph{prefix-free} if no codeword is a prefix of another. Institutions in the sense of64this paper (standardized names, units, legal forms, accounting categories, calendars) are shared codes in exactly65this sense \citep{arrow1974,cremer2007}.66\end{definition}6768\begin{assumption}69\label{ass:settle}70Settling a task means that the initiating participant identifies the required plan $x \in S$ to the other $k-1$71participants without ambiguity; messages to different participants are counted separately in messages but72identically in bits per recipient, so the result is stated per recipient.73\end{assumption}7475\begin{theorem}[coding gain]76\label{thm:coding}77\begin{enumerate}[label=(\roman*)]78\item Without a shared code, identifying an arbitrary element of $S$ without error requires at least79$\lceil \log_2 s \rceil$ bits per recipient in the worst case.80\item With a shared prefix-free code that is optimal for $p$, the expected length per recipient81$L_p^{*} = \min_{c\ \text{prefix-free}} \sum_x p(x)\ell(x)$ satisfies82\begin{equation}83H(p) \le L_p^{*} < H(p) + 1 .84\end{equation}85\item Hence the expected saving per recipient, $G(p) = \lceil\log_2 s\rceil - L_p^{*}$, obeys86$G(p) > \lceil\log_2 s\rceil - H(p) - 1 \ge 0$ whenever $\lceil\log_2 s\rceil \ge H(p)+1$, and $G(p)$ is strictly87positive as soon as $p$ is sufficiently concentrated ($H(p) < \lceil\log_2 s\rceil - 1$).88\end{enumerate}89\end{theorem}9091\begin{proof}92(i) Without a shared code there is no prior agreement on which binary string denotes which plan, so the only93error-free strategy is a fixed-length enumeration of $S$ agreed by the trivial convention of listing; any scheme94that distinguishes $s$ alternatives with binary strings of a fixed length $m$ needs $2^m \ge s$, i.e.\95$m \ge \lceil\log_2 s\rceil$. (The point is not that $\log_2 s$ is a lower bound for \emph{all} conceivable codes;96it is the cost of the \emph{codeless} baseline, in which the message must be self-describing.)9798(ii) \emph{Lower bound.} For any prefix-free code the Kraft inequality holds: $\sum_x 2^{-\ell(x)} \le 1$99\citep[Thm.~5.2.1]{cover2006}. Put $q(x) = 2^{-\ell(x)}/Z$ with $Z = \sum_x 2^{-\ell(x)} \le 1$. Then100\[101\sum_x p(x)\ell(x) - H(p) = \sum_x p(x)\log_2\frac{p(x)}{2^{-\ell(x)}}102= \sum_x p(x)\log_2\frac{p(x)}{q(x)} - \log_2 Z = D_{KL}(p\|q) - \log_2 Z \ge 0,103\]104because $D_{KL} \ge 0$ (Gibbs' inequality) and $-\log_2 Z \ge 0$. \emph{Upper bound.} Choose105$\ell(x) = \lceil -\log_2 p(x)\rceil$. Then $\sum_x 2^{-\ell(x)} \le \sum_x p(x) = 1$, so by the converse Kraft106theorem \citep[Thm.~5.2.1]{cover2006} a prefix-free code with these lengths exists, and107$\sum_x p(x)\ell(x) < \sum_x p(x)(-\log_2 p(x) + 1) = H(p) + 1$. The optimal code does at least as well.108109(iii) Subtract (ii) from (i).110\end{proof}111112\noindent\emph{Remark.} Because the fixed-length code of $\lceil\log_2 s\rceil$ bits is itself prefix-free, the optimal code satisfies $L_p^* \le \lceil\log_2 s\rceil$ and the saving $G(p)$ is never negative; (iii) is only a lower bound. A suboptimal prefix code with Shannon lengths $\lceil -\log_2 p(x)\rceil$ also satisfies $H(p) \le L < H(p)+1$ but can exceed $\lceil\log_2 s\rceil$ when $p$ is nearly uniform; the simulations of Section~\ref{sec:sim} use Shannon lengths so that Theorem~\ref{thm:stale} applies verbatim, and this is why one cell there shows a code longer than raw description.113114115\paragraph{Remark.}116Theorem~\ref{thm:coding} is the source-coding theorem of \citet{shannon1948} applied to the plan set. Its content117for the paper is only this: \emph{a shared code converts a worst-case identification cost into an average-case118cost that is lower exactly to the extent that tasks are predictable.} The ``compression'' in the name of the119hypothesis refers to this step and to nothing else. The same economy appears in the mechanism-design bounds on120message-space size \citep{mount1974,segal2007,nisan2006} and in communication complexity121\citep{yao1979,kushilevitz1997}; the rate--distortion view of \citet{marzen2017} and the coarse-graining view of122\citet{dedeo2017} are the closest conceptual neighbours.123124\subsection{Break-even scale of an institution}125\label{sec:theory-breakeven}126127\begin{definition}[institutional cost structure]128\label{def:costs}129Adopting a code costs a fixed set-up $F \ge 0$ (designing and promulgating the standard), a learning cost130$\ell \ge 0$ per agent (each of the $N$ agents must learn the code), and an administrative overhead $a \ge 0$ bits131per task (verification, record-keeping, enforcement). It saves $G > 0$ bits per task (Theorem~\ref{thm:coding}, per132recipient; multiply by $k-1$ if desired; the argument is unchanged).133\end{definition}134135\begin{theorem}[break-even]136\label{thm:breakeven}137Over a horizon with $T$ tasks, the institution lowers total communication cost if and only if138\begin{equation}139T\,(G - a) > F + N\ell .140\end{equation}141With $T = N\lambda\tau$ over $\tau$ periods, this is equivalent to142\begin{equation}143\lambda\tau > \frac{F/N + \ell}{G - a}\quad\text{when } G > a,144\end{equation}145and the inequality can never hold when $a \ge G$.146\end{theorem}147148\begin{proof}149Total cost without the institution is $C_0 = T\lceil\log_2 s\rceil$. With it,150$C_1 = F + N\ell + T(L_p^{*} + a)$. Then $C_1 < C_0$ if and only if151$F + N\ell < T(\lceil\log_2 s\rceil - L_p^{*} - a) = T(G - a)$. If $a \ge G$ the right-hand side is152$\le 0 < F + N\ell$ unless $F = \ell = 0$ and $a = G$, in which case the costs are equal, so strict improvement is153impossible. If $G > a$, substitute $T = N\lambda\tau$ and divide both sides by $N(G-a) > 0$.154\end{proof}155156\begin{corollary}[scale advantage]157\label{cor:scale}158The threshold $(F/N + \ell)/(G-a)$ is decreasing in $N$: holding $\lambda,\tau$ fixed, a larger society crosses the159break-even point sooner, because the set-up cost is spread over more agents. It is \emph{not} decreasing in $N$160through $\ell$: per-agent learning costs do not benefit from scale.161\end{corollary}162163This is the formal version of the observation that institutions ``pay off at scale''164\citep{johnson1982,garicano2000,bolton1994}, and it also makes precise when they do not: when overhead $a$ is close165to the coding gain $G$, or when horizons $\tau$ are short.166167\subsection{Mismatched code penalty: institutional rigidity}168\label{sec:theory-stale}169170\begin{theorem}[cost of a stale code]171\label{thm:stale}172Let the code be optimal for $p$ in the sense of using lengths $\ell_p(x) = \lceil -\log_2 p(x) \rceil$, and let the173true task distribution be $q$ with $q \ll p$ (every plan with positive $q$-probability has positive174$p$-probability). Then the expected length under $q$ satisfies175\begin{equation}176H(q) + D_{KL}(q\|p) \le \mathbb{E}_q[\ell_p(X)] < H(q) + D_{KL}(q\|p) + 1 .177\end{equation}178The excess cost of using the $p$-code when the world is $q$, relative to the $q$-optimal code, lies in179$(D_{KL}(q\|p) - 1,\ D_{KL}(q\|p) + 1)$ bits per task.180\end{theorem}181182\begin{proof}183Since $-\log_2 p(x) \le \ell_p(x) < -\log_2 p(x) + 1$,184\[185\mathbb{E}_q[\ell_p(X)] \ge \sum_x q(x)\bigl(-\log_2 p(x)\bigr)186= \sum_x q(x)\log_2\frac{q(x)}{p(x)} + \sum_x q(x)\bigl(-\log_2 q(x)\bigr) = D_{KL}(q\|p) + H(q),187\]188and the same computation with the $+1$ gives the strict upper bound. The $q$-optimal code has expected length in189$[H(q), H(q)+1)$ by Theorem~\ref{thm:coding}(ii), which yields the stated interval for the excess. This is the190``wrong code'' result of \citet[Thm.~5.4.3]{cover2006}.191\end{proof}192193\begin{corollary}[fragility]194\label{cor:fragility}195A shock that moves the task distribution from $p$ to $q$ raises the per-task communication cost of an existing196institution by about $D_{KL}(q\|p)$ bits. Combined with Theorem~\ref{thm:breakeven}, the institution remains197worthwhile after the shock only if $G(p) - D_{KL}(q\|p) > a$ (up to the $\pm 1$ rounding).198\end{corollary}199200An institution can therefore \emph{lower} efficiency after a shock, and the more specialized the code (the smaller201$H(p)$ relative to $\log_2 s$), the larger the divergence a given shock can produce. This is the precise sense in202which standardization trades efficiency for rigidity; it echoes the declining marginal returns of203\citet{tainter1988} but gives the mechanism a unit (bits per task) and a magnitude (a divergence).204205\subsection{Hierarchical routing bound and root load}206\label{sec:theory-hierarchy}207208\begin{definition}[hierarchy]209\label{def:hierarchy}210A hierarchy is a rooted tree whose $N$ leaves are the agents and whose internal nodes are administrative units;211every internal node has at most $b \ge 2$ children; the depth (leaf-to-root distance) is $d = \lceil \log_b N \rceil$212for a complete $b$-ary tree. A task between two leaves is \emph{settled by relay} when each participant sends a213message up to their lowest common ancestor (LCA) and the LCA sends the settlement down.214\end{definition}215216\begin{theorem}[routing and root load]217\label{thm:routing}218\begin{enumerate}[label=(\roman*)]219\item \emph{Message bound.} Any pairwise task can be settled with at most $2d$ messages (up-path plus down-path220through the LCA), versus up to $N-1$ messages for a flat broadcast in which the initiator must reach every other221agent. For $M$ pairwise tasks the total is at most $2M\lceil\log_b N\rceil$.222\item \emph{Root load.} In a complete $b$-ary tree with $N = b^d$ leaves, if the two participants of a task are223drawn uniformly at random among leaves (independently, with replacement), the probability that their LCA is the224root is $1 - 1/b$. Hence the expected number of tasks that pass through the root is $M(1 - 1/b)$.225\item \emph{Federated or modular coordination.} If a fraction $\theta \in [0,1]$ of tasks is \emph{local} (both226participants belong to the same first-level module, i.e.\ child subtree of the root, by construction of the task227process) and the remaining $1-\theta$ are drawn uniformly as in (ii), the expected root load is228$M(1-\theta)(1 - 1/b)$.229\end{enumerate}230\end{theorem}231232\begin{proof}233(i) In a tree there is a unique simple path between two leaves; it passes through their LCA and has length at most234$2d$ since each leaf is at depth $\le d$ and the LCA is at depth $\ge 0$. Sending one message per edge along the two235leaf-to-LCA paths settles the task in $\le 2d$ messages. A flat broadcast from one initiator to the other $N-1$236agents uses $N-1$ messages. Summing over $M$ tasks gives (i).237238(ii) The root has $b$ children, each the root of a subtree with $b^{d-1}$ leaves. The first participant lies in some239child subtree $j$; the second participant, drawn uniformly and independently, lies in the same subtree $j$ with240probability $b^{d-1}/b^d = 1/b$. The LCA is the root exactly when the two participants lie in different child241subtrees, which has probability $1 - 1/b$. Expectation of a sum of $M$ indicators gives $M(1-1/b)$.242243(iii) Local tasks never reach the root (their LCA lies inside a module); the non-local fraction behaves as in (ii).244By linearity of expectation the load is $M(1-\theta)(1-1/b)$.245\end{proof}246247\paragraph{Remark.}248Theorem~\ref{thm:routing} formalizes the trade-off between centralized administration and federated coordination249\citep{radner1993,bolton1994}: a hierarchy cuts the \emph{message} cost of coordination from linear to logarithmic in250$N$, but concentrates a constant fraction $1 - 1/b$ of all non-local tasks on the apex, whose processing rate then251bounds $K$ (Definition~\ref{def:capacity}). Modularity reduces apex load by the locality fraction $\theta$ at the252price of \emph{not} reconciling across modules. These are the five regimes compared in the pre-registered253simulations (no institution, shared norms, standardized records, centralized administration, federated), and the254simulations measure messages and bits directly rather than assuming the bounds.255256\subsection{Backlog dynamics under mismatch}257\label{sec:theory-backlog}258259\begin{definition}[backlog]260\label{def:backlog}261Let $D_t$ be the number of tasks arriving in period $t$ and $K_t$ the number the system can settle in period $t$.262The backlog of unsettled tasks follows the Lindley recursion263\begin{equation}264B_{t+1} = \max\{0,\ B_t + D_t - K_t\},\qquad B_0 \ge 0 .265\end{equation}266\end{definition}267268\begin{theorem}[backlog dynamics]269\label{thm:backlog}270\begin{enumerate}[label=(\roman*)]271\item \emph{Deterministic case.} If $D_t = D$ and $K_t = K$ for all $t$, then: if $D \le K$,272$B_t \le \max\{B_0 - t(K-D), 0\}$, so the backlog reaches $0$ after at most $\lceil B_0/(K-D) \rceil$ periods when273$D < K$ and stays constant at $B_0$ when $D = K$; if $D > K$, $B_t = B_0 + t(D-K) \to \infty$ linearly.274\item \emph{Stochastic case.} If $(D_t - K_t)_{t\ge0}$ is i.i.d.\ with finite mean, then when275$\mathbb{E}[D_t] < \mathbb{E}[K_t]$ the backlog has a proper stationary distribution to which $B_t$ converges from276any $B_0$, and when $\mathbb{E}[D_t] > \mathbb{E}[K_t]$, $B_t \to \infty$ almost surely.277\item \emph{Delay under a Markovian approximation.} If arrivals are Poisson with rate $\lambda_D$ and settlements278are exponential with rate $\mu$ (an M/M/1 queue), with $\rho = \lambda_D/\mu < 1$, the expected sojourn time is279$W = 1/(\mu - \lambda_D) = (1/\mu)/(1-\rho)$, which is increasing and convex in $\rho$ on $[0,1)$ and diverges as280$\rho \uparrow 1$.281\end{enumerate}282\end{theorem}283284\begin{proof}285(i) If $D \le K$, then $B_{t+1} = \max\{0, B_t - (K-D)\}$; by induction $B_t \le \max\{0, B_0 - t(K-D)\}$, with286equality as long as the argument is nonnegative, which gives the hitting time when $D < K$ and constancy when287$D = K$. If $D > K$, then $B_t + D - K > 0$ for all $t$ (since $B_t \ge 0$), so the maximum is never binding and288$B_{t+1} = B_t + (D-K)$, hence $B_t = B_0 + t(D-K)$.289290(ii) The recursion is the single-server waiting-time recursion of \citet{lindley1952}, and the stability dichotomy291is the theorem of \citet{loynes1962}; the proof is not reproduced here and the result is cited.292293(iii) The M/M/1 stationary number in system is geometric with parameter $\rho$, so the mean is $\rho/(1-\rho)$ and294Little's law gives $W = \rho/((1-\rho)\lambda_D) = 1/(\mu(1-\rho)) = 1/(\mu - \lambda_D)$. The function295$f(\rho) = (1-\rho)^{-1}$ has $f' = (1-\rho)^{-2} > 0$ and $f'' = 2(1-\rho)^{-3} > 0$ on $[0,1)$, and296$f(\rho) \to \infty$ as $\rho \uparrow 1$.297\end{proof}298299\paragraph{Scope of Theorem~\ref{thm:backlog}.}300This is a \emph{mechanism}, not evidence. Polities are not single-server queues: capacity is endogenous301(institutions are built in response to backlog), arrivals are correlated across periods and agents, and302``unsettled tasks'' are not observed in the historical record. The theorem is used only to motivate two qualitative303predictions, that costs rise smoothly and convexly in utilization and that sustained $\rho > 1$ is qualitatively304different from $\rho < 1$, and both are stated as conjectures below.305306\subsection{Conjectures}307\label{sec:theory-conjectures}308309The following statements are unproved. Each is given with its reason and its falsifier.310311\begin{conjecture}[proxy monotonicity]312\label{conj:c1}313The pre-registered Seshat and V-Dem proxies are monotone in the model's primitives: the scale variables314(population, territory, largest settlement, settlement hierarchy) increase with $D$ of Definition~\ref{def:demand},315and the four indicator blocks (administration, standardized rules and money, communication infrastructure,316record-keeping) increase with $K$ of Definition~\ref{def:capacity}, so that $P = z(\log D) - z(K)$ is monotone in317$\log\rho$ up to measurement error.318\end{conjecture}319320\noindent\emph{Reason:} population and territory raise the number and the dispersion of participants; written321records, legal codes, money and measurement systems are shared codes (Theorem~\ref{thm:coding}); roads, couriers322and postal systems raise bits per period; administrative levels implement the routing of323Theorem~\ref{thm:routing}. \emph{Falsifier:} the proxies for $D$ and $K$ are so collinear (as Seshat's first324principal component suggests; \citealp{shin2020}) that $P$ retains almost no variance, or alternative defensible $K$325weightings reverse the sign of $P$ for a substantial share of polities; both are reported in the measurement326section.327328\begin{conjecture}[sustained overload raises breakdown probability]329\label{conj:c2}330Periods with $\rho > 1$ sustained longer than the time needed to build capacity raise the probability of331institutional breakdown.332\end{conjecture}333334\noindent\emph{Mechanism family (not proved):} unsettled tasks accumulate (Theorem~\ref{thm:backlog}), so335participants cannot form the common knowledge required for coordinated action \citep{chwe2000} and local defection336spreads by contagion through the interaction network \citep{morris2000}; institutions whose codes are stale after a337shock lose their cost advantage (Theorem~\ref{thm:stale}) and are abandoned. \emph{Falsifier:} the pre-registered338H2 tests find $\beta_P$ not positive, or $P$ does not beat the population-only and capacity-only baselines out of339sample.340341\begin{conjecture}[smooth, not step]342\label{conj:c3}343The relation between $P$ and instability is convex and smooth away from $\rho \approx 1$ and steepens near it,344rather than a discrete threshold.345\end{conjecture}346347\noindent\emph{Reason:} Theorem~\ref{thm:backlog}(iii) gives a smooth, convex delay in $\rho$; discontinuities348would require an additional tipping mechanism (e.g.\ a critical mass in a contagion model) that the framework does349not supply. \emph{Falsifier:} the pre-registered H4 comparison finds that a hinge or step model beats the smooth350model out of sample in at least 80\% of folds, which would favour a tipping mechanism; conversely, if neither351nonlinear form beats the linear model, the ``steepening'' clause of the conjecture is unsupported.352353\subsection{What is not claimed, and which test speaks to each statement}354\label{sec:theory-notclaimed}355356\begin{itemize}357\item \textbf{No Shannon measurement of history.} Theorems~\ref{thm:coding} and~\ref{thm:stale} are about codes for358a plan set with a known distribution. Seshat indicators (presence of writing, coins, courts, roads) are not message359lengths, and no entropy or divergence is computed from historical data anywhere in this paper. Shannon quantities360appear only in the simulations, where the plan distribution $p$ and the messages are defined by the experimenter and361can be measured.362\item \textbf{No causal claim.} The theorems describe an idealized communication system; the empirical tests are363associational and were pre-registered as such; the identification discussion is in Section~\ref{sec:identification}.364\item \textbf{Endogeneity of capacity.} The framework takes $K$ as given within a period; historically, $K$ is built365in response to $D$ \citep{johnson1982,shin2020}. This is why H1 is pre-registered as an association without a366direction.367\end{itemize}368369\begin{table}[htbp]370\centering371\caption{Which evidence can speak to each statement of the framework.}372\label{tab:theory-map}373\small374\begin{tabular}{p{3.2cm} p{6.2cm} p{5.2cm}}375\toprule376Statement & What can speak to it & Pre-registered test \\377\midrule378Theorem~\ref{thm:coding} (coding gain) & Simulated messages and bits with vs.\ without a shared code & H3 (regimes b, c vs.\ a) \\379Theorem~\ref{thm:breakeven} (break-even) & Simulated total cost across $N$, $\lambda$, horizon, overhead $a$ & H3 factorial cells; search for cells where institutions lose \\380Theorem~\ref{thm:stale} (stale code) & Simulated shocks to the task distribution & H3 adverse regimes (shock treatment) \\381Theorem~\ref{thm:routing} (routing, root load) & Simulated messages and apex load under centralized vs.\ federated regimes & H3 (regimes d, e) \\382Theorem~\ref{thm:backlog} (backlog) & Not testable with the historical data (backlog unobserved); illustrated in simulation only & none (stated as mechanism) \\383Conjecture~\ref{conj:c1} & Measurement diagnostics: collinearity of $D$ and $K$ proxies, sensitivity of $P$ to weights & Measurement section \\384Conjecture~\ref{conj:c2} & Lagged $P$ and subsequent instability vs.\ baselines, out of sample & H2 (Seshat Y1--Y3; V-Dem Y4--Y6), H5 \\385Conjecture~\ref{conj:c3} & Linear vs.\ hinge vs.\ spline in $P$, out of sample & H4 \\386\bottomrule387\end{tabular}388\end{table}389