\begin{table}[p]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.05}
\setlength{\tabcolsep}{4pt}
\begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}p{0.155\linewidth}>{\raggedright\arraybackslash}p{0.2\linewidth}>{\raggedright\arraybackslash}p{0.205\linewidth}@{}}
\toprule
Principle & Reverse mathematics & Weihrauch & Constructive (BISH) \\
\midrule
\multicolumn{4}{@{}l}{\emph{Existence}}\\
Maximum of a continuous $f$ on $[0,1]$ & $\WKL$ & $\seqv\WKLp$ & positive infimum $\Leftrightarrow\FAN$ \\
Minimum of a convex $f$ on $[0,1]$ / on $[0,1]^n$ with a modulus & $\RCA$ (\ref{thm:convex-rca}) & $\seqv\XC_n$ (\ref{prop:cmin-weihrauch}) & quasi-convex: no $\FAN$ needed \\
Polynomial minimum, fixed format & $\RCA$ (\ref{prop:tarski}) & non-unif.\ computable & \\
Unique minimizer: modulus of uniqueness & $\WKL$ & computable & existence $\Leftrightarrow\FAN$; convex: free \\
Brouwer fixed point, $n\ge2$ & $\WKL$ & $\seqv\WKLp$ & \\
Minimum of an lsc $f$ on $[0,1]$ & $\ACA$ & & \\
Projection onto a closed subspace; norm of a functional & $\ACA$ & & \\
Ekeland: free, continuous $f$, compact space & $\WKL$ & & \\
Ekeland: two of the three weakenings, or compact space alone & $\ACA$ & & \\
Ekeland: free; localized; localized for continuous $f$ & $\PiCA$ & & \\
Caristi for Borel maps & $\mathsf{TLPP}_0$ & & \\
\midrule
\multicolumn{4}{@{}l}{\emph{Convergence}}\\
Rates for $f(x_k)-f^\ast$ and $\nabla f(x_k)$ & $\RCA$ & computable, uniform & constructive \\
Gradient descent, PPA, KM converge (on $\R$) & $\ACA$ (\ref{thm:A}) & limit $\seqv\lm$ & \\
Projection of an anchor onto the solution set (Halpern, Tikhonov) & implies $\ACA$ (\ref{thm:D}) & $\seqv\lm$ & \\
Modulus of regularity exists / regularity & $\ACA$ / $\WKL$ & $\eqv\widehat\LPO\eqv\lm$ & \\
Mean ergodic theorem & $\ACA$ & & \\
Gradient descent in $\R^d$: at most $2C_dD/\eps$ $\eps$-fluctuations (\ref{thm:E}) & (quantitative) & limits on computable instances d.c.e.\ (\ref{thm:F}) & \\
\midrule
\multicolumn{4}{@{}l}{\emph{Duality}}\\
LP duality, minimax (real data, all sizes) & $\RCA$ (\ref{thm:lp-rca}) & linear inequalities: $\eqv\pr{AoUC}^\ast_{[0,1]}$ & Farkas $\Leftrightarrow\LPO$ \\
Nash equilibria, bimatrix / $n$ players & $\le\WKL$ & $\eqv\pr{AoUC}^\ast_{[0,1]}$ / open & existence not constructive \\
Hahn--Banach; separation of open convex sets & $\WKL$ & HB: $\eqv\WKLp$ & HB $\Leftrightarrow\LLPO$; a separation form $\Leftrightarrow$ MP \\
Strict separation from a separably closed set & $\ACA$ & & \\
K\H{o}nig duality (countable bipartite graphs) & $\ATR$ & arithm.\ $\equiv\pr{ATR}_2$ & \\
Menger (countable graphs) & $\ATR\le\cdot\le\mathsf{TLPP}_0$ & & \\
\midrule
\multicolumn{4}{@{}l}{\emph{Feasible world}}\\
\multicolumn{4}{@{}p{\linewidth}@{}}{Local optima: $\PLS$, the $\NP$ search problems provably total in $T^1_2$. Approximate KKT points of smooth functions on the square (given by circuits):
$\CLS=\PPAD\cap\PLS$. Nash, Brouwer, local min-max: $\PPAD$. Exact KKT points on $[0,1]$: $\KKT_1\seqv\IVT$
(\ref{prop:kkt1}).}\\
\bottomrule
\end{tabularx}
\caption{An atlas of the calibrations discussed in this paper. Numbers refer to results of this paper; sources for
the other entries are given in \cref{sec:existence,sec:convergence,sec:duality,sec:feasible}. Empty cells are not
claims of anything; several are open. ``Non-unif.\ computable'' means that computable instances have computable
solutions.}
\label{tab:atlas}
\end{table}