\section{Conventions and dictionary}
\label{app:conventions}

The radius of the sphere is one. The gauge group is $SU(N)$, and we work at leading order in $1/N$. The Yang--Mills coupling $g_{\rm YM}$ enters through the 't Hooft coupling $\lambda=g_{\rm YM}^2N$. Table~\ref{tab:couplings} collects the couplings that appear in the text and in the papers we compare with.

\begin{table}[ht]
\centering
\begin{tabular}{lll}
\toprule
symbol & definition & used by \\
\midrule
$\lambda$ & $g_{\rm YM}^2N$ & this paper, \cite{SpradlinVolovich} \\
$g^2$ & $\lambda/16\pi^2$ & this paper, \cite{HW1,HW2} \\
$G$ & $2g^2\beta=\beta\lambda/8\pi^2$ & this paper; the combination in \eqref{eq:master} \\
$\lambda_A$ & $\lambda/2$ & \cite{AMMPV1,AMMPV2} \\
$\lambda_{\rm HO}$ & $\lambda/4\pi^2=4g^2$ & \cite{HO1,HO2} \\
\bottomrule
\end{tabular}
\caption{Couplings.}
\label{tab:couplings}
\end{table}

The dilatation operator is $D=D_0+g^2D_2+g^4D_4+\ldots$ On two adjacent letters of a single-trace operator the planar part of $g^2D_2$ acts as $2g^2\sum_j h(j)P_j$, where $P_j$ projects onto total spin $j$ of $psu(2,2|4)$ and $h(j)=\sum_{k=1}^j1/k$. We write $\cH$ for the complete one-loop operator in the normalisation $g^2D_2=(\lambda/8\pi^2)\cH$. On $\mathbb{R}\times S^3$ the dilatation operator is the Hamiltonian, and $h_2$, $h_4$ are the terms of order $g^2$ and $g^4$ of the effective Hamiltonian in the degenerate subspaces of $D_0$.

The Polyakov loops are $u_n=\tr\,U^n/N$, with $U$ the holonomy of the gauge field around the thermal circle and $u_{-n}=u_n^*$. The effective action is normalised as $Z=\int dU\,e^{-S_{\rm eff}(U)}$, and $\vev{\ldots}_U$ is the average over free thermal states twisted by $U$, defined in section~\ref{sec:oneloop}. Table~\ref{tab:symbols} lists the functions of temperature that enter it. All of them are evaluated at the free Hagedorn point $x_H=7-4\sqrt3$ unless an argument is shown.

\begin{table}[ht]
\centering
\begin{tabular}{llr}
\toprule
symbol & meaning & value at $x_H$ \\
\midrule
$z(x)$ & single-letter partition function, \eqref{eq:z} & $1$ \\
$\beta_0$ & $-\ln x_H$ & $2.6339158$ \\
$A_2$ & $1-z_2(x_H)$, twice the mass squared of $u_2$ & $0.9742786$ \\
$d_1$ & coefficient of $|u_1|^2$ in $\vev{\cH}_U/N^2$, \eqref{eq:H-avg} & $\tfrac32$ \\
$c_3$ & coefficient of $u_1^2u_{-2}+{\rm c.c.}$ in $\vev{\cH}_U/N^2$ & $0.16821$ \\
$Q_4$ & coefficient of $|u_1|^4$ in $\vev{\cH^2}_{U,\rm conn}/N^2$, \eqref{eq:Q4q4} & $2.1222$ \\
$I$ & $\half Q_4+2c_3^2/A_2$ & $1.11918$ \\
$C_{41}$ & coefficient of $|u_1|^2$ in $\vev{\cH^2}_{U,\rm conn}/N^2$ & $8.353829$ \\
$d_4$ & coefficient of $|u_1|^2$ in $\vev{D_4}_U/N^2$ & $-10.71$ \\
$q_4$ & coefficient of $|u_1|^4$ in $\vev{D_4}_U/N^2$ & $-2.234$ \\
$m_1^2$ & coefficient of $|u_1|^2$ in $S_{\rm eff}/N^2$ & $0$ at $T_H$ \\
$b$ & coefficient of $|u_1|^4$ in $S_{\rm eff}/N^2$ after integrating out $u_2$ & $-36.94\,g^4$ \\
\bottomrule
\end{tabular}
\caption{Functions of temperature. Values of $c_3$, $Q_4$ are in the literal scheme.}
\label{tab:symbols}
\end{table}

We use short names for the lightest letters. $Z$ and $\phi$ are two of the three complex scalars in their lowest harmonic, with energy one. $\psi$ is a fermion in its lowest harmonic, with energy $\tfrac32$. $F$ is the lowest mode of the gauge field, with energy two. $DZ$, $D\psi$ and $D^2Z$ are the first and second excited harmonics, which in the language of local operators are letters with one or two covariant derivatives. The letter $DZ$ has spins $(\half,\half)$, and is the lightest letter with both spins non-zero.

The cut-off $J$ is an upper limit on $2j_L$ for every virtual mode.