\section{Introduction}\label{sec:intro}

The Hopfield model \cite{Hopfield82} stores $M$ random patterns $\xi^1,\dots,\xi^M\in\{-1,1\}^N$ in the energy
\begin{equation}\label{eq:hamiltonian}
  H_N(\sigma)=-\frac1{2N}\sum_{\mu\le M}(\xi^\mu\cdot\sigma)^2,\qquad \sigma\in\Cube:=\{-1,1\}^N.
\end{equation}
Each pattern is meant to sit at the bottom of a well of $H_N$, and one asks how many patterns the landscape can hold. We let $M/N\to\alpha$ and call this limit the load, the number of patterns per neuron. In this regime, Amit, Gutfreund and Sompolinsky answered the question with the replica method \cite{AGS85,AGS87}: these wells exist up to a critical load $\alc(T)$, with $\alc(0)\approx0.138$. Steffan and K\"uhn later computed the corrections due to replica symmetry breaking, and found $0.138186$ at one step and $0.138187733$ at two steps \cite{SK94}. Rigorous results have remained well below these values. Newman proved an energy barrier around each pattern up to load $0.056$ \cite{Newman88}, and this was improved to $0.071$ by Loukianova \cite{Loukianova97} and to $0.113$ by Feng, Shcherbina and Tirozzi \cite{FST01}.

We study the largest load at which the free energy at fixed overlap with one pattern has a well. Our starting point is a formula for this free energy. For one binary pattern in a bulk of Gaussian rows, it is the minimum of an explicit functional over the law of the overlap path, and this functional is strictly convex, while for binary patterns it is a supremum of such formulas over finitely many retained patterns. This gives a variational characterization of the critical load at every temperature, the same for binary patterns and for Gaussian rows. At zero temperature, we then prove $\alo\le\alc(0)\le\ahi$ in ball arithmetic, where $\alo:=0.138198129$ and $\ahi:=0.13820106$. The lower end is above both published replica symmetry breaking values (Subsection~\ref{ssec:physics}). Following \cite{FST01}, we call $\alc(T)$ the critical capacity. It is the threshold for wells in the free energy landscape at fixed overlap: the supremum of the loads at which, for some $0<m_1<m_2<1$, the limiting free energy at overlap $m_2$ exceeds the one at overlap $m_1$. The definition is \eqref{eq:well-def}, and at $T=0$ the energy takes the place of the free energy, with $m_2=1$ allowed.

Two obstacles stand in the way of a direct argument. The first is that the wells cannot be seen at equilibrium. Above load about $0.05$, the replica symmetric computation of \cite{AGS85} places the global minimum of $H_N$ at a spin glass state, and the wells around the patterns become metastable. At load $0.1$, for instance, the well around a pattern lies above the global minimum. The capacity therefore cannot be read off the equilibrium free energy, although this free energy is now known \cite{ABCHIM}, and this is why we fix the overlap: a well then appears as a local maximum, visible at every load.

The second obstacle is that at fixed overlap one cannot simply replace all the patterns but one by Gaussian rows, because a state can be correlated with several patterns. Such mixtures do exist, and they raise the free energy at low overlap. Informally, a state of the binary model with overlap $m$ with one pattern and overlaps $\ov$ with others is never better than a state of the Gaussian-row model with the single overlap $V=(m^2+|\ov|^2)^{1/2}$. This is Jensen's inequality on the sphere of overlap vectors, and its field form is due to \cite{ABCHIM}. The inequality identifies the maximum of the free energy over all overlaps, which is all that the equilibrium problem sees, but it still allows the free energy at a given overlap to rise up to the maximum over larger overlaps and fill a well. The capacity is therefore decided by a strict form of the inequality, with a gap that does not depend on the number of retained patterns, which we prove. Together with the inequality itself, this strict form shows that binary patterns and Gaussian rows have the same wells.

\subsection{Free energy at fixed overlap}
Let $\xi^1,\dots,\xi^M$ be i.i.d.\ uniform on $\Cube$, with $M/N\to\alpha>0$, and let $\beta=1/T>0$. We write $m_\mu(\sigma)=\xi^\mu\cdot\sigma/N$ for the overlap, which is the fraction of sites where the two configurations agree minus the fraction where they differ. Its possible values are $m_k=2k/N-1$, and $\Slice(k)$ is the set of $\sigma$ with exactly $k$ plus signs. The free energy at fixed overlap is
\begin{equation}\label{eq:pN}
  p_N(k):=\frac1N\E\log 2^{-N}\sum_{\sigma:\,m_1(\sigma)=m_k}e^{-\beta H_N(\sigma)}.
\end{equation}
The comparison model keeps one binary pattern and replaces the others by Gaussian rows. With the pattern gauged to the all-ones vector and $G$ an $(M-1)\times N$ matrix of i.i.d.\ standard Gaussians, its free energy at fixed overlap is
\begin{equation}\label{eq:pNG}
  p^{G}_N(k):=\frac{\beta m_k^2}{2}+\frac1N\E\log Y_{N,k},\qquad
  Y_{N,k}:=2^{-N}\sum_{\sigma\in\Slice(k)}\exp\Bigl(\frac{\beta}{2N}|G\sigma|^2\Bigr).
\end{equation}
Only the limit of the ratio of rows to columns enters the results, so in later sections we will also use $M\times N$ matrices.

Our formula involves the Parisi functional on a slice of Boursier and Mourrat \cite{BM}. Let $\Qone$ be the set of nondecreasing right-continuous paths $x:[0,1)\to[0,1]$, and $\Qinf$ the set of bounded nondecreasing right-continuous paths with values in $\R_+$. For a law $\nu$ on $\R_+$ with bounded support, we let $\phi_\nu$ solve
\begin{equation}\label{eq:parisi-pde}
  -\partial_s\phi_\nu=\partial_x^2\phi_\nu+\nu([0,s])(\partial_x\phi_\nu)^2,\qquad \phi_\nu(S,\cdot)=\log\cosh,
\end{equation}
with $S\ge\sup\supp\nu$. We set $\resp_\nu:=\partial_x\phi_\nu(0,\cdot)$, and $\lambda(\nu,m)$ denotes the solution of $\resp_\nu(\lambda)=m$. We also put $g_m(\nu):=S+\sup_u\{um-\phi_\nu(0,u)\}$, which does not depend on $S$. For $a\in\Qinf$, we write $g_m(a)$ for $g_m$ at the law of $a(U)$, with $U$ uniform on $[0,1)$. The conjugate is $J_m(x):=\sup_{a\in\Qinf}\{\int_0^1xa-g_m(a)\}$. For $x\in\Qone$, let
\begin{equation}\label{eq:sbeta-def}
  \delta_x(w):=\int_0^1(1-x(u)\vee w)\dd u,\qquad D_x(w):=1-\beta\delta_x(w),\qquad
  \Sbeta(x):=\frac\beta2\int_0^1\frac{\dd w}{D_x(w)}
\end{equation}
if $D_x(0)>0$, and $\Sbeta(x):=+\infty$ otherwise. For $|m|<1$, we set
\begin{equation}\label{eq:main-formula}
  \Phi(m)=\min_{x\in\Qone}\bigl\{J_m(x)+\alpha\,\Sbeta(x)\bigr\},\qquad\Lam(m)=\frac{\beta m^2}2+\Phi(m),
\end{equation}
extended continuously to $m=\pm1$. Since the objective is strictly convex in the law of $x$, the minimizer $x^*$ is unique. Let $\nu^*(m)=\nu^*(m;\alpha,\beta)$ be the law of $a^*=\frac{\alpha\beta^2}{2}\int_0^{x^*}\dd s/D_{x^*}(s)^2$. We then put $h(m)=h(m;\alpha,\beta):=\lambda(\nu^*(m),m)$.

To describe states correlated with several patterns, let $r\ge1$ and $d_r=2^{r-1}$, and let $\eta$ run over $\{-1,1\}^{r-1}$, with $\Eeta$ the uniform average. For $\tv=(t_\eta)\in[-1,1]^{d_r}$, we put
\begin{equation}\label{eq:V-def}
  m=\Eeta t_\eta,\qquad \ov(\tv)=\Eeta[\eta\,t_\eta]\in\R^{r-1},\qquad V(\tv)^2=m^2+|\ov(\tv)|^2,
\end{equation}
and $P_r(m)=\{\tv\in[-1,1]^{d_r}:\Eeta t_\eta=m\}$. Given $r-1$ further patterns, the sites split into $d_r$ types according to their values, and $t_\eta$ is the magnetization on type $\eta$. We denote by $\Phi^{(r)}(\tv)$ the limit free energy of the Gaussian rows on the corresponding type slice. Its formula is given in Theorem~\ref{thm:gaussian-slice}, and $\Phi^{(1)}=\Phi$.

\begin{theorem}[Free energy at fixed overlap]\label{thm:main-free-energy}
Let $\alpha,\beta>0$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item \textup{(Gaussian rows.)} $\sup_k|p^G_N(k)-\Lam(m_k)|\to0$, with $\Lam$ given by \eqref{eq:main-formula}. $\Lam$ is $C^1$ on $(-1,1)$ with $\Lam'(m)=\beta m-h(m)$.
\item \textup{(Binary patterns.)} $\sup_k|p_N(k)-\LamR(m_k)|\to0$, where
\begin{equation}\label{eq:main-sup-r}
  \LamR(m)=\sup_{r\ge1}\ \sup_{\tv\in P_r(m)}\Fr(\tv),\qquad \Fr(\tv):=\frac\beta2V(\tv)^2+\Phi^{(r)}(\tv).
\end{equation}
\item \textup{(Merging.)} For all $r$ and all $\tv\in[-1,1]^{d_r}$,
\begin{equation}\label{eq:main-merging}
  \Fr(\tv)\le\Lam(V(\tv)).
\end{equation}
Hence for $m\in[0,1]$
\begin{equation}\label{eq:main-envelope}
  \Lam(m)\le\LamR(m)\le\max_{[m,1]}\Lam,\qquad \LamR(0)=\max_{[0,1]}\Lam,
\end{equation}
and $\LamR(m)=\Lam(m)$ wherever $\Lam(m)=\max_{[m,1]}\Lam$. If $0<m<1$ and $\Lam(m)<\max_{[m,1]}\Lam$, then $\LamR(m)<\max_{[m,1]}\Lam$.
\end{enumerate}
\end{theorem}

The free energy at fixed overlap does not reduce to $r=1$ everywhere: by \eqref{eq:main-envelope}, $\LamR(0)>\Lam(0)$ as soon as a state at positive overlap beats $m=0$. The last statement of part (c) is what keeps this from changing the wells.

Since $\Lam$ is the logarithm of a partition function, a well of the free energy is a local maximum of $\Lam$, and it requires $\Lam$ to rise at positive overlap. For $m_s\in(0,1)$, we define
\begin{equation}\label{eq:well-def}
  \Well_T(m_s):=\bigl\{\alpha>0:\ \Lam(m_2)>\Lam(m_1)\text{ for some }m_s\le m_1<m_2<1\bigr\},
\end{equation}
and $\WellR_T(m_s)$ is the same set with $\LamR$. We set $\Well_T=\bigcup_{m_s}\Well_T(m_s)$, $\alc(T;m_s)=\sup\Well_T(m_s)$ and $\alc(T)=\sup\Well_T$, with $\sup\emptyset=0$, and similarly with the superscript $\mathrm R$. The point $m=0$ is a strict local maximum of $\Lam$ at every $T>0$ (Theorem~\ref{thm:uniqueness-derivative}) and $\Lam'\to-\infty$ at $m=1$, so a point where $\Lam'>0$ always lies between a local minimum and a local maximum at positive overlap.

\subsection{Energy at fixed overlap}
At zero temperature, the role of the free energy is played by the energy at fixed overlap,
\begin{equation}\label{eq:eN}
  e_N(k):=\frac1N\E\max_{m_1(\sigma)=m_k}\bigl(-H_N(\sigma)\bigr),
\end{equation}
whose limit is $\eR$ for binary patterns and $e$ for Gaussian rows (Theorem~\ref{thm:main-zero}). With this sign, larger values mean lower energy. The set $\Well_0(m_s)$ consists of the $\alpha>0$ with $e(m_2)>e(m_1)$ for some $m_s\le m_1<m_2\le1$, and $\WellR_0(m_s)$ is the same set with $\eR$. The numbers $\alc(0;m_s)$, $\alc(0)$, $\alcR(0;m_s)$, $\alcR(0)$ are defined as before, so that $\alc(0)=\sup_{m_s>0}\alc(0;m_s)$.

The zero-temperature formula involves the following recursion. Let $\mathcal U$ be the set of nondecreasing right-continuous $\kappa:[0,1)\to[0,\infty)$ with $\int_0^1\kappa<\infty$, and for $\kappa\in\mathcal U$ and $C\ge0$, put $\delta(w)=C+\int_w^1\kappa$. Let $\Psi_\kappa$ solve $-\partial_w\Psi_\kappa=\frac12\bigl(\partial_x^2\Psi_\kappa+\kappa(w)(\partial_x\Psi_\kappa)^2\bigr)$ with $\Psi_\kappa(1,x)=|x|$. This is the recursion at unit scale. In Section~\ref{sec:zero} we will also use a version scaled by $\alpha$, and there this one carries a tilde.

\begin{theorem}[Zero temperature]\label{thm:main-zero}
\begin{enumerate}[label=\textup{(\alph*)}]
\item The energy at fixed overlap converges uniformly in $k$: to $\eR(m_k)$ for binary patterns, with $\eR=\lim_{\beta\to\infty}\LamR/\beta$, and to $e(m_k)$ for Gaussian rows, where $e=m^2/2+L^2/2$ and
\begin{equation}\label{eq:main-L}
  L(\alpha,m)=\inf_{\kappa,C,\lambda}\Bigl\{\frac C2+\frac\alpha2\int_0^1\frac{\dd w}{\delta(w)}+\Psi_\kappa(0,\lambda)-\lambda m\Bigr\}.
\end{equation}
For $|m|<1$ the infimum is attained at a unique point. $L$ is jointly concave and $C^1$ on $(0,\infty)\times(-1,1)$, with $\partial_mL=-\lambda^*$, and $\partial_m e=m-L\lambda^*$.
\item $\WellR_0(m_s)=\Well_0(m_s)$ for every $m_s\in(0,1)$, and $\alpha\in\Well_0(m_s)$ if and only if $m>L(\alpha,m)\lambda^*(\alpha,m)$ for some $m\in(m_s,1)$.
\item $\alc(0;m_s)=\alcR(0;m_s)$ is at least $\alo$ for every $m_s\in(0,0.9666236]$ and at most $\ahi$ for every $m_s\in(0,1)$.
\item For $\alpha\ge\ahi$, $\eR=e$ on $[0,1]$ and $\eR$ is strictly decreasing on $(0,1)$. Every $\alpha\in(0,\alo]$ lies in $\WellR_0(m_s)$ for all $m_s\le0.9616$, so there are no holes below $\alo$.
\item If $\eR(m_2)-\eR(m_1)=\Delta>0$ with $m_1<m_2$, and $m_{k_i}\to m_i$, then for large $N$, with probability at least $1-e^{-cN}$, for all patterns simultaneously,
\begin{equation}\label{eq:main-landscape}
  \min\{H_N(\sigma):m_\mu(\sigma)=m_{k_1}\}\ge\min\{H_N(\sigma):m_\mu(\sigma)=m_{k_2}\}+\frac\Delta3N.
\end{equation}
\item Every limit point of $\alc(T)$ as $T\to0$ lies in $[\alo,\ahi]$, and so does every limit point of $\alc(T;m_s)$ for $m_s\in(0,0.9666236]$.
\end{enumerate}
\end{theorem}

Parts (c), (d) and (f) have computer-assisted proofs, while parts (a), (b) and (e) do not use a computer.

\subsection{The capacity at positive temperature}
The next theorem characterizes the wells at positive temperature and gives them their meaning at finite $N$.

\begin{theorem}[Capacity at positive temperature]\label{thm:main-capacity}
Let $T>0$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item $\alpha\in\Well_T$ if and only if $\beta m>h(m;\alpha,\beta)$ for some $m\in(0,1)$, if and only if $m<\resp_{\nu^*(m)}(\beta m)$ for some $m\in(0,1)$. The stationary points $m>0$ of $\Lam$ are exactly the solutions of the generalized Mattis equation $m=\resp_{\nu^*(m)}(\beta m)$.
\item $\Well_T$ is open. It is empty for $T\ge1$. For $T<1$ it contains an interval $(0,\alpha_0(T))$ with $\alpha_0(T)>0$. For every $T$, $\Well_T\subset(0,2)$.
\item At a positive boundary point of $\Well_T$ there is $m_c\in(0,1)$ with $h(m_c)=\beta m_c$, and $\Lam'\le0$ on $[0,1)$. So the well disappears at positive overlap, with no collapse to $m=0$ or $m=1$.
\item If $\alpha\in\Well_T(m_s)$, there are $m_s\le m_b<m_*<1$ and $\Delta>0$ such that for large $N$, with probability at least $1-e^{-cN}$, for all patterns $\mu$ simultaneously,
\begin{equation}\label{eq:main-all-patterns}
  Z^\mu(k_b)\le e^{-N\Delta/3}\,Z^\mu(k_*),\qquad Z^\mu(k):=2^{-N}\sum_{m_\mu(\sigma)=m_k}e^{-\beta H_N(\sigma)},
\end{equation}
where $m_{k_b}\to m_b$, $m_{k_*}\to m_*$.
\item If $\alpha\notin\Well_T$, then $\LamR=\Lam$ on $[0,1]$ and $\LamR$ is nonincreasing there. So above $\alc(T)$ there is no well for binary patterns at any overlap.
\end{enumerate}
\end{theorem}

By part (d), a well implies that, around every pattern at once, the slice at overlap $m_{k_b}$ carries exponentially less weight than the slice at the larger overlap $m_{k_*}$, and \eqref{eq:main-landscape} is the analogue for the minimum energy at zero temperature.

\subsection{Main theorem}
Our main result collects the identity of the well sets with the bounds on the capacity from Theorems~\ref{thm:main-zero} and~\ref{thm:main-capacity}.

\begin{theorem}[Critical capacity]\label{thm:main}
Let $\alo=0.138198129$ and $\ahi=0.13820106$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item For every $T>0$ and $m_s\in(0,1)$, $\WellR_T(m_s)=\Well_T(m_s)$: binary patterns and Gaussian rows have the same wells. So $\alcR(T;m_s)=\alc(T;m_s)$, $\alcR(T)=\alc(T)$, and
\begin{equation}\label{eq:main-alc}
  \alc(T)=\sup\bigl\{\alpha>0:\ \exists\, m\in(0,1),\ \beta m>h(m;\alpha,\beta)\bigr\}.
\end{equation}
\item $\alc(T)=0$ for $T\ge1$, and $0<\alc(T)\le2$ for $T<1$.
\item $\alo\le\alc(0)=\alcR(0)\le\ahi$. So the capacity at $T=0$ is $0.13820$ to five digits.
\item At $T=0$, in both models, there is no well at any overlap for $\alpha\ge\ahi$, and every $\alpha\in(0,\alo]$ has a well.
\end{enumerate}
\end{theorem}

\subsection{Relation to prior work}
The equilibrium free energy is now known. The Hamilton--Jacobi method of Boursier--Mourrat, Chen--Mourrat and Chen--Ho--Issa \cite{BM,CM,CHI} handles Ising and spherical spins with a non-convex coupling. Camilli, Contucci and Mingione announced the formula for Gaussian patterns \cite{CCM}, Barra and Durante reduced binary patterns to finitely many retained patterns in a Gaussian bulk \cite{BD}, and Ali, Boursier, Chen, Ho, Issa and Mourrat proved the limit for all $\alpha,\beta>0$, with one retained pattern \cite{ABCHIM}.

The fixed-overlap formula, the capacity characterization, the zero-temperature results and the certified bounds are not in \cite{ABCHIM}. The proofs given here do, however, rest on that preprint, which has not been refereed, mainly in two places. The limit of the mixed model at fixed radius in Section~\ref{sec:gaussian} is their Theorem~3.3 with their Proposition~4.1, and we also take from them the facts on the Ising and spherical terms in their Lemmas~2.1 and~2.3 and (4.5). The comparison of Rademacher and Gaussian disorder in Section~\ref{sec:reduction} uses their Lemmas~6.1 and~6.2, their estimate (6.20), whose proof also uses their Lemma~6.3, and their concentration bounds (6.3)--(6.5). We first obtained these two steps without \cite{ABCHIM}, by rerunning the Hamilton--Jacobi argument of \cite{BM} on the mixed slice and by extending the sector comparison of Barra and Durante \cite{BD}. That longer version is in the repository given under Code and data, as \texttt{paper/long\_version.pdf}. The Jensen step behind Lemma~\ref{lem:jensen-sphere} is due to \cite[Prop.~5.2]{ABCHIM}, although the lemma has its own short proof. That lemma, the merging theorems built on it, and the extension of the capacity statements to all overlaps came after we had read \cite{ABCHIM}.

The upper bound \eqref{eq:main-merging} is not stated in \cite{ABCHIM}, but it follows from their Proposition~5.2 in a few lines (Remark~\ref{rem:abchim-link}). They also state that a refinement of their analysis gives a strict inequality at a fixed number of fields, without giving the refinement. Theorem~\ref{thm:merging-strict} proves the strict inequality at a fixed slice with a gap uniform in $r$, and the identity of the well sets needs this uniformity.

Stojnic \cite{Stojnic24,Stojnic26} studies the capacity through lifted random duality and computes finite-level values of the same kind as those in \cite{SK94}. Talagrand proved the replica symmetric formula in a region of parameters \cite{Talagrand98}. Compared with \cite{FST01}, \eqref{eq:main-landscape} implies their event for each pattern, at every load up to $\alo$, with an inner point that is not specified.

\subsection{Comparison with physics}\label{ssec:physics}
At $T=0$, replica symmetry gives $0.137906$ \cite{AGS85,SK94}, and \cite{SK94} gives $0.1381864895$ at one step and $0.138187733$ at two steps. The one-step value $0.144$ of \cite{CAG86} is still widely quoted, although \cite{SK94} regards it as an error. Finite-size simulations started from a stored pattern report $0.145\pm0.009$ (Amit, Gutfreund and Sompolinsky, as quoted in \cite{SK94}), about $0.143\pm0.001$ \cite{Kohring90} and $0.141\pm0.0015$ \cite{SMK96}. All of these lie above our interval, but they measure retrieval under a dynamics at finite $N$, whereas $\alc$ measures wells of the limiting profile at fixed overlap.

\subsection{Outline of the proof}
We first treat Gaussian rows at fixed overlap. Writing $\exp(\frac\beta{2N}|G\sigma|^2)$ as a Gaussian integral over $z\in\R^{M-1}$ and separating the radius of $z$ from its direction, we obtain an Ising slice coupled bilinearly to a sphere. At fixed radius, this mixed model is solved by the Hamilton--Jacobi theorem of \cite{ABCHIM}, uniformly in the slice, and a Laplace principle in the radius gives the slice formula (Section~\ref{sec:gaussian}).

To pass from binary patterns to retained patterns, we note that after a gauge a slice is a deterministic weight on the cube, and that the comparison lemmas of \cite{ABCHIM} between Rademacher and Gaussian disorder hold for any weight. Large hidden coordinates go into the weight, and the others are compared under a cutoff. Convex concentration on the cube then gives a union over all patterns and slices (Section~\ref{sec:reduction}), and hence Theorem~\ref{thm:main-free-energy}(b).

After the sphere and the radius are eliminated in closed form, the slice problem becomes \eqref{eq:main-formula}, which is strictly convex in the law of $x$. Uniqueness gives differentiability and the multiplier $h$, and the capacity follows from the shape of $\Lam$ (Section~\ref{sec:convex}). Section~\ref{sec:zero} does the same at zero temperature.

The merging step rests on the fact that $q\mapsto\phi_\nu(0,\sqrt q)$ is concave for every $\nu$. Applied to the effective fields of a state with several overlaps, Jensen's inequality gives \eqref{eq:main-merging}, and a fourth-moment bound for Rademacher sums gives a strict form that is uniform in $r$. Together they give Theorem~\ref{thm:main}(a) (Section~\ref{sec:merging}).

Finally, the zero-temperature recursion can be evaluated in ball arithmetic. A trial bounds $L$ from above for all $\alpha$ and a tangent of the sphere problem bounds it from below, and together they certify a well at $\alo$ and $e'<0$ on $[0.89,1)$ at $\ahi$. A separate bound on the slope of the response, $h'\ge(1+m^2/\wf(m)^2)^{1/2}$ almost everywhere, excludes $(0,0.89)$ (Section~\ref{sec:digits}). Appendix~\ref{app:cert} gives the error terms of the ball arithmetic and describes what was computed.