From 79f9c65c6b60f989a0ac0a702cadde3a59d7ae0f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 18 Sep 2024 00:04:52 +0200
Subject: Lots of writing and rewriting.

---
 topology.tex | 587 ++++++++++++++++++++++++++++++++---------------------------
 1 file changed, 321 insertions(+), 266 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 08c3a0b..d6fb36f 100644
--- a/topology.tex
+++ b/topology.tex
@@ -151,14 +151,14 @@ Gaussian random functions with covariance
 \end{equation}
 for some choice of function $f$. When the covariance function $f$ is polynomial, the
 $V_k$ are also polynomial, with a term of degree $p$ in $f$ corresponding to
-all possible terms of degree $p$ in $V_k$. In particular, taking
+all possible terms of degree $p$ in $V_k$. In particular, one can explicitly construct functions that satisfy \eqref{eq:covariance} by taking
 \begin{equation}
   V_k(\mathbf x)
   =\sum_{p=0}^\infty\frac1{p!}\sqrt{\frac{f^{(p)}(0)}{N^p}}
   \sum_{i_1\cdots i_p}^NJ^{(k,p)}_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
 \end{equation}
 with the elements of the tensors $J^{(k,p)}$ as independently distributed
-unit normal random variables satisfies \eqref{eq:covariance}. The size of the
+unit normal random variables. The size of the
 series coefficients of $f$ therefore control the variances in the coefficients
 of random polynomial constraints. When $M=1$, this problem corresponds to the
 level set of a spherical spin glass with energy density $E=V_0/\sqrt{N}$.
@@ -199,6 +199,8 @@ putting strong constraints on the resulting topology and geometry.
 
 \section{The average Euler characteristic}
 
+\subsection{Definition and derivation}
+
 The Euler characteristic $\chi$ of a manifold is a topological invariant \cite{Hatcher_2002_Algebraic}. It is
 perhaps most familiar in the context of connected compact orientable surfaces, where it
 characterizes the number of handles in the surface: $\chi=2(1-\#)$ for $\#$
@@ -226,8 +228,7 @@ the sign of the determinant, we arrive at the Euler characteristic, or
 \begin{equation} \label{eq:kac-rice}
   \chi(\Omega)=\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)\det\operatorname{Hess}H(\mathbf x)
 \end{equation}
-When the Kac--Rice formula is used to \emph{count} stationary points, the sign
-of the determinant is a nuisance that one must take pains to preserve
+When the Kac--Rice formula is used to \emph{count} stationary points, one must take pains to preserve the sign of the determinant
 \cite{Fyodorov_2004_Complexity}. Here we are correct to exclude it.
 
 We need to choose a function $H$ for our calculation. Because $\chi$ is
@@ -236,7 +237,7 @@ symmetry with the underlying manifold, i.e., that $H$ satisfies the Smale condit
 constraints has no symmetries, we can take a simple height function $H(\mathbf
 x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
 $\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is
-used as the polar axis, $H$ gives the height on the sphere.
+used as the polar axis, $H$ gives the height on the sphere relative to the equator.
 
 We treat the integral over the implicitly defined manifold $\Omega$ using the
 method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to
@@ -258,7 +259,7 @@ is the vector of partial derivatives with respect to all $N+M+1$ variables.
 This integral is now in a form where standard techniques from mean-field theory
 can be applied to calculate it.
 
-To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
+To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$-functions and determinant to integral form, with
 \begin{align}
   \label{eq:delta.exp}
   \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
@@ -273,7 +274,7 @@ To evaluate the average of $\chi$ over the constraints, we first translate the $
 where $\hat{\mathbf x}$ and $\hat{\pmb\omega}$ are ordinary vectors and
 $\bar{\pmb\eta}$, $\pmb\eta$, $\bar{\pmb\gamma}$, and $\pmb\gamma$ are
 Grassmann vectors. With these expressions substituted into
-\eqref{eq:kac-rice.lagrange}, the result is a integral over an exponential
+\eqref{eq:kac-rice.lagrange}, the result is an integral over an exponential
 whose argument is linear in the random functions $V_k$. These functions can
 therefore be averaged over, and the resulting expression treated with standard
 methods. Details of this calculation can be found in Appendix~\ref{sec:euler}.
@@ -296,7 +297,7 @@ where $g$ is a prefactor of $o(N^0)$, and $\mathcal S_\chi$ is an effective acti
     \right)
   \end{aligned}
 \end{equation}
-The remaining order parameters defined by the scalar products
+The remaining order parameters are defined by the scalar products
 \begin{align}
   R=-i\frac1N\mathbf x\cdot\hat{\mathbf x}
   &&
@@ -307,15 +308,17 @@ The remaining order parameters defined by the scalar products
   \hat m=-i\frac1N\hat {\mathbf x}\cdot\mathbf x_0
 \end{align}
 
+\subsection{Features of the effective action}
+
 This integral can be evaluated to leading order by a saddle point method. For reasons we will
 see, it is best to extremize with respect to $R$, $D$, and $\hat m$, leaving a
 new effective action of $m$ alone. This can be solved to give
 \begin{equation}
-  D=-\frac{m+R^*}{1-m^2}R^* \qquad \hat m=0
+  D=-\frac{m+R_*}{1-m^2}R_* \qquad \hat m=0
 \end{equation}
-\begin{equation}
+\begin{equation} \label{eq:rs}
   \begin{aligned}
-    R^*
+    R_*
     =\frac{-m(1-m^2)}{2[f(1)-(1-m^2)f'(1)]^2}
     \Bigg[
       \alpha V_0^2f'(1)
@@ -328,23 +331,23 @@ new effective action of $m$ alone. This can be solved to give
   \end{aligned}
 \end{equation}
 with the resulting effective action as a function of $m$ alone given by
-\begin{equation}
+\begin{equation} \label{eq:S.m}
   \mathcal S_\chi(m)
   =-\frac\alpha2\bigg[
     \log\left(
-      1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R^*}}{1-m^2}
+      1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R_*}}{1-m^2}
     \right)
     +\frac{V_0^2}{f(1)}\left(
-      1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R^*}}
+      1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R_*}}
     \right)^{-1}
   \bigg]
-  +\frac12\log\left(-\frac m{R^*}\right)
+  +\frac12\log\left(-\frac m{R_*}\right)
 \end{equation}
 This function is plotted as a function of $m$ in Fig.~\ref{fig:action} for a variety of $V_0$ and $f$.
 To finish evaluating the integral, this expression should be maximized with
-respect to $m$. The order parameter $m$ is both physical and interpretable, as
+respect to $m$. If $m_*$ is such a maximum, then $\overline{\chi(\Omega)}\propto e^{N\mathcal S_\chi(m_*)}$. The order parameter $m$ is both physical and interpretable, as
 it gives the overlap of the configuration $\mathbf x$ with the height axis
-$\mathbf x_0$. Therefore, the value $m^*$ that maximizes this action can be
+$\mathbf x_0$. Therefore, the value $m$ that maximizes this action can be
 understood as the latitude on the sphere where most of the contribution to the
 Euler characteristic is made.
 
@@ -355,29 +358,30 @@ Euler characteristic is made.
 
   \caption{
     \textbf{Effective action for the Euler characteristic.}
-    The effective action governing the average Euler characteristic as a function of the overlap
-    $m=\frac1N\mathbf x\cdot\mathbf x_0$ with the height direction for two
-    different homogeneous polynomial functions and a variety of target values $V_0$. In both
+    The effective action \eqref{eq:S.m} governing the average Euler characteristic as a function of the overlap
+    $m=\frac1N\mathbf x\cdot\mathbf x_0$ with the height axis for two
+    different homogeneous polynomial constraints and a variety of target values $V_0$. Dashed lines depict $\operatorname{Re}\mathcal S_\chi$ when its imaginary part is nonzero. In both
     plots $\alpha=\frac12$. \textbf{Left:} With linear functions there are two
     regimes. For small $V_0$, there are maxima at $m=\pm m^*$ where the action
-    is zero, while after the satisfiability transition at $V_0=V_\text{\textsc{sat}}=1$, $m^*$
-    goes to zero and the action becomes negative. \textbf{Right:} With nonlinear
-    functions there are four regimes. For small $V_0$ the behavior is the same
-    as in the linear case, with zero action. After an onset transition at
-    $V_0=V_\text{on}\simeq1.099$ the maxima are at the edge of validity of the
-    action and the action is positive. At a shattering transition at
-    $V_0=V_\text{sh}\simeq1.394$, the maximum goes to zero and the action is positive.
-    Finally, at the satisfiability transition
-    $V_0=V_\text{\textsc{sat}}\simeq1.440$ the action becomes negative.
+    is zero, while after the satisfiability transition at $V_0=V_{\text{\textsc{sat}}\ast}=1$, $m_*$
+    goes to zero and the action becomes negative everywhere. \textbf{Right:} With nonlinear
+    functions there are other possible regimes. For small $V_0$, there are maxima at $m=\pm m^*$ where the action is zero but the real part of the action is maximized at $m=0$ where the action is complex. For larger $V_0\geq V_\text{on}\simeq1.099$ the maxima at $m=\pm m^*$ disappear. For $V_0\geq V_\text{sh}\simeq1.394$ larger still, the action becomes real everywhere.
+    Finally, at a satisfiability transition
+    $V_0=V_\text{\textsc{sat}}\simeq1.440$ the action becomes negative everywhere.
   } \label{fig:action}
 \end{figure}
 
-The action $\mathcal S_\chi$ is extremized with respect to $m$ at $m=0$ or $m=\pm m^*=\mp R^*(m^*)$.
-In the latter case, $m^*$ takes the value
+The action $\mathcal S_\chi$ is extremized with respect to $m$ at $m=0$ or $m=\pm m_*=\mp R_*(m_*)$ for
 \begin{equation}
   m^*=\sqrt{1-\frac{\alpha}{f'(1)}\big(V_0^2+f(1)\big)}
 \end{equation}
-and $\mathcal S_\chi(m^*)=0$. If this solution were always well-defined, it would vanish when the argument of the square root vanishes for
+and $\mathcal S_\chi(m^*)=0$. Zero action implies that
+$\overline{\chi(\Omega)}$ does not vary exponentially with $N$, and in fact we
+show in Appendix~\ref{sec:prefactor} that the contribution from each of these
+maxima is $(-1)^{N-M-1}+o(N^0)$, so that their sum is $2$ in even dimensions and $0$ in odd dimensions. This result is consistent with the topology of an $N-M-1$ sphere.
+
+If this solution were always well-defined, it would
+vanish when the argument of the square root vanishes at
 \begin{equation}
   V_0^2>V_{\text{\textsc{sat}}\ast}^2\equiv\frac{f'(1)}\alpha-f(1)
 \end{equation}
@@ -385,19 +389,50 @@ This corresponds precisely to the satisfiability transition found in previous wo
 However, the action becomes complex in the region $m^2<m_\text{min}^2$ for
 \begin{equation}
   m_\text{min}^2
-  =1-\frac{f(1)^2}{f'(1)}\frac{V_0^2[2(1+\sqrt{1-\alpha})-\alpha]-\alpha f(1)}{
-    2V_0^2f(1)(2-\alpha)-\alpha(V_0^4+f(1)^2)
+  =1-\frac{f(1)^2}{f'(1)}\frac{V_0^2(1+\sqrt{1-\alpha})^2-\alpha f(1)}{
+    4V_0^2f(1)-\alpha[V_0^2+f(1)]^2
   }
 \end{equation}
-When $m_*^2<m_\text{min}^2$, this solution is no longer valid. Likewise, when
-$m_\text{min}^2>0$, the solution at $m=0$ is also not valid. In fact, it is not
-clear what the average value of the Euler characteristic should be at all when
+When $m_*^2<m_\text{min}^2$, the solution at $m=\pm m_*$ no longer maximizes
+the action. This happens when the target value is larger than an onset value $V_\text{on}$ defined by
+\begin{equation}
+  V_0^2>V_\text{on}^2\equiv\frac{f(1)}\alpha\left(1-\alpha+\sqrt{1-\alpha}\right)
+\end{equation}
+Comparing this with the satisfiability transition associated with $m_*$ going to zero, one sees
+\begin{equation}
+  V_\text{on}^2-V_{\text{\textsc{sat}}\ast}^2
+  =\frac1\alpha\left(f'(1)-f(1)-f(1)\sqrt{1-\alpha}\right)
+\end{equation}
+If $f(q)$ is purely linear, then $f'(1)=f(1)$ and
+$V_\text{on}^2>V_{\text{\textsc{sat}}\ast}^2$, so the naïve satisfiability
+transition happens first. On the other hand, when $f(q)$ contains powers of $q$
+strictly greater than 1, then $f'(1)\geq 2f(1)$ and $V_\text{on}^2\leq
+V_{\text{\textsc{sat}}\ast}^2$, so the onset happens first. In situations with
+mixed constant, linear, and nonlinear terms in $f$, the order of the
+transitions depends on the precise form of $f$.
+
+Likewise, the solution at $m=0$ is sometimes complex-valued and sometimes real-valued. For $V_0$ less than a shattering value $V_\text{sh}$ defined by
+\begin{equation}
+  V_0^2<V_\text{sh}^2\equiv\frac{f(1)}\alpha\left(1-\frac{f(1)}{f'(1)}\right)\left(1+\sqrt{1-\alpha}\right)^2
+\end{equation}
+the maximum at $m=0$ is complex while for $V_0$ greater than this value the action is real. For purely
+linear $f(q)$, $V_\text{sh}=0$ and the action at $m=0$ is always real, though
+for $V_0^2<V_{\text{\textsc{sat}}\ast}^2$ it is a minimum rather than a
+maximum. Finally, there is another satisfiability transition at
+$V=V_\text{\textsc{sat}}$ corresponding to the vanishing of the effective
+action at the $m=0$ solution, with $\mathcal S(0)=0$. For generic covariance
+function $f$ it is not possible to write an explicit formula for
+$V_\text{\textsc{sat}}$, and we usually calculate it through a numeric
+root-finding algorithm.
+
+When $m_\text{min}^2>0$, the solution at $m=0$ is difficult to interpret, since the action takes a complex value. In fact, it is not
+clear what the contribution to the average value of the Euler characteristic should be at all when
 there is some range $-m_\text{min}<m<m_\text{min}$ where the effective action
 is complex. Such a result could arise from the breakdown of the large-deviation
 principle behind the calculation of the effective action, or it could be the
 result of a negative Euler characteristic.
 
-To address this problem, we compute also the average of the square of the Euler
+To address this ambiguity, we compute also the average of the square of the Euler
 characteristic, $\overline{\chi(\Omega)^2}$, with details in
 Appendix~\ref{sec:rms}. This has the benefit of always being positive, so that
 the saddle-point approach to the calculation at large $N$ does not produce
@@ -410,14 +445,75 @@ and one at $m=0$ where
   \frac1N\log\overline{\chi(\Omega)^2}=2\operatorname{Re}\mathcal S_\chi(0)
 \end{equation}
 which is consistent with
-$\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$. Such a
+$\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$. It is therefore possible to interpret the average Euler characteristic in the
+regime where its effective action is complex-valued as being negative, with a
+magnitude implied by the real part of the action.
+
+Such a
 correspondence, which indicates that the `annealed' calculation here is also
 representative of typical realizations of the constraints, is not always true.
 With average squared Euler characteristic we find instabilities of the solution
-described here to replica symmetry breaking (\textsc{rsb}). We do not explore
+at $m=0$ to replica symmetry breaking (\textsc{rsb}). We do not explore
 these \textsc{rsb} solutions here, except in the context of $M=1$ in
 Section~\ref{sec:ssg}. However, in the following Figures \ref{fig:phases} and
-\ref{fig:crossover} we depict the unstable region with shading.
+\ref{fig:crossover} we shade the unstable region.
+
+\subsection{Topological phases and their interpretation}
+
+The results of the previous section allow us to unambiguously define distinct
+topological phases, which differ depending on the presence or absence of the
+local maxima at $m=\pm m^*$, on the presence or absence of the local maximum
+at $m=0$, on the real or complex nature of this maximum, and finally on
+whether the action is positive or negative. Below we enumerate these regimes,
+which are schematically represented in Fig.~\ref{fig:cartoons}.\footnote{
+  In the following we characterize regimes by values of
+  $\overline{\chi(\Omega)}$. These should be understood as their values in
+  \emph{even} dimensions, since in odd dimensions the Euler characteristic is
+  always identically zero. We do not expect the qualitative results to change
+  depending on the evenness or oddness of the manifold dimension.
+}
+
+\paragraph{Regime I: \boldmath{$\overline{\chi(\Omega)}=2$}.}
+
+This regime is found when the magnitude of the target value $V_0$ is less than
+the onset $V_\text{on}$ and $\operatorname{Re}\mathcal S(0)<0$, so that the
+maxima at $m=\pm m^*$ exist and are the dominant contributions to the average
+Euler characteristic. Here, $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$,
+strongly indicating a topology homeomorphic to the $S^{N-M-1}$ sphere. This regime is the only nontrivial one found with linear covariance $f(q)$, where the solution manifold must be a sphere if it is not empty.
+
+\paragraph{Regime II: \boldmath{$\overline{\chi(\Omega)}$} large and negative, isolated contributions at \boldmath{$m=\pm m^*$}.}
+
+This regime is found when the magnitude of the target value $V_0$ is less than the onset $V_\text{on}$, $\operatorname{Re}\mathcal S(0)>0$, and the value of the action at $m=0$ is complex. The dominant contribution to the average Euler characteristic comes from the equator at $m=0$, but the complexity of the action implies that the Euler characteristic is negative.
+While the topology of the manifold is not necessarily connected in this
+regime, holes are more numerous than components. Since $V_0^2<V_\text{on}^2$,
+there are isolated contributions to $\overline{\chi(\Omega)}$ at $m=\pm m^*$.
+This implies a temperate band of relative simplicity: given a random point on
+the sphere, the nearest parts of the solution manifold do not have holes or
+disconnected components.
+
+\paragraph{Regime III: \boldmath{$\overline{\chi(\Omega)}$} large and negative, no contribution at \boldmath{$m=\pm m^*$}.}
+
+The same as Regime II, but with $V_0^2>V_\text{on}^2$. The solutions at
+$m=\pm m_*$ no longer exist, and nontrivial contributions to the Euler
+characteristic are made all the way to the solution manifold's boundary.
+
+\paragraph{Regime IV: \boldmath{$\overline{\chi(\Omega)}$} large and positive.}
+
+This regime is found when the magnitude of the target value $V_0$ is greater
+than the shattering value $V_\text{sh}$ and $\mathcal S(0)>0$. Above the
+shattering transition the effective action is real everywhere, and its value at
+the equator is the dominant contribution. Large connected components of the
+manifold may or may not exist, but small disconnected components outnumber
+holes.
+
+\paragraph{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.}
+
+Here $\frac1N\log\overline{\chi(\Omega)}<0$, indicating that the average
+Euler characteristic shrinks exponentially with $N$. Under most conditions
+we conclude this is the \textsc{unsat} regime where no manifold exists, but
+there may be circumstances where part of this regime is characterized by
+nonempty solution manifolds that are overwhelmingly likely to have Euler
+characteristic zero.
 
 \begin{figure}
   \includegraphics[width=0.196\textwidth]{figs/connected.pdf}
@@ -451,52 +547,6 @@ Section~\ref{sec:ssg}. However, in the following Figures \ref{fig:phases} and
   } \label{fig:cartoons}
 \end{figure}
 
-We therefore have five possible topological regimes:
-\begin{itemize}
-  \item \textbf{Regime I:
-    \boldmath{$\overline{\chi(\Omega)}=2$}.}
-
-    $\frac1N\log\overline{\chi(\Omega)^2}=0$,
-    $\frac1N\log\overline{\chi(\Omega)}=0$
-    This regime occurs when $m_*^2>0$
-    and $\operatorname{Re}\mathcal S_\chi(0)<0$. Here,
-    $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$, strongly indicating a
-    topology homeomorphic to the $S^{N-M-1}$ sphere.
-
-  \item \textbf{Regime II: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, isolated contribution at \boldmath{$m=\pm m^*$}.}
-
-    Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}<0$.
-    This regime occurs when $m_\text{min}^2>0$ and $\operatorname{Re}\mathcal
-    S_\chi(0)>0$. Here the average Euler characteristic is large and negative.
-    While the topology of the manifold is not necessarily connected in this
-    regime, holes are more numerous than components. In addition, here
-    $m_*^2>m_\text{min}^2$, meaning that the solutions at $m=\pm m_*$ are still
-    present. This indicates that the manifold has topologically simple
-    boundaries with some separation from the sea of holes.
-
-  \item \textbf{Regime III: \boldmath{$\overline{\chi(\Omega)}$} very large and negative, no contribution at \boldmath{$m=\pm m^*$}.}
-
-    The same as Regime II, but with $m_*^2<m_\text{min}^2$. The solutions at
-    $m=\pm m_*$ no longer exist, and nontrivial contributions to the Euler
-    characteristic are made all the way to the solution manifold's boundary.
-
-  \item \textbf{Regime IV: \boldmath{$\overline{\chi(\Omega)}$} very large and positive.}
-
-    Here $\frac1N\log\overline{\chi(\Omega)^2}>0$ and $\overline{\chi(\Omega)}>0$.
-    This regime occurs when $m_\text{min}^2<0$ and $\mathcal S_\chi(0)>0$. Here the
-    average Euler characteristic is large and positive. Large connected
-    components of the manifold may or may not exist, but small disconnected
-    components outnumber holes.
-
-  \item \textbf{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.}
-
-    Here $\frac1N\log\overline{\chi(\Omega)}<0$, indicating that the average
-    Euler characteristic shrinks exponentially with $N$. Under most conditions
-    we conclude this is the \textsc{unsat} regime where no manifold exists, but
-    there may be circumstances where part of this regime is characterized by
-    nonempty solution manifolds that are overwhelmingly likely to have Euler
-    characteristic zero.
-\end{itemize}
 
 \begin{figure}
   \includegraphics{figs/phases_1.pdf}
@@ -508,57 +558,50 @@ We therefore have five possible topological regimes:
   \caption{
     \textbf{Topological phase diagram.}
     Topological phases of the model for three different homogeneous covariance
-    functions. The onset transition $V_\text{on}$, shattering transition
-    $V_\text{sh}$, and satisfiability transition $V_\text{\textsc{sat}}$
-    are indicated when they exist. In the limit of $\alpha\to0$, the behavior
-    of level sets of the spherical spin glasses are recovered: the final plot
-    shows how in the pure cubic model the ground state energy $E_\text{gs}$ and threshold energy
-    $E_\text{th}$ correspond with the limits of the satisfiability and
-    shattering transitions, respectively. Note that for mixed models with
-    inhomogeneous covariance functions, $E_\text{th}$ is not the lower limit of
-    $V_\text{sh}$.
+    functions. The regimes are defined in the text and depicted as cartoons in
+    Fig.~\ref{fig:cartoons}. The shaded region in the center panel shows where
+    these results are unstable to \textsc{rsb}. In the limit of $\alpha\to0$,
+    the behavior of level sets of the spherical spin glasses are recovered: the
+    righthand plot shows how in the pure cubic model the ground state energy
+    $E_\text{gs}$ and threshold energy $E_\text{th}$ correspond with the limits
+    of the satisfiability and shattering transitions, respectively. Note that
+    for mixed models with inhomogeneous covariance functions, $E_\text{th}$ is
+    not the lower limit of $V_\text{sh}$.
   } \label{fig:phases}
 \end{figure}
 
-However, when the magnitude of $V_0$ is sufficiently large, with
-\begin{equation}
-  V_0^2>V_\text{on}^2\equiv\frac{1-\alpha+\sqrt{1-\alpha}}\alpha f(1)
-\end{equation}
-$R^*(m^*)$ becomes complex and this solution is no longer valid. Since
-\begin{equation}
-  V_\text{on}^2-V_{\text{\textsc{sat}}\ast}^2
-  =\frac{f'(1)-f(1)}\alpha-\frac{\sqrt{1-\alpha}}\alpha f(1)
-\end{equation}
-If $f(q)$ is purely linear, then $f'(1)=f(1)$ and
-$V_\text{on}^2>V_{\text{\textsc{sat}}\ast}^2$, so the naive satisfiability
-transition happens first. On the other hand, when $f(q)$ contains powers of $q$
-strictly greater than 1, then $f'(1)\geq 2f(1)$ and $V_\text{on}^2\leq
-V_{\text{\textsc{sat}}\ast}^2$, so the onset happens first. In situations with
-mixed constant, linear, and nonlinear terms in $f$, the order of the
-transitions depends on the precise form of $f$.
+\paragraph{}
 
-Likewise, the solution at $m=0$ is not always valid. For $V_0$ of magnitude sufficiently small, with
-\begin{equation}
-  V_0^2<V_\text{sh}^2\equiv\frac{2(1+\sqrt{1-\alpha})-\alpha}{\alpha}f(1)\left(1-\frac{f(1)}{f'(1)}\right)
-\end{equation}
-the maximum at $m=0$ becomes complex and that solution is invalid. For purely
-linear $f(q)$, $V_\text{sh}=0$ and the solution at $m=0$ is always real, though
-for $V_0^2<V_{\text{\textsc{sat}}\ast}^2$ it is a minimum rather than a
-maximum.
-These transition values of the target $V_0$ correspond with transition values in $\alpha$ of
-\begin{align}
-  \alpha_\text{on}=1-\left(\frac{V_0^2}{V_0^2+f(1)}\right)^2
-  &&
-  \alpha_\text{sh}=4V_0^2f(1)f'(1)\frac{f'(1)-f(1)}{\big((V_0^2+f(1))f'(1)-f(1)^2\big)^2}
-\end{align}
-Finally, there is another satisfiability transition at
-$V=V_\text{\textsc{sat}}$ corresponding to the vanishing of the effective
-action at the $m=0$ solution. For generic covariance function $f$ it is not
-possible to write an explicit formula for $V_\text{\textsc{sat}}$, and we
-usually calculate it through a numeric root-finding algorithm. However, for
-linear $f(q)$ one can see that
-$V_\text{\textsc{sat}}=V_{\text{\textsc{sat}}\ast}$, and indeed this is the
-case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$.
+The distribution of these phases for situations with homogeneous polynomial
+constraint functions is shown in Fig.~\ref{fig:phases}. For purely linear
+models, the only two regimes are I and V, separated by a satisfiability
+transition at $V_{\text{\textsc{sat}}\ast}$. This is expected: the intersection
+of a plane and a sphere is another sphere, and therefore a model of linear
+constraints in a spherical configuration space can only produce a solution
+manifold consisting of a single sphere, or the empty set. For purely nonlinear
+models, regime I does not appear, while the other three nontrivial regimes do.
+Regimes II and III are separated by the onset transition at $V_\text{on}$,
+while III and IV are separated by the shattering transition at $V_\text{sh}$.
+Finally, IV and V are now separated by the satisfiability transition at
+$V_\text{\textsc{sat}}$.
+
+One interesting
+feature occurs in the limit of $\alpha$ to zero. If $V_0$ is likewise rescaled
+in the correct way, the limit of these phase boundaries approaches known
+landmark energy values in the pure spherical spin glasses. In particular, the
+limit to zero $\alpha$ of the scaled satisfiability transition
+$V_\text{\textsc{sat}}\sqrt\alpha$ approaches the ground state energy
+$E_\text{gs}$, while the limit to zero $\alpha$ of the scaled shattering
+transition $V_\text{sh}\sqrt\alpha$ approaches the threshold energy
+$E_\text{th}$. The correspondence between ground state and satisfiability is
+expected: when the energy of a level set is greater in magnitude than the
+ground state, the level set will usually be empty. The correspondence between
+the threshold and shattering energies is also intuitive, since the threshold
+energy is typically understood as the point where the landscape fractures into
+pieces. However, this second correspondence is only true for the pure spherical
+models with homogeneous $f(q)$. For any other model with an inhomogeneous
+$f(q)$, $E_\text{sh}^2<E_\text{th}^2$. This may have implications for dynamics
+in such mixed models, and we discuss it at length in Section~\ref{sec:ssg}.
 
 \begin{figure}
   \includegraphics{figs/phases_12_1.pdf}
@@ -573,118 +616,28 @@ case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$.
     \textbf{Linear--quadratic crossover.}
     Topological phases for models with a covariance function
     $f(q)=(1-\lambda)q+\lambda\frac12q^2$ for several values of $\lambda$,
-    interpolating between homogeneous linear ($\lambda=0$) and quadratic ($\lambda=1$) constraints.
+    interpolating between homogeneous linear ($\lambda=0$) and quadratic
+    ($\lambda=1$) constraints. The regimes are defined in the text and depicted
+    as cartoons in Fig.~\ref{fig:cartoons}. The shaded region on each plot
+    shows where these results are unstable to \textsc{rsb}.
   } \label{fig:crossover}
 \end{figure}
 
-The phase diagram implied by these transitions is shown in
-Fig.~\ref{fig:phases} for three different homogeneous $f(q)$. One interesting
-feature occurs in the limit of $\alpha$ to zero. If $V_0$ is likewise rescaled
-in the correct way, the limit of these phase boundaries approaches known
-landmark energy values in the pure spherical spin glasses. In particular, the
-limit to zero $\alpha$ of the scaled satisfiability transition
-$V_\text{\textsc{sat}}\sqrt\alpha$ approaches the ground state energy
-$E_\text{gs}$, while the limit to zero $\alpha$ of the scaled shattering
-transition $V_\text{sh}\sqrt\alpha$ approaches the threshold energy
-$E_\text{th}$. The correspondence between ground state and satisfiability is
-expected: when the energy of a level set is greater in magnitude than the
-ground state, the level set will usually be empty. The correspondence between
-the threshold and shattering energies is also sensible, since the threshold
-energy is typically understood as the point where the landscape fractures into
-pieces. However, this second correspondence is only true for the pure spherical
-models with homogeneous $f(q)$. For any other model with an inhomogeneous
-$f(q)$, $E_\text{sh}^2<E_\text{th}^2$. This may have implications for dynamics
-in such mixed models, see the discussion in Section~\ref{sec:ssg}.
-
-Interpreting the specific topology implied by the Euler characteristic is difficult when the prediction is positive, but
-a larger problem lies with the many negative values it can take. A manifold with many
-handles will have a very negative Euler characteristic, and our calculation
-relying on the saddle point of an exponential integral would be invalid. In the
-regime of $\mathcal S_\chi(m)$ where the action is complex-valued, is this
-breakdown the result of a negative average characteristic, or is it the result
-of another reason? It is difficult to answer this with the previous calculation
-alone. However, in the next section we will make a complementary calculation
-that rules out the negative Euler characteristic picture. Instead, we will see
-that in the complex region, the large-deviation function in $m$ that $\mathcal
-S_\chi(m)$ represents breaks down due to the vanishingly small probability of
-finding any stationary points.
-
-
-\section{Implications for the topology of solutions}
-
-It is not straightforward to directly use the average Euler characteristic to
-infer something about the number of connected components in the set of
-solutions. To understand why, a simple example is helpful. Consider the set of
-solutions on the sphere $\|\mathbf x\|^2=N$ that satisfy the single quadratic
-constraint
-\begin{equation}
-  0=\sum_{i=1}^N\sigma_ix_i^2
-\end{equation}
-where each $\sigma_i$ is taken to be $\pm1$ with equal probability. If we take $\mathbf x$ to be ordered such that all terms with $\sigma_i=+1$ come first, this gives
-\begin{equation}
-  0=\sum_{i=1}^{N_+}x_i^2-\sum_{i=N_++1}^Nx_i^2
-\end{equation}
-where $N_+$ is the number of terms with $\sigma_i=+1$. The topology of the resulting manifold can be found by adding and subtracting this constraint from the spherical one, which gives
-\begin{align}
-  \frac12=\sum_{i=1}^{N_+}x_i^2
-  \qquad
-  \frac12=\sum_{i=N_++1}^{N}x_i^2
-\end{align}
-These are two independent equations for spheres of radius $1/\sqrt2$, one of
-dimension $N_+$ and the other of dimension $N-N_+$. Therefore, the topology of
-the configuration space is that of $S^{N_+-1}\times S^{N-N_+-1}$. The Euler
-characteristic of a product space is the product of the Euler characteristics,
-and so we have $\chi(\Omega)=\chi(S^{N_+-1})\chi(S^{N-N_+-1})$.
-
-What is the average value of the Euler characteristic over values of
-$\sigma_i$? First, recall that the Euler characteristic of a sphere $S^d$ is 2
-when $d$ is even and 0 when $d$ is odd. When $N$ is odd, any value
-of $N_+$ will result in one of the two spheres in the product to be
-odd-dimensional, and therefore $\chi(\Omega)=0$, as is always true for
-odd-dimensional manifolds. When $N$ is even, there are two possibilities: when $N_+$ is even then both spheres are odd-dimensional, while when $N_+$ is odd then both spheres are even-dimensional.
-The number of terms $N_+$ with $\sigma_i=+1$ is distributed with the binomial distribution
-\begin{equation}
-  P(N_+)=\frac1{2^N}\binom{N}{N_+}
-\end{equation}
-Therefore, the average Euler characteristic for even $N$ is
-\begin{equation}
-  \overline{\chi(\Omega)}
-  =\sum_{N_+=0}^NP(N_+)\chi(S^{N_+-1})\chi(S^{N-N_+-1})
-  =\frac4{2^N}\sum_{n=0}^{N/2}\binom{N}{2n}
-  =2
-\end{equation}
-Thus we find the average Euler characteristic in this simple example is 2
-despite the fact that the possible manifolds resulting from the constraints
-have characteristics of either 0 or 4.
-
-
-\paragraph{}
-Fig.~\ref{fig:cartoons} shows a low-dimensional facsimile of what these
-difference regimes look like. Surprisingly, this approach sees no signal of a
-replica symmetry breaking (\textsc{rsb}) transition previously found in
-\cite{Urbani_2023_A}. The
-instability is predicted to occur when
-\begin{equation} \label{eq:vrsb}
-  V_0^2>V_\text{\textsc{rsb}}^2
-  \equiv\frac{[f(1)-f(0)]^2}{\alpha f''(0)}
-  -f(0)-\frac{f'(0)}{f''(0)}
-\end{equation}
-Some preliminary attempts to find an instability towards finite-\textsc{rsb} in
-this region in the calculation of the Euler characteristic did not turn up anything. We conjecture that the \textsc{rsb}
-instability found in \cite{Urbani_2023_A} is a trait of the cost function
-\eqref{eq:cost}, and is not inherent to the structure of the solution manifold.
-Perhaps the best evidence for this is to consider the limit of $M=1$, or
-$\alpha\to0$ with $E=V_0\sqrt\alpha$ held fixed, where this problem reduces to
-the level sets of the spherical spin glasses. The instability \eqref{eq:vrsb}
-implies for the pure spherical 2-spin model with $f(q)=\frac12q^2$ that
-$E_\textsc{rsb}=\frac12$, though nothing of note is known to occur in the level
-sets of 2-spin model at such an energy.
+Rich coexistence between all four regimes occurs in models with mixed linear
+and nonlinear constraints. Fig.~\ref{fig:crossover} shows examples of the phase
+diagrams for models with a covariance function that interpolates between pure
+linear ($\lambda=0$) and pure quadratic ($\lambda=1$). A new phase boundary
+appears separating regimes I and II, defined as the point where the real part
+of the action at $m=0$ changes from negative to positive. In purely quadratic
+case, and in mixed linear and nonlinear cases, there is a substantial region of
+the phase diagram shown in Appendix~\ref{sec:rms} to be susceptible to
+\textsc{rsb}, especially for small $V_0$ and large $\alpha$. Future research
+into the structure of solutions in this regime is merited.
 
 \section{Implications for the dynamics of spherical spin glasses}
 \label{sec:ssg}
 
-
-As indicated earlier, for $M=1$ the solution manifold corresponds to the energy
+When $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=\sqrt NV_0$. All the
 results from the previous sections follow, and can be translated to the spin
 glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{-1/2}$ fixed. With a little algebra this procedure yields
@@ -694,11 +647,9 @@ glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{-1/2}$ fixe
   E_\text{sh}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
   \label{eq:ssg.energies}
 \end{align}
-for the energies at which level sets of the spherical spin glasses have
-disconnected pieces appear, and that at which a large connected component
-vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
+for the onset and shattering energies. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
 $E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
-models \cite{Castellani_2005_Spin-glass}. This is expected, since threshold energy, defined as the place where
+models \cite{Castellani_2005_Spin-glass}. This is intuitive, since threshold energy, defined as the place where
 marginal minima are dominant in the landscape, is widely understood as the
 place where level sets are broken into pieces.
 
@@ -708,7 +659,7 @@ However, for general mixed models the threshold energy is
 \end{equation}
 which satisfies $|E_\text{sh}|\leq|E_\text{th}|$. Therefore, as one
 descends in energy in generic models one will meet the shattering energy before
-the threshold energy. This is perhaps unexpected, since one wight imagine that
+the threshold energy. This is perhaps unexpected, since one might imagine that
 where level sets of the energy break into many pieces would coincide with the
 largest concentration of shallow minima in the landscape. We see here that this isn't the case.
 
@@ -774,8 +725,7 @@ descent from a random initial condition? The evidence in Fig.~\ref{fig:ssg} is
 compelling but inconclusive. The difference between the shattering energy and
 the extrapolated \textsc{dmft} data is about the same as the difference between
 the values predicted by the two extrapolation methods. If both extrapolation
-methods suffer from similar systematic biases, it is plausible they true value
-corresponds with the shattering energy. However, better estimates of the
+methods suffer from similar systematic biases, it is plausible the true value is the shattering energy. However, better estimates of the
 asymptotic values are needed to support or refute this conjecture. This
 motivates working to integrate the \textsc{dmft} equations to longer times, or
 else look for analytic asymptotic solutions that approach $E_\text{sh}$.
@@ -786,9 +736,7 @@ We have shown how to calculate the average Euler characteristic of the solution
 manifold in a simple model of random constraint satisfaction. The results
 constrain the topology and geometry of this manifold, revealing when it is
 connected and trivial, when it is extensive but topologically nontrivial, and
-when it is shattered into disconnected pieces. These inferences are supported
-by a auxiliary calculation of the complexity of stationary points for a test
-function on the same solution manifold.
+when it is shattered into disconnected pieces.
 
 This calculation has novel implications for the geometry of the energy
 landscape in the spherical spin glasses, where it reveals a previously unknown
@@ -838,7 +786,7 @@ with associated measures
   d\pmb\sigma=d\pmb\omega\,\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}\,d\bar{\pmb\gamma}\,d\pmb\gamma
 \end{align}
 The Euler characteristic can be expressed using these supervectors as
-\begin{align}
+\begin{align} \label{eq:kac-rice.super}
   &\chi(\Omega)
   =\int d\pmb\phi\,d\pmb\sigma\,e^{\int d1\,L(\pmb\phi(1),\pmb\sigma(1))} \\
   &=\int d\pmb\phi\,d\pmb\sigma\,\exp\left\{
@@ -1201,11 +1149,11 @@ orders in $N$, since for odd-dimensional manifolds the Euler characteristic is
 always zero. When $N+M+1$ is even, we have $\overline{\chi(\Omega)}=2$ to
 leading order in $N$, as specified in the main text.
 
-\section{The root mean square Euler characteristic}
+\section{The average squared Euler characteristic}
 \label{sec:rms}
 
 Here we calculate $\overline{\chi(\Omega)^2}$, the average of the squared Euler
-characteristic. This is accomplished by taking two copies of the integral, with
+characteristic. This is accomplished by taking two copies of the integral \eqref{eq:kac-rice.super}, with
 \begin{equation}
   \chi(\Omega)^2
   =\int d\pmb\phi_1\,d\pmb\sigma_1\,d\pmb\phi_2\,d\pmb\sigma_2\,
@@ -1232,8 +1180,12 @@ where we have defined
 Expanding the superindices and applying the Dirac $\delta$-functions implied by
 the Lagrange multipliers associated with the spherical constraint, we arrive at an expression
 \begin{equation}
-  \frac1N\log\overline{\chi(\Omega)^2}
-  =-\hat m_1-\hat m_2
+  \overline{\chi(\Omega)^2}
+  \simeq\int dC_{12}\,dR_{11}\,dR_{12}\,dR_{21}\,dR_{22}\,dD_{11}\,dD_{12}\,dD_{22}\,dG_{12}\,dG_{21}\,dm_1\,dm_2\,d\hat m_1\,d\hat m_2\,e^{N\mathcal S_{\chi^2}}
+\end{equation}
+where we have defined another effective action by
+\begin{equation}
+  \mathcal S_{\chi^2}=-\hat m_1-\hat m_2
   -\frac\alpha2\log\frac{\det A_1}{\det A_2}
   -\frac{\alpha V_0^2}2\begin{bmatrix}
     0 & 1 & 0 & 1
@@ -1245,40 +1197,139 @@ the Lagrange multipliers associated with the spherical constraint, we arrive at
   +\frac12\log\frac{\det A_3}{\det A_4}
 \end{equation}
 with the matrices $A_1$, $A_2$, $A_3$, and $A_4$ defined by
-\begin{align}
-  A_1&=\begin{bmatrix}
+\begin{equation}
+  A_1=\begin{bmatrix}
     D_{11}f'(1) & iR_{11}f'(1) & D_{12}f'(C_{12})+\Delta_{12}f''(C_{12}) & i R_{21}f'(C_{12}) \\
     i R_{11}f'(1) & f(1) & i R_{12}f'(C_{12}) & f(C_{12}) \\
     D_{12}f'(C_{12}) + \Delta_{12}f''(C_{12}) & iR_{12}f'(C_{12}) & D_{22} & iR_{22}f'(1) \\
     iR_{21}f'(C_{12}) & f(C_{12}) & iR_{22}f'(1) & f(1)
   \end{bmatrix}
-  \\
-  A_2&=\begin{bmatrix}
+\end{equation}
+\begin{equation}
+  A_2=\begin{bmatrix}
     0 & R_{11}f'(1) & 0 & -G_{21}f'(C_{12}) \\
     -R_{11}f'(1) & 0 & G_{12}f'(C_{12}) & 0 \\
     0 & -G_{12}f'(C_{12}) & 0 & R_{22}f'(1) \\
     G_{21}f'(C_{12}) & 0 & -R_{22}f'(1) & 0
   \end{bmatrix}
-  \\
-  A_3&=\begin{bmatrix}
+\end{equation}
+\begin{equation}
+  A_3=\begin{bmatrix}
     1-m_1^2 & i(R_{11}-m_1\hat m_1) & C_{12}-m_1m_2 & i(R_{21}-m_1\hat m_2) \\
     i(R_{11}-m_1\hat m_1) & D_{11}+\hat m_1^2 & i(R_{12}-m_2\hat m_1) & D_{12}+\hat m_1\hat m_2 \\
     C_{12}-m_1m_2 & i(R_{12}-m_2\hat m_1) & 1-m_2^2 & i(R_{22}-m_2\hat m_2) \\
     i(R_{21}-m_1\hat m_2) & D_{12}+\hat m_1\hat m_2 & i(R_{22}-m_2\hat m_2) & D_{22}+\hat m_2^2
   \end{bmatrix}
-  \\
-  A_4&=\begin{bmatrix}
+\end{equation}
+\begin{equation}
+  A_4=\begin{bmatrix}
     0 & R_{11} & 0 & -G_{21} \\
     -R_{11} & 0 & G_{12} & 0 \\
     0 & -G_{12} & 0 & R_{22} \\
     G_{21} & 0 & -R_{22} & 0
   \end{bmatrix}
-\end{align}
+\end{equation}
 and where $\Delta_{12}=G_{12}G_{21}-R_{12}R_{21}$. The expression must be
 extremized over all the order parameters. We look for solutions in two regimes
 that are commensurate with the solutions found for the Euler characteristic.
-These correspond to $m_1=m_2=0$ and $C_{12}=0$, and $m_1=m_2=m^*$ and
-$C_{12}=1$.
+These correspond to $m_1=m_2=0$ and $C_{12}=0$, and $m_1=m_2=\pm m_*$ and
+$C_{12}=1$. We find such solutions, and in all cases they have
+\begin{align}
+  G_{12}=G_{21}=R_{12}=R_{21}=D_{12}=\hat m_1=\hat m_2=0
+  \\
+  D_{ii}=-\frac{m+R_{ii}}{1-m^2}R_{ii}
+  \hspace{2em}
+  R_{22}=R_{11}^\dagger
+  \hspace{2em}
+  R_{11}=R_*
+\end{align}
+where $\dagger$ denotes the complex conjugate and $R_*$ is the saddle point
+solution of \eqref{eq:rs}. Upon substituting these solutions into the
+expressions above, we find in both cases that
+\begin{equation}
+  \mathcal S_{\chi^2}=2\operatorname{Re}\mathcal S_\chi
+\end{equation}
+as referenced in the main text. This corresponds with
+$\overline{\chi(\Omega)^2}\simeq[\overline{\chi(\Omega)}]^2$, justifying the
+`annealed' approach we have taken here.
+
+However, these solutions are not always the correct saddle point for evaluating
+the average squared Euler characteristic. When another solution is dominant,
+the dissonance between the average square and squared average indicates the
+necessity of a quenched calculation to determine the behavior of typical
+samples, and likely instability to \textsc{rsb}. We can find these points of
+instability by examining the Hessian of the action of the average square of the
+Euler characteristic at $m=0$. The stability of this matrix is not sufficient
+to determine if our solution is stable, since the many $\delta$-functions
+employed in our derivation ensure that the resulting saddle point never at a
+true maximum with respect to some combinations of variables. We rather look for
+places where the stability of this matrix changes, which represent places where
+another solution branches from the existing one. However, we must neglect the
+branching of trivial solutions, which occur when $R_*$ goes from real- to
+complex-valued.
+
+By examination of the results, it appears that nontrivial \textsc{rsb}
+instabilities occur along eigenvectors of the Hessian of $\mathcal S_{\chi^2}$
+in the subspace spanned by $C_{12}$, $R_{12}$, $R_{21}$, and $D_{12}$. This may
+not be surprising, since these are the parameters that represent nontrivial
+correlations between the two copies of the system. We can therefore find the \textsc{rsb} instability by looking for nontrivial zeros of
+\begin{equation}
+  \det\frac{\partial^2\mathcal S_{\chi^2}}{\partial[C_{12},R_{12},R_{21},D_{12}]^2}
+\end{equation}
+evaluated at the $m=0$ solution described above. The resulting expression is
+usually quite heinous, but there is a regime where a dramatic simplification is
+possible. The instability always occurs along the direction
+$R_{21}=R_{12}^\dagger$, but when $R_*$ is real, $R_{11}=R_{22}$ and the instability occurs along
+the direction $R_{21}=R_{12}$. This allows us to examine a simpler action, and
+we find the determinant is proportional to two nontrivial factors. If we let
+$r_*=\lim_{m\to0}R_*/m$, then these are
+\begin{align}
+  [(1+r_*)f(1)]^3-3r_*[(1+r_*)f(1)]^2f'(1)
+  +\alpha V_0^2\big[2(1+r_*)f'(0)^2+r_*f'(1)f''(0)\big]
+  \quad
+  \notag \\
+  +\alpha r_*f'(0)^2f'(1)-[r_*f'(1)]^3
+  -(1+r_*)f(1)\Big(\alpha\big[f'(0)^2+V_0^2f''(0)\big]-3[r_*f'(1)]^2\Big)
+\end{align}
+\begin{align}
+  &\big[(1+r_*)f(1)-r_*f'(1)\big]^3[f'(1)^2-\alpha f'(0)^2]
+  \Big(f'(1)\big[(1+r_*)f(1)-r_*f'(1)\big]-\alpha f'(0)^2\Big)
+  \notag \\
+  &\qquad-[\alpha V_0^2r_*f'(1)]^2f''(0)\big[(1+r_*)f'(0)^2+r_*f'(1)f''(0)\big]
+  \notag \\
+  &\qquad-\alpha V_0^2\big[(1+r_*)f(1)-r_*f'(1)\big]^2
+  \bigg[
+    (1+r_*)f'(0)^2\big[\alpha f'(0)^2-f'(1)^2\big]
+    \notag \\
+  &\qquad\qquad+
+    r_*f'(1)f''(0)\bigg(
+      \alpha f'(0)^2\frac{(1+r_*)f(1)-2r_*f'(1)}{(1+r_*)f(1)-r_*f'(1)}
+      -(1-r_*)f'(1)^2
+    \bigg)
+  \bigg]
+\end{align}
+As $\alpha$ is increased from zero, the first of these factors to go through
+zero represents the instability point. These formulas are responsible for the
+shaded regions in Fig.~\ref{fig:phases} and Fig.~\ref{fig:crossover}.
+
+Surprisingly, this approach sees no signal of a
+replica symmetry breaking (\textsc{rsb}) transition previously found in
+\cite{Urbani_2023_A}. The
+instability is predicted to occur when
+\begin{equation} \label{eq:vrsb}
+  V_0^2>V_\text{\textsc{rsb}}^2
+  \equiv\frac{[f(1)-f(0)]^2}{\alpha f''(0)}
+  -f(0)-\frac{f'(0)}{f''(0)}
+\end{equation}
+We conjecture that the \textsc{rsb}
+instability found in \cite{Urbani_2023_A} is a trait of the cost function
+\eqref{eq:cost}, and is not inherent to the structure of the solution manifold.
+Perhaps the best evidence for this is to consider the limit of $M=1$, or
+$\alpha\to0$ with $E=V_0\sqrt\alpha$ held fixed, where this problem reduces to
+the level sets of the spherical spin glasses. The instability \eqref{eq:vrsb}
+implies for the pure spherical 2-spin model with $f(q)=\frac12q^2$ that
+$E_\textsc{rsb}=\frac12$, though nothing of note is known to occur in the level
+sets of 2-spin model at such an energy.
 
 \section{The quenched shattering energy}
 \label{sec:1frsb}
@@ -1416,7 +1467,11 @@ shattering energy, since it is the point where the $m=0$ solution vanishes.
 Starting from this point, we take small steps in $s$ and $\lambda_s$, again
 simultaneously extremizing, ensuring continuity, and maximizing $E$. This draws
 out the shattering energy across the entire range of $s$ plotted in
-Fig.~\ref{fig:ssg}. The transition to the \textsc{rs} solution occurs when the value $q_0$ that maximizes $E$ hits zero.
+Fig.~\ref{fig:ssg}. The transition to the \textsc{rs} solution occurs when the
+value $q_0$ that maximizes $E$ hits zero. We find that the transition between
+\textsc{rs} and \textsc{frsb} is consistent with the \textsc{rsb} instability
+predicted in Appendix~\ref{sec:rms} by analyzing the solution to the average
+square of the Euler characteristic.
 
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