From 5c7c86b56e35fbf9d639e0daebb4b9f65a90013e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 18 Sep 2024 21:27:18 +0200
Subject: Full pass of edits.

---
 topology.tex | 183 +++++++++++++++++++++++++++++++----------------------------
 1 file changed, 97 insertions(+), 86 deletions(-)

diff --git a/topology.tex b/topology.tex
index 2ed25aa..bbd978a 100644
--- a/topology.tex
+++ b/topology.tex
@@ -66,8 +66,9 @@ this behavior into five phases with different implications for the topology of
 the solution manifold. When $M=1$ there is a correspondence between this
 problem and level sets of the energy in the spherical spin glasses. We
 conjecture that the transition energy dividing two of the topological phases
-corresponds to the asymptotic limit of gradient descent from a random initial
-condition, possibly resolving an open problem in out-of-equilibrium dynamics.
+corresponds to the energy asymptotically reached by gradient descent from a
+random initial condition, possibly resolving an open problem in
+out-of-equilibrium dynamics.
 }
 }
 
@@ -127,13 +128,13 @@ set of solutions, which can influence the behavior of algorithms trying
 to find them \cite{Baldassi_2016_Unreasonable, Baldassi_2019_Properties,
 Beneventano_2023_On}. Here, we show how topological information about
 the set of solutions can be calculated in a simple problem of satisfying random
-nonlinear equalities. This allows us to reason about the connectivity of the
+nonlinear equalities. This allows us to reason about the connectivity and structure of the
 solution set. The topological properties revealed by this calculation yield
 surprising results for the well-studied spherical spin glasses, where a
 topological transition thought to occur at a threshold energy $E_\text{th}$
 where marginal minima are dominant is shown to occur at a different energy
 $E_\text{sh}$. We conjecture that this difference resolves an outstanding
-problem in gradient descent dynamics in these systems.
+problem with the out-of-equilibrium dynamics in these systems.
 
 We consider the problem of finding configurations $\mathbf x\in\mathbb R^N$
 lying on the $(N-1)$-sphere $\|\mathbf x\|^2=N$ that simultaneously satisfy $M$
@@ -146,16 +147,16 @@ 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 $V_k$. One can explicitly construct functions that satisfy \eqref{eq:covariance} by taking
+all possible terms of degree $p$ in the $V_k$. 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
+where the elements of the tensors $J^{(k,p)}$ are independently distributed
 unit normal random variables. The
 series coefficients of $f$ therefore control the variances of the random coefficients
-in the random polynomials $V_k$. When $M=1$, this problem corresponds to
+in the polynomials $V_k$. When $M=1$, this problem corresponds to
 finding the level set of a spherical spin glass at energy density
 $E=V_0/\sqrt{N}$.
 
@@ -171,7 +172,7 @@ studied properties of the cost function
   \mathscr C(\mathbf x)=\frac12\sum_{k=1}^M\big[V_k(\mathbf x)-V_0\big]^2
 \end{equation}
 which achieves zero only for configurations that satisfy all the constraints.
-From the perspective of the cost function, the set of solutions looks like a network of flat canyons at the bottom of the landscape.
+From the perspective of the cost function, the set of solutions looks like a network of flat canyons at the bottom of the cost landscape.
 Here we dispense with the cost function and study the set of solutions
 directly. This set can be written as
 \begin{equation}
@@ -220,7 +221,7 @@ a manifold, the Euler characteristic of its product with the circle $S^1$ is
 zero.
 
 The canonical method for computing the Euler characteristic is to construct a
-complex on the manifold in question, essentially a higher-dimensional
+complex on the manifold in question, which is a higher-dimensional
 generalization of a polygonal tiling. Then $\chi$ is given by an alternating
 sum over the number of cells of increasing dimension, which for 2-manifolds
 corresponds to the number of vertices, minus the number of edges, plus the
@@ -298,7 +299,7 @@ form
   \overline{\chi(\Omega)}
   =\left(\frac N{2\pi}\right)^2\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\chi(R,D,m,\hat m)}
 \end{equation}
-where $g$ is a prefactor of $o(N^0)$, and $\mathcal S_\chi$ is an effective action defined by
+where $g$ is a prefactor of $o(N^0)$, $\mathcal S_\chi$ is an effective action defined by
 \begin{equation} \label{eq:euler.action}
   \begin{aligned}
     \mathcal S_\chi(R,D,m,\hat m)
@@ -322,20 +323,22 @@ The remaining order parameters are 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 in $N$ by a saddle point approximation. First
-we extremize with respect to $R$, $D$, and $\hat m$, which take the saddle-point
-values
-\begin{equation}
+between the configurations $\mathbf x$, the auxiliary configurations
+$\hat{\mathbf x}$, and the height axis $\mathbf x_0$.
+The integral \eqref{eq:pre-saddle.characteristic} can be evaluated to leading
+order in $N$ by a saddle point approximation. First we extremize with respect
+to $R$, $D$, and $\hat m$, which take the saddle-point values
+\begin{align}
+  R=R^*
+  &&
   D=-\frac{m+R_*}{1-m^2}R_*
-  \hspace{8em}
+  &&
   \hat m=0
-\end{equation}
+\end{align}
+where we have defined
 \begin{equation} \label{eq:rs}
   \begin{aligned}
-    R=R_*
+    R_*
     \equiv\frac{-m(1-m^2)}{2[f(1)-(1-m^2)f'(1)]^2}
     \Bigg[
       \alpha V_0^2f'(1)
@@ -347,7 +350,8 @@ values
   \Bigg]
   \end{aligned}
 \end{equation}
-This results in an effective action as a function of $m$ alone given by
+Upon substitution of these solutions into \eqref{eq:euler.action}, we find an
+effective action as a function of $m$ alone given by
 \begin{equation} \label{eq:S.m}
   \mathcal S_\chi(m)
   =-\frac\alpha2\bigg[
@@ -360,14 +364,12 @@ This results in an effective action as a function of $m$ alone given by
   \bigg]
   +\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
-selection of sample parameters. To finish evaluating the integral, this
+This function is plotted in Fig.~\ref{fig:action} for a
+selection of parameters.
+To finish evaluating the integral, this
 expression should be maximized with respect to $m$. If $m_*$ is such a maximum,
 then the resulting Euler characteristic is $\overline{\chi(\Omega)}\propto
-e^{N\mathcal S_\chi(m_*)}$. The order parameter $m$ is the overlap of the
-configuration $\mathbf x$ with the height axis $\mathbf x_0$. 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.
+e^{N\mathcal S_\chi(m_*)}$.
 
 \begin{figure}[tbh]
   \includegraphics{figs/action_1.pdf}
@@ -393,6 +395,12 @@ sphere where most of the contribution to the Euler characteristic is made.
   } \label{fig:action}
 \end{figure}
 
+\subsection{Features of the effective action}
+
+The order parameter $m$ is the overlap of the
+configuration $\mathbf x$ with the height axis $\mathbf x_0$. Therefore, the
+value $m$ that maximizes this action can be understood as the latitude on the
+sphere at which most of the contribution to the Euler characteristic is made.
 The action $\mathcal S_\chi$ is extremized with respect to $m$ at $m=0$ or at $m=\pm m_*$ for
 \begin{equation}
   m_*=\sqrt{1-\frac{\alpha}{f'(1)}\big(V_0^2+f(1)\big)}
@@ -457,7 +465,7 @@ function $f$ it is not possible to write an explicit formula for
 $V_\text{\textsc{sat}}$, and we calculate it through a numeric
 root-finding algorithm.
 
-When $m_\text{min}^2>0$, the solution at $m=0$ is difficult to interpret, since
+When $V_0^2<V_\text{sh}^2$, the solution at $m=0$ is difficult to interpret, since
 the action takes a complex value. 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.
@@ -471,7 +479,7 @@ complex values even when $\overline{\chi(\Omega)}$ is negative. Under the restri
   present the replica symmetric (\textsc{rs}) description of this problem can
   have $q_0>0$, and $\overline{\chi(\Omega)^2}\neq[\overline{\chi(\Omega)}]^2$
   always.
-} we find three
+} we identify three
 saddle points that could contribute to the value of
 $\overline{\chi(\Omega)^2}$: two at $\pm m_*$ where
 $\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$,
@@ -537,7 +545,7 @@ which are schematically represented in Fig.~\ref{fig:cartoons}.\footnote{
     The solution manifold is shown as a shaded region, and the height axis
     $\mathbf x_0$ is a black arrow. In Regime I, the average Euler
     characteristic is consistent with a manifold with a single simply-connected
-    component. In Regime II, holes occupy the equator but the most polar
+    component. In Regime II, holes occupy the equator but the temperate
     regions are topologically simple. In Regime III, holes dominate and the
     edge of the manifold is not necessarily simple. In Regime IV, disconnected
     components dominate. In Regime V, the manifold is empty.
@@ -550,7 +558,7 @@ 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.
+strongly indicating a topology homeomorphic to the $S^{N-M-1}$ sphere. This regime is the only nontrivial one found with linear covariance $f(q)=q$, 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{$m=\pm m_*$}.}
 
@@ -596,14 +604,14 @@ characteristic zero.
 
   \caption{
     \textbf{Topological phase diagram.}
-    Topological phases of the model for three different homogeneous covariance
+    Topological phases of the problem for three different homogeneous covariance
     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 the ground state energy
     $E_\text{gs}$ and threshold energy $E_\text{th}$ of the 3-spin spherical model correspond with the limits
-    of the satisfiability and shattering transitions in the pure cubic model. Note that
+    of the satisfiability and shattering transitions in the pure cubic problem. Note that
     for mixed models with inhomogeneous covariance functions, $E_\text{th}$ is
     not the lower limit of $V_\text{sh}$.
   } \label{fig:phases}
@@ -624,13 +632,13 @@ 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
+An 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
+limit $\alpha\to0$ 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
+$E_\text{gs}$, while the limit $\alpha\to0$ 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
@@ -640,7 +648,7 @@ 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}.
+in these mixed models, and we discuss them at length in Section~\ref{sec:ssg}.
 
 \begin{figure}
   \includegraphics{figs/phases_12_1.pdf}
@@ -688,18 +696,22 @@ glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{-1/2}$ fixe
 \end{align}
 for the onset and shattering energies. The same limit taken for
 $V_\text{\textsc{sat}}\alpha^{-1/2}$ coincides with the ground state energy
-$E_\text{gs}$.
+$E_\text{gs}$. In fact, for all energies below the threshold energy
+$E_\text{th}$ (where minima become more numerous than saddle points in the
+spin glass energy function) the logarithm of the average Euler characteristic
+is precisely the complexity of stationary points of the spin glass energy. In
+this regime, the Euler characteristic is dominated by contributions coming from
+the sphere-like slices of the energy basins directly above minima.
 
 For the pure $p$-spin spherical spin glasses, which have homogeneous covariance functions $f(q)=\frac12q^p$,
-the shattering energy is $E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the same as the threshold energy $E_\text{th}$ \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
+the shattering energy is $E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the same as the threshold energy $E_\text{th}$ \cite{Castellani_2005_Spin-glass}. This is intuitive, since threshold energy is widely understood as the
 place where level sets are broken into pieces.
 However, for general mixed models with inhomogeneous covariance functions the threshold energy is
 \begin{equation}
   E_\mathrm{th}=\pm\frac{f(1)[f''(1)-f'(1)]+f'(1)^2}{f'(1)\sqrt{f''(1)}}
 \end{equation}
 which satisfies $|E_\text{sh}|\leq|E_\text{th}|$. Therefore, as one
-descends in energy in one will generically meet the shattering energy before
+descends in energy one will generically meet the shattering energy before
 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.
@@ -708,11 +720,10 @@ This fact mirrors another another that was made clear recently: when gradient
 decent dynamics are run on these models, they will asymptotically reach an
 energy above the threshold energy \cite{Folena_2020_Rethinking,
 Folena_2021_Gradient, Folena_2023_On}. The old belief that the threshold energy
-qualitatively coincides with a kind of shattering is one source of the
+qualitatively coincides with a kind of shattering of the landscape is one source of the
 expectation that the it should coincide with the dynamic limit. Motivated by
 our discovery that the actual shattering energy is different from the threshold
-energy, we make a comparison with existing data on the asymptotic limits of
-dynamics.
+energy, we make a comparison of it with existing data on asymptotic dynamics
 
 \begin{figure}
   \includegraphics{figs/dynamics_2.pdf}
@@ -727,22 +738,21 @@ dynamics.
     shown is the annealed threshold energy $E_\text{th}$, where marginal minima
     are the most common type of stationary point. The section of $E_\text{sh}$
     that is dashed on the left plot indicates the continuation of the annealed
-    result, whereas the solid portion gives the value calculated with a
-    {\oldstylenums 1}\textsc{frsb} ansatz.
+    result, whereas the solid portion gives the quenched prediction.
   } \label{fig:ssg}
 \end{figure}
 
-Measurements of the asymptotic limits of dynamics were recently taken in
+Measurements of the asymptotic energies reached by dynamics were recently taken in
 \cite{Folena_2023_On} for two different classes of models with inhomogeneous
 $f(q)$, with
 \begin{equation}
   f(q)=\frac12\big[\lambda q^p+(1-\lambda)q^s\big]
 \end{equation}
 The authors of \cite{Folena_2023_On} studied models with this covariance for $p=2$ and $p=3$ while varying $s$.
-In both cases, the relative weight $\lambda$ varies with $s$ and was chosen to
+In both cases, the relative weight $\lambda$ between the two terms varies with $s$ and was chosen to
 maximize a heuristic to increase the chances of seeing nontrivial behavior. The
 authors numerically integrated the dynamic mean field theory (\textsc{dmft})
-equations for gradient descent on these models from a random initial condition to large but finite time, then attempted to
+equations for gradient descent in these models from a random initial condition to large but finite time, then attempted to
 extrapolate the infinite-time behavior by two different methods. The black
 symbols in Fig.~\ref{fig:ssg} show the measurements taken from
 \cite{Folena_2023_On}. The difference between the two extrapolations is not
@@ -775,7 +785,7 @@ else look for analytic asymptotic solutions that approach $E_\text{sh}$.
 \label{sec:conclusion}
 
 We have shown how to calculate the average Euler characteristic of the solution
-manifold in a simple model of random constraint satisfaction. The results
+manifold in a simple model of random continuous constraint satisfaction. The results
 constrain the topology 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.
@@ -785,11 +795,11 @@ landscape in the spherical spin glasses, where it reveals a previously unknown
 landmark energy $E_\text{sh}$. This shattering energy is where the topological
 calculation implies that the level set of the energy breaks into disconnected
 pieces, and differs from the threshold energy $E_\text{th}$ in mixed models.
-It's possible that $E_\text{sh}$ is the asymptotic limit of gradient descent
+It's possible that $E_\text{sh}$ is the asymptotic energy reached by gradient descent
 from a random initial condition in such models, but the quality of the
 currently available data makes this conjecture inconclusive.
 
-Our work also highlights a severe limitation of using the statistics of
+Our work also highlights a limitation of using the statistics of
 stationary points of an energy or cost function to infer topological properties
 of the level sets. In the mixed spherical spin glasses, neither one nor two
 stationary point statistics reveal the presence of the topologically significant
@@ -852,7 +862,7 @@ The Euler characteristic can be expressed using these supervectors as
     \right]
   \right\} \notag
 \end{align}
-where $d1=d\bar\theta_1\,d\theta_1$ is the integral measure over both Grassmann
+where $d1=d\bar\theta_1\,d\theta_1$ is the integration measure over both Grassmann
 indices. Since this is an exponential integrand linear in the Gaussian
 functions $V_k$, we can take their average to find
 \begin{equation} \label{eq:χ.post-average}
@@ -975,7 +985,7 @@ We can treat the integral over $\sigma_0$ immediately. It gives
   =2\times2\pi\delta(C-1)\,\delta(G+R)\,\bar HH
 \end{equation}
 This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well
-as removing all dependence on $\bar H$ and $H$. With these solutions inserted, the remaining terms in the exponential give
+as removing all dependence on $\bar H$ and $H$. With these solutions inserted, the remaining terms in the exponential expand to give
 \begin{align} \label{eq:sdet.q}
   &\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)
   =1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
@@ -992,7 +1002,7 @@ as removing all dependence on $\bar H$ and $H$. With these solutions inserted, t
   +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H
   \\ \label{eq:inv.q}
   &\int d1\,d2\,f(\mathbb Q)^{-1}(1,2)
-  =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1}
+  =\frac1{f(1)}\left(1+\frac{R^2f'(1)}{Df(1)}\right)^{-1}
   +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H
 \end{align}
 The Grassmann terms in these expressions do not contribute to the effective
@@ -1010,7 +1020,7 @@ introduction of $\delta$-functions to define the order parameters, integrals
 over Grassmann order parameters, and from the saddle point approximation to the
 large-$N$ integral. In addition, there are important contributions of a sign of the magnetization at the solution that arise from our super-Gaussian integrations.
 
-\subsection{Contribution from the Hubbard--Stratonovich transformations}
+\subsection{Contribution from the Hubbard--Stratonovich transformation}
 \label{sec:prefactor.hs}
 
 First, we examine the factors arising from the definition of order parameters. This begins by introducing to the integral \eqref{eq:pre.hubbard-strat} the factor of one
@@ -1044,7 +1054,7 @@ analogously to those of $\mathbb Q$ and $\mathbb M$. This is now a super-Gaussia
   \Bigg\}
   \end{aligned}
 \end{equation}
-We can perform the remaining Gaussian integral in $\tilde{\mathbb M}$ to find
+We can perform the remaining super-Gaussian integral in $\tilde{\mathbb M}$ to find
 \begin{equation}
   \begin{aligned}
     \int d\pmb\phi\,1=
@@ -1053,7 +1063,7 @@ We can perform the remaining Gaussian integral in $\tilde{\mathbb M}$ to find
     \exp\Bigg\{
       -\frac N2\log\operatorname{sdet}\tilde{\mathbb Q}
       \hspace{5em} \\
-      \frac N2\int d1\,d2\,\tilde{\mathbb Q}(1,2)\big[
+      +\frac N2\int d1\,d2\,\tilde{\mathbb Q}(1,2)\big[
         \mathbb Q(1,2)-\mathbb M(1)\mathbb M(2)
       \big]
     \Bigg\}
@@ -1085,16 +1095,16 @@ are important to getting correctly the sign of the prefactor. For instance, cons
 \begin{equation}
   \operatorname{sdet}\mathbb Q=\frac{CD+R^2}{G^2}
 \end{equation}
-The numerator and denominator arise from the determinant in the sector of ordinary number and Grassmann number basis elements for the superoperator, respectively. In our calculation, such superdeterminants appear after Gaussian integrals, which produce
+The numerator and denominator arise from the determinant in the sector of ordinary number and Grassmann number basis elements for the superoperator, respectively. In our calculation, such superdeterminants appear after Gaussian integrals, like
 \begin{equation}
   \int d\pmb\phi\,\exp\left\{-\frac12\int d1\,d2\,\pmb\phi(1)\mathbb Q(1,2)\pmb\phi(2)\right\}
   =(\operatorname{sdet}\mathbb Q)^{-\frac12}
   =(CD+R^2)^{-\frac12}G
 \end{equation}
-Here we emphasize that in the expanded result of the integral, the term from
-the denominator of the square root enters not as $(G^2)^{\frac12}=|G|$ but as
+Here we emphasize that in the expanded result of the integral, the factor from
+the denominator of the superdeterminant enters not as $(G^2)^{\frac12}=|G|$ but as
 $G$, including its sign. Therefore, when we write in the effective action
-$\frac12\log\operatorname{sdet}\mathbb Q$, we should really be writing
+$-\frac12\log\operatorname{sdet}\mathbb Q$, we should really be writing
 \begin{equation}
   \int d\pmb\phi\,\exp\left\{-\frac12\int d1\,d2\,\pmb\phi(1)\mathbb Q(1,2)\pmb\phi(2)\right\}
   =\operatorname{sign}(G)e^{-\frac12\log\operatorname{sdet}\mathbb Q}
@@ -1113,7 +1123,7 @@ After integrating out the Lagrange multiplier enforcing the spherical
 constraint in \eqref{eq:sigma0.integral}, the Grassmann variables $\bar H$ and
 $H$ are eliminated from the integrand. This leaves dependence on $\bar{\hat
 H}$, $\hat H$, $\bar H_0$, and $H_0$. Expanding the contributions from
-\eqref{eq:sdet.q}, \eqref{eq:sdet.fq}, and \eqref{eq:inv.q}, the contribution to the action is given by
+\eqref{eq:sdet.q}, \eqref{eq:sdet.fq}, and \eqref{eq:inv.q}, the total contribution to the action is given by
 \begin{equation}
   \int d\bar{\hat H}\,d\hat H\,d\bar H_0\,dH_0\,\exp\left\{
     N\begin{bmatrix}
@@ -1144,7 +1154,7 @@ where
   \\
   &h_4=-\frac1{R^2}\frac1{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}
 \end{align}
-The contribution at leading order in $N$ is therefore
+The contribution to the prefactor at leading order in $N$ is therefore
 \begin{equation}
   \frac{N^2}{R^2[D(1-m^2)+R^2-2Rm\hat m+\hat m^2]}\left(
     \alpha\frac{(D+R^2)[Df(1)^2+R^2f'(1)[V_0^2+f(1)]]}{
@@ -1191,7 +1201,7 @@ vector of derivatives with respect to the remaining order parameters. For both o
 whereas
 \begin{equation}
   g(\mp m_*,0,\pm m_*,0)
-  =\frac{(\mp1)^{N+M+1}}{|m_*|^4}\left(1-\frac{\alpha[V_0^2+f(1)]}{f'(1)}\right)
+  =\frac{(\mp1)^{N+M+1}}{(m_*)^4}\left(1-\frac{\alpha[V_0^2+f(1)]}{f'(1)}\right)
 \end{equation}
 The saddle point at $m=-m_*$, characterized by minima of the height function, always contributes with a positive term. On the other hand, the saddle point with $m=+m_*$, characterized by maxima of the height function, contributes with a sign depending on if $N+M+1$ is even or odd. This follows from the fact that minima, with an index of 0, have a positive contribution to the sum over stationary points, while maxima, with an index of $N-M-1$, have a contribution that depends on the dimension of the manifold.
 
@@ -1242,7 +1252,7 @@ expression
   \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
+with another effective action defined by
 \begin{equation}
   \mathcal S_{\chi^2}=-\hat m_1-\hat m_2
   -\frac\alpha2\log\frac{\det A_1}{\det A_2}
@@ -1255,39 +1265,40 @@ where we have defined another effective action by
   \end{bmatrix}
   +\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{equation}
+with the matrices $A_1$, $A_2$, $A_3$, and $A_4$ given by
+\begin{align}
+  &
   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}
-\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}
-\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}
-\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{equation}
+\end{align}
 and where $\Delta_{12}=G_{12}G_{21}-R_{12}R_{21}$. The effective action 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.
@@ -1310,7 +1321,7 @@ expressions above, we find in both cases that
 \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.
+`annealed' approach we have taken in the rest of the paper.
 
 \subsection{Instability to replica symmetry breaking}
 \label{sec:rsb.instability}
@@ -1319,14 +1330,14 @@ 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 also a likely instability to \textsc{rsb}. We can find these points of
+samples, and also indicates a 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 is never at a
 true maximum with respect to some combinations of variables. We rather look for
-places where the stability of this matrix changes, where
-another solution branches from the existing one. However, we must neglect the
+places where the stability of this matrix changes, indicating 
+another solution branching from the existing one. However, we must neglect the
 branching of trivial solutions, which occur when $R_*$ goes from real- to
 complex-valued.
 
@@ -1379,7 +1390,7 @@ zero represents the instability point. These formulas are responsible for
 defining the boundaries of the shaded regions in Fig.~\ref{fig:phases} and
 Fig.~\ref{fig:crossover}.
 
-Surprisingly, this approach sees no signal of a
+Surprisingly, this approach sees no signal of the
 replica symmetry breaking (\textsc{rsb}) transition previously found in
 \cite{Urbani_2023_A}. The
 instability is predicted to occur when
@@ -1406,7 +1417,7 @@ Here we share how the quenched shattering energy is calculated under a
 the spherical spin glasses, we start with \eqref{eq:χ.post-average}. The
 formula in a quenched calculation is almost the same as that for the annealed,
 but the order parameters $C$, $R$, $D$, and $G$ must be understood as $n\times
-n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\omega_0$, $\omega_1$, and $\hat\omega_1$ should be considered $n$-dimensional vectors, but since in our ansatz replica vectors are constant we can take them to be constant from the start. Expanding the superspace notation, we have
+n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\omega_0$, $\omega_1$, and $\hat\omega_1$ should be considered $n$-dimensional vectors, but since in our ansatz replica vectors are constant we can take them to be constant from the start. Expanding the superspace notation, setting $V_0=E\sqrt{N/M}$, and taking $M=1$, we have
 \begin{align}
   &\overline{\log\chi(\Omega)}
   =\lim_{n\to0}\frac\partial{\partial n}\int dC\,dR\,dD\,dG\,dm\,d\hat m\,d\omega_0\,d\hat\omega_0\,d\omega_1\,d\hat\omega_1\,
@@ -1415,7 +1426,7 @@ n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\o
     +\frac i2\hat\omega_0\operatorname{Tr}(C-I)
     \notag \\
   &\hspace{2em}-\omega_0\operatorname{Tr}(G+R)
-    -in\hat\omega_1E_0
+    -in\hat\omega_1E
     +\frac12\log\det\begin{bmatrix}
       C-m^2 & i(R-m\hat m) \\ i(R-m\hat m) & D-\hat m^2
     \end{bmatrix}
@@ -1428,7 +1439,7 @@ n$ matrices rather than scalars. In principle $m$, $\hat m$, $\omega_0$, $\hat\o
     \right]
   \Bigg\}
 \end{align}
-which is completely general for the spherical spin glasses with $M=1$. We now
+We now
 make a series of simplifications. Ward identities associated with the BRST
 symmetry possessed by the original action \cite{Annibale_2003_The, Annibale_2003_Supersymmetric, Annibale_2004_Coexistence} indicate that
 \begin{align}
@@ -1478,7 +1489,7 @@ If we redefine $\hat\beta=-i\hat\omega_1$ and $\tilde r_d=\omega_1 r_d$, we find
 \end{equation}
 which is exactly the effective action for the supersymmetric complexity in the
 spherical spin glasses when in the regime where minima dominate
-\cite{Kent-Dobias_2023_How}. As the effective action for the Euler characteristic, this expression is valid whether minima dominate or not. Following the same steps as in
+\cite{Kent-Dobias_2023_How}. As the effective action for the Euler characteristic, this expression is always valid. Following the same steps as in
 \cite{Kent-Dobias_2023_How}, we can write the continuum version of this action
 for arbitrary \textsc{rsb} structure in the matrix $C$ as
 \begin{equation} \label{eq:cont.action}
@@ -1508,7 +1519,7 @@ where
 \begin{equation}
   \chi_0(q)=\frac1{\hat\beta}[f''(q)^{-1/2}-\tilde r_d]
 \end{equation}
-is the function implied by extremizing \eqref{eq:cont.action} over $\chi$. The
+is the function implied by extremizing \eqref{eq:cont.action} over $\chi$ ignoring the continuity and other constraints. The
 variable $q_0$ must be chosen so that $\chi$ is continuous. The key difference
 between \textsc{frsb} and {\oldstylenums1}\textsc{frsb} in this setting is that
 in the former case the ground state has $q_0=1$, while in the latter the ground
@@ -1545,7 +1556,7 @@ shattering energy is found by slowly lowering $q_0$ and solving the combined
 extremal and continuity problem for $\hat\beta$, $\tilde r_d$, and $E$ until
 $E$ reaches a maximum value and starts to decrease. This maximum is the
 shattering energy, since it is the point where the $m=0$ solution becomes complex.
-Starting from this point, we take small steps in $s$ and $\lambda_s$, again
+Starting from this point, we take small steps in $s$ and $\lambda_s$,
 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
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