From 65e831dd2e347f0c960974e8bb59d83543060c16 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 3 Dec 2023 14:23:11 +0100
Subject: Lots of small writing changes.

---
 2-point.tex | 256 ++++++++++++++++++++++++++++++++++++++----------------------
 1 file changed, 164 insertions(+), 92 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 8a5a7f3..267229e 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -84,7 +84,7 @@
 \begin{abstract}
   The mixed spherical models were recently found to violate long-held
   assumptions about mean-field glassy dynamics. In particular, the threshold
-  energy, where most stationary points are marginal and which in the simpler
+  energy, where most stationary points are marginal and that in the simpler
   pure models attracts long-time dynamics, seems to lose significance. Here,
   we compute the typical distribution of stationary points relative to each
   other in mixed models with a replica symmetric complexity. We examine the
@@ -102,7 +102,8 @@
 
 Many systems exhibit ``glassiness,'' characterized by rapid slowing of dynamics
 over a short parameter interval. These include actual (structural) glasses,
-spin glasses, certain inference and optimization problems, and more. Glassiness
+spin glasses, certain inference and optimization problems, and more 
+\cite{Stein_1995_Broken, Krzakala_2007_Landscape, Altieri_2021_Properties, Yang_2023_Stochastic}. Glassiness
 is qualitatively understood to arise from structure of an energy or cost
 landscape, whether due to the proliferation of metastable states, or to the
 raising of barriers which cause effective dynamic constraints
@@ -126,7 +127,7 @@ percolation transition. In fact, this threshold energy is significant in other
 ways: it attracts the long-time dynamics after quenches in temperature to below
 the dynamical transition from any starting temperature
 \cite{Biroli_1999_Dynamical, Sellke_2023_The}. All of this can be understood in terms of the
-landscape structure.
+landscape structure, and namely in the statistics of stationary points of the energy.
 
 In slightly less simple models, the mixed spherical models, the story changes.
 There are now a range of energies with exponentially many marginal minima. It
@@ -135,7 +136,7 @@ common type of stationary point would play the same role as the threshold
 energy in the pure models (in fact we will refer to this energy level as the
 threshold energy in the mixed models). However, recent work has shown that
 this is incorrect. Quenches from different starting temperatures above the
-dynamical transition temperature result in dynamics that approach different
+dynamical transition temperature result in dynamics that approach marginal minima at different
 energy levels, and the purported threshold does not attract the long-time
 dynamics in most cases \cite{Folena_2020_Rethinking, Folena_2021_Gradient}.
 
@@ -143,42 +144,32 @@ This paper studies the two-point structure of stationary points in the mixed
 spherical models, or their arrangement relative to each other, previously
 studied only for the pure models \cite{Ros_2019_Complexity}. This gives various
 kinds of information. When one point is a minimum, we see what other kinds of
-minima are nearby, and what kind of saddle points (barriers) separate them.
+minima are nearby, and the height of the saddle points that separate them.
 When both points are saddles, we see the arrangement of barriers relative to
-each other, perhaps learning something about the geometry of the basins of
-attraction that they surround.
+each other.
 
 More specifically, one \emph{reference} point is fixed with certain properties.
 Then, we compute the logarithm of the number of other points constrained to lie
-at a fixed overlap from the reference point. The fact of constraining the count
-to a fixed overlap produces constrained points with atypical properties. For
-instance, we will see that when the constrained overlap is made sufficiently
-large, the constrained points tend to have an isolated eigenvalue pulled out
-of their spectrum, and the associated eigenvector is correlated with the
+at a fixed overlap from the reference point. Constraining the count
+to points of a fixed overlap from the reference point produces constrained points with atypical properties. For
+instance, when the constrained overlap is made sufficiently
+large, typical constrained points tend to have an isolated eigenvalue pulled out
+of their spectrum, and its associated eigenvector is correlated with the
 direction of the reference point. Without the proximity constraint, such an
-isolated eigenvalue amounts to a large deviation from the typical spectrum of
-stationary points.
+isolated eigenvalue amounts to a large deviation from the spectrum of
+typical stationary points.
 
-In order to address the open problem of what attracts the long-time dynamics,
+In order to address the open problem of what energies attract the long-time dynamics,
 we focus on the neighborhoods of the marginal minima, to see if there is
 anything interesting to differentiate sets of them from each other. Though we
 find rich structure in this population, their properties pivot around the
 debunked threshold energy, and the apparent attractors of long-time dynamics
-are not distinguished by this measure. Moreover, we show that the usual picture of a
+are not distinguished. Moreover, we show that the usual picture of a
 marginal `manifold' of inherent states separated by subextensive barriers \cite{Kurchan_1996_Phase} is only true
 at the threshold energy, while at other energies typical marginal minima are far apart
 and separated by extensive barriers. Therefore, with respect to the problem of
 dynamics this paper merely deepens the outstanding issues.
 
-In \S\ref{sec:model}, we introduce the mixed spherical models and discuss their
-properties, defining many of the symbols and concepts relied on in the rest of
-the paper. In \S\ref{sec:results}, we share the main results of the paper, with
-a large discussion of the neighborhood of marginal states. In
-\S\ref{sec:complexity} we detail the calculation of the two-point complexity,
-and in \S\ref{sec:eigenvalue} and \S\ref{sec:franz-parisi} we do the same for
-the properties of an isolated eigenvalue and for the zero-temperature
-Franz--Parisi potential. Finally in \S\ref{sec:conclusion} we draw some conclusions from the study.
-
 \section{Model}
 \label{sec:model}
 
@@ -204,16 +195,8 @@ where the function $f$ is defined from the coefficients $a_p$ by
 \begin{equation}
   f(q)=\frac12\sum_pa_pq^p
 \end{equation}
-In this paper, we will focus on models with a replica symmetric complexity, but
-many of the intermediate formulae are valid for arbitrary replica symmetry
-breakings. At most {\oldstylenums1}\textsc{rsb} in the equilibrium is guaranteed if the function
-$\chi(q)=f''(q)^{-1/2}$ is convex \cite{Crisanti_1992_The}. The complexity at the ground state must
-reflect the structure of equilibrium, and therefore be replica symmetric. We
-are not aware of any result guaranteeing this for the complexity away from the
-ground state, but we check that our replica-symmetric solutions satisfy the
-saddle point equations at {\oldstylenums1}\textsc{rsb}.
-
-\cite{Kent-Dobias_2023_When}
+The choice of $f$ has significant effect on the form of order in the model, and
+this likewise influences the geometry of stationary points.
 
 To enforce the spherical constraint at stationary points, we make use of a Lagrange multiplier $\omega$. This results in the extremal problem
 \begin{equation}
@@ -309,14 +292,16 @@ one uniquely fixes the other. This property leads to the great simplification
 of these models: marginal minima exist \emph{only} at one energy level, and
 therefore only that energy has the possibility of trapping the long-time
 dynamics. In generic mixed models this is not the case and at a given energy
-level $E$ there are many stabilities for which exponentially many stationary
+level $E$ there are many stabilities for which exponentially many marginal
 points are found. We define the threshold energy $E_\mathrm{th}$ as the energy
-at which most stationary points are marginal. Note that crucially this is
+at which most stationary points are marginal.\footnote{
+  Note that crucially this is
 \emph{not} the energy that has the most marginal stationary points: this energy
 level with the largest number of marginal points has even more saddles of
 extensive index. So $E_\mathrm{th}$ contains a \emph{minority} of the
 marginal points, even if those marginal points are the \emph{majority} of
 stationary points with energy $E_\mathrm{th}$.
+}
 
 \begin{figure}
   \centering
@@ -324,7 +309,7 @@ stationary points with energy $E_\mathrm{th}$.
   \caption{
     Plot of the complexity (logarithm of the number of stationary points) for
     the $3+4$ mixed spherical model studied in this paper. Energies and stabilities
-    of interest are marked, including the ground state energy and stability
+    of interest are marked, including the ground state energy
     $E_\mathrm{gs}$, the marginal stability $\mu_\mathrm
     m$, and the threshold energy $E_\mathrm{th}$. The blue line shows the location
     of the most common type of stationary point at each energy level. The
@@ -334,33 +319,44 @@ stationary points with energy $E_\mathrm{th}$.
   } \label{fig:complexities}
 \end{figure}
 
-In this study, we focus exclusively on the model studied in
+In this study, our numeric examples are drawn exclusively from the model studied in
 \cite{Folena_2020_Rethinking}, whose covariance function is given by
 \begin{equation}
   f_{3+4}(q)=\frac12\big(q^3+q^4\big)
 \end{equation}
-First, it has convex $f''(q)^{-1/2}$, so at least the ground state complexity
-must be replica symmetric, as in the pure spherical models. Second, properties
-of its long-time dynamics have been extensively studied. The annealed one-point
+First, the ordering of its stationary points is like that of the pure spherical models, without any clustering \cite{Kent-Dobias_2023_When}. Second, properties
+of its long-time dynamics have been extensively studied. Though the numeric examples all come from the $3+4$ model, the results apply to any model sharing its simple order. The annealed one-point
 complexity of these models was calculated in \cite{BenArous_2019_Geometry}, and
-for this model the annealed is expected to be correct. The complexity of this
-model is plotted in Fig.~\ref{fig:complexities}.
+for this model the annealed is expected to be correct.
+
+The one-point complexity of this model as a function of energy $E$ and
+stability $\mu$ is plotted in Fig.~\ref{fig:complexities}. The same plot for a
+pure $p$-spin model would consist of only a line, because $E$ and $\mu$ cannot
+be varied independently. Several important features of the complexity are
+highlighted: the energies of the ground state $E_\text{gs}$ and the threshold
+$E_\text{th}$, along with the line of marginal stability $\mu_\text m$. Along
+the line of marginal stability, energies which attract aging dynamics from
+different temperatures are highlighted in red. One might expect something
+quantitative to mark the ends of this range, something that would differentiate
+marginal minima that support aging dynamics from those that do not. As
+indicated in the introduction, the two-point complexity studied in this paper
+does not produce such a result.
 
 \section{Results}
 \label{sec:results}
 
-Our results are in the form of the two-point complexity $\Sigma_{12}$, which is defined as
-the logarithm of the number of stationary points with energy $E_1$ and
-stability $\mu_1$ that lie at an overlap $q$ with a typical reference stationary point
-whose energy is $E_0$ and stability is $\mu_0$. When the complexity is
-positive, there are exponentially many stationary points with the given
-properties conditioned on the existence of the reference one. When it is zero,
-there are only order-one such points, and when it is negative there are
+Our results stem from the two-point complexity $\Sigma_{12}$, which is defined
+as the logarithm of the number of stationary points with energy $E_1$ and
+stability $\mu_1$ that lie at an overlap $q$ with a typical reference
+stationary point whose energy is $E_0$ and stability is $\mu_0$. When the
+complexity is positive, there are exponentially many stationary points with the
+given properties conditioned on the existence of the reference one. When it is
+zero, there are only order-one such points, and when it is negative there are
 exponentially few (effectively, none). In the examples below, the boundary of
-zero complexity between exponentially many and few points is often highlighted.
+zero complexity between exponentially many and few points is often highlighted, with parameter regions that have negative complexity having no color.
 Finally, as a result of the condition that the counted points lie with a given
 proximity to the reference point, their spectrum can be modified by the
-presence of an isolated eigenvalue, which can change the stability as in
+presence of an isolated eigenvalue, which can change the stability as shown in
 Fig.~\ref{fig:spectra}.
 
 \subsection{Barriers around deep states}
@@ -393,50 +389,87 @@ Fig.~\ref{fig:spectra}.
   } \label{fig:min.neighborhood}
 \end{figure}
 
-If the reference configuration is restricted to stable minima, then there is a
-gap in the overlap between those minima and their nearest neighbors in
+If the reference configuration is a stable minimum, then there is a
+gap in the overlap between it and its nearest neighbors in
 configuration space. We can characterize these neighbors as a function of their
 overlap and stability, with one example seen in
 Fig.~\ref{fig:min.neighborhood}. For stable minima, the qualitative results for
 the pure $p$-spin model continue to hold, with some small modifications
 \cite{Ros_2019_Complexity}.
 
-First, the nearest neighbor points are always oriented saddles, sometimes
-saddles with an extensive index and sometimes index-one saddles (Fig.~\ref{fig:spectra}(d,
-f)). Like in the pure models, the emergence of oriented index-one saddles along
-the line of lowest-energy states at a given overlap occurs at the local minimum
-of this line. Unlike the pure models, neighbors exist for independent $\mu_1$
-and $E_1$, and the line of lowest-energy states at a given overlap is different
-from the line of maximally-stable states at a given overlap.
-
+The largest different is the decoupling of nearby
+stable points from nearby low-energy points: in the pure $p$-spin model, the
+left and right panels of Fig.~\ref{fig:min.neighborhood} would be identical up
+to a constant factor $-p$. Instead, for mixed models they differ substantially,
+as evidenced by the dot-dashed lines in both plots that in the pure models
+would correspond exactly with the solid lines. One significant consequence of
+this difference is the diminished significance of the threshold energy
+$E_\text{th}$: in the left panel, marginal minima of the threshold energy are
+the most common among unconstrained points, but marginal minima of lower energy
+are more common in the near vicinity of the example reference minimum, whose energy is lower than the threshold energy.
+
+The nearest neighbor points are always oriented saddles, sometimes
+saddles with an extensive index and sometimes index-one saddles
+(Fig.~\ref{fig:spectra}(d, f)). Like in the pure models, the minimum energy and
+maximum stability of nearby points are not monotonic: there is a range of
+overlap where the minimum energy of neighbors decreases with proximity. The
+emergence of oriented index-one saddles along the line of lowest-energy states
+at a given overlap occurs at the local minimum of this line, another similarity with the pure models
+\cite{Ros_2019_Complexity}. It is not clear why this should be true or what implications it has for behavior.
+
+\subsection{Grouping of saddle points}
+
+At stabilities lower than the marginal stability one finds saddles with an
+extensive index. Though, being unstable, saddles are not attractors of
+dynamics, their properties influence out-of-equilibrium dynamics. For example,
+high-index saddle points are stationed at the boundaries between different
+basins of attraction of gradient flow, and for a given basin the flow between
+adjacent saddle points defines a complex with implications for the landscape
+topology \cite{Audin_2014_Morse}.
+
+Other stationary points are found at arbitrarily small distances from a
+reference extensive saddle point. The energy and stability of these near
+neighbors approach that of the reference point as the difference in overlap
+$\Delta q$ is brought to zero. However, the approach of the energy and
+stability are at different rates: the energy difference between the reference
+and its neighbors shrinks like $\Delta q^2$, while the stability difference
+shrinks like $\Delta q$. This means that the near neighborhood of saddle points
+is dominated by the presence of other saddle points at very similar energy, but
+varied index. It makes it impossible to draw conclusions about the way saddle
+points are connected by gradient flow from the properties of nearest neighbors.
+Descending between saddles must lower the index -- and therefore the stability
+-- and the energy, but if the energy and stability change with the same order
+of magnitude the connected saddle points must lie at a macroscopic distance
+from each other.
 
 \subsection{Geometry of marginal states}
 
-The set of marginal states is of special interest. First, it has more structure
-than in the pure models, with different types of marginal states being found at
-different energies. Second, marginal states are known to attract physical and
-algorithmic dynamics \cite{Folena_2023_On}. We find, surprisingly, that the
-properties of marginal states pivot around the threshold energy, the energy at
-which most stationary points are marginal.
+The set of marginal states is of special interest. First, marginal states are
+known to attract physical and algorithmic dynamics \cite{Folena_2023_On}.
+Second, they have more structure than in the pure models, with different types
+of marginal states being found at different energies.  We find, surprisingly,
+that the properties of marginal states pivot around the threshold energy, the
+energy at which most stationary points are marginal, but which is not
+significant for aging dynamics.
 
 \begin{itemize}
-  \item \textbf{Energies below the threshold.} Marginal states have a
-    macroscopic gap in their overlap with nearby minima and saddles. The
-    nearest stationary points are saddles with an oriented direction,
-    and always have a higher energy density than the reference state.
+  \item \textbf{Energies below the threshold.} These marginal states have a
+    macroscopic gap in their overlap with nearby minima and saddles. Their
+    nearest-neighbor stationary points are saddles with an oriented direction,
+    and their nearest neighbors always have a higher energy density than the reference state.
     Fig.~\ref{fig:marginal.prop.below} shows examples of the neighborhoods of
     these marginal minima.
 
-  \item \textbf{Energies above the threshold.} Marginal states have neighboring
+  \item \textbf{Energies above the threshold.} These marginal states have neighboring
     stationary points at arbitrarily close distance, with a quadratic pseudogap in
-    their complexity. The nearest ones are \emph{strictly} saddle points with
-    an extensive number of downward directions and always have a higher energy
+    their complexity. Their nearest neighbors are \emph{strictly} saddle points with
+    an extensive number of downward directions and their nearest neighbors always have a higher energy
     density than the reference state. The nearest neighboring marginal states
     have an overlap gap with the reference state.
     Fig.~\ref{fig:marginal.prop.above} shows examples of the neighborhoods of
     these marginal minima.
 
-  \item \textbf{At the threshold energy.} Marginal states have neighboring
+  \item \textbf{At the threshold energy.} These marginal states have neighboring
     stationary points at arbitrarily close distance, with a cubic pseudogap
     in their complexity. The nearest ones include oriented saddle
     points with an extensive number of downward directions, and oriented stable
@@ -517,9 +550,8 @@ except at the threshold energy, \emph{typical marginal minima are separated by
 extensive energy barriers}. Therefore, the picture of a marginal
 \emph{manifold} of many (even all) marginal states lying arbitrarily close and
 being connected by subextensive energy barriers can only describe the
-collection of marginal minima at the threshold energy, or an atypical population of marginal minima. At energies both below and above the threshold energy,
-typical marginal minima are isolated from each other.
-
+collection of marginal minima at the threshold energy, which is an atypical population of marginal minima. At energies both below and above the threshold energy,
+typical marginal minima are isolated from each other.\footnote{
 We must put a small caveat here: in \emph{any} situation, this calculation
 admits order-one other marginal minima to lie a subextensive distance from the
 reference point. For such a population of points, $\Sigma_{12}=0$ and $q=1$,
@@ -530,6 +562,7 @@ points is isolated by extensive barriers from each other cluster in the way
 described above. To move on a `manifold' of nearby marginal minima within such a
 cluster cannot describe aging, since the overlap with the initial condition
 will never change from one.
+}
 
 This has implications for how quench dynamics should be interpreted. When
 typical marginal states are approached above the threshold energy, they must have been
@@ -539,7 +572,7 @@ energy must  be reached after an extensive distance in configuration space
 without encountering any stationary point. The geometric conditions of the
 neighborhoods above and below are quite different, but the observed aging
 dynamics don't appear to qualitatively change \cite{Folena_2020_Rethinking,
-Folena_2021_Gradient}. Therefore, if the marginal minima attracting dynamics are typical, the conditions in the neighborhood of the
+Folena_2021_Gradient}. Therefore, if the marginal minima attracting dynamics are typical ones, the conditions in the neighborhood of the
 marginal minimum eventually reached at infinite time appear to be irrelevant
 for the nature of aging dynamics at any finite time.
 
@@ -566,19 +599,31 @@ scope of this paper, but is clear fodder for future research.
 \section{Calculation of the two-point complexity}
 \label{sec:complexity}
 
-We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
+To calculate the two-point complexity, we extend a common method for counting
+stationary points: the Kac--Rice method \cite{Kac_1943_On,
+Rice_1944_Mathematical}. The basic idea is that stationary points of a function
+can be counted by integrating a Dirac $\delta$-function containing the
+function's gradient over its domain. Because the argument of the
+$\delta$-function is nonlinear in the integration variable, it must be weighted
+by the determinant of the Jacobian of the argument, which happens to be the
+Hessian of the function. It is not common that this procedure can be
+analytically carried out for an explicit function. However, in the spherical
+models it can be carried out \emph{on average}.
+
+In order to lighten notation, we introduce the Kac--Rice  measure
 \begin{equation}
   d\nu_H(\mathbf s,\omega)
   =2\,d\mathbf s\,d\omega\,\delta(\|\mathbf s\|^2-N)\,
   \delta\big(\nabla H(\mathbf s,\omega)\big)\,
   \big|\det\operatorname{Hess}H(\mathbf s,\omega)\big|
 \end{equation}
-which counts stationary points of the function $H$. If integrated over
+containing the gradient and Hessian of the Hamiltonian, along with a $\delta$-function enforcing the spherical constraint.
+If integrated over
 configuration space, $\mathcal N_H=\int d\nu_H(\mathbf s,\omega)$ gives the
-total number of stationary points in the function. The Kac--Rice method has been used by in many studies to analyze the geometry of random functions \cite{Cavagna_1998_Stationary, Fyodorov_2007_Density, Bray_2007_Statistics}. More interesting is the
+total number of stationary points in the function. The Kac--Rice method has been used by in many studies to analyze the geometry of random functions \cite{Cavagna_1998_Stationary, Fyodorov_2007_Density, Bray_2007_Statistics, Kent-Dobias_2023_How}. More interesting is the
 measure conditioned on the energy density $E$ and stability $\mu$ of the
 points,
-\begin{equation}
+\begin{equation} \label{eq:measure.energy}
   d\nu_H(\mathbf s,\omega\mid E,\mu)
   =d\nu_H(\mathbf s,\omega)\,
   \delta\big(NE-H(\mathbf s)\big)\,
@@ -587,7 +632,7 @@ points,
 While $\mu$ is strictly the trace of the Hessian, we call it the stability
 because in this family of models all stationary points have a bulk spectrum of
 the same shape, shifted by different constants. The stability $\mu$ sets this
-shift, and therefore determines if the spectrum has bulk support on zero. See
+shift, and therefore determines the stiffness of minima and the typical index of saddle points. See
 Fig.~\ref{fig:spectra} for examples.
 
 We want the typical number of stationary points with energy density
@@ -602,15 +647,32 @@ therefore defined by
   =\frac1N\overline{\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,
   \log\bigg(\int d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s)\bigg)}
 \end{equation}
-Both the denominator and the logarithm are treated using the replica trick, which yields
+Inside the logarithm, the measure \eqref{eq:measure.energy} is integrated with
+the further condition that $\mathbf s$ has a certain overlap with $\pmb\sigma$.
+The entire expression is then integrated over $\pmb\sigma$ again by
+the Kac--Rice measure, then divided by a normalization. This is equivalent to
+summing the logarithm over all stationary points $\pmb\sigma$ with the given
+properties, then dividing by their total number, i.e., an average.
+
+It is difficult to take the disorder average of anything that is not an
+exponential integral. The normalization integral over $\pmb\sigma'$ in the
+denominator and the integral inside the logarithm both pose a problem. Each can
+be treated using the replica trick: $\lim_{m\to0} x^{m-1}=\frac1x$ and $\lim_{n\to0}\frac\partial{\partial n}x^n=\log x$. Applying these transformations, we have
 \begin{equation}
   \Sigma_{12}
   =\frac1N\lim_{n\to0}\lim_{m\to0}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b,\varsigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb \sigma_1\cdot \mathbf s_a)\right)}
 \end{equation}
-Note that because of the structure of \eqref{eq:complexity.definition},
-$\pmb\sigma_1$ is special among the set of $\pmb\sigma$ replicas, since it alone
-is constrained to lie a given overlap from the $\mathbf s$ replicas. This
-replica asymmetry will be important later.
+Note that among the $\pmb\sigma$ replicas, $\pmb\sigma_1$ is special. The $m-1$
+replicas $\pmb\sigma_2,\ldots,\pmb\sigma_m$ correspond to the replicated
+normalization integral over $\pmb\sigma'$, which is completely uncoupled from
+$\mathbf s$. The variable $\pmb\sigma_1$ is not a replica: it is the same as
+$\pmb\sigma$ in \eqref{eq:complexity.definition}, and is the only of the $\pmb\sigma$s that couples with $\mathbf s$.
+
+This expression can now be averaged over the disordered couplings, and its
+integration evaluated using the saddle point method. We must assume the form of
+order among the replicas $\mathbf s$ and $\pmb\sigma$, and we take them to be
+replica symmetric. For the $3+4$ model that is our immediate interest and other
+models like it this choice is well-motivated. Details of this calculation can be found in Appendix~\ref{sec:complexity-details}.
 
 The resulting expression for the complexity, which must
 still be extremized over the parameters $\hat\beta_1$, $r^{01}$,
@@ -1227,7 +1289,17 @@ not constrained) and then the saddle point taken.
 
 In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
 and $D^{01}$ to have constant \emph{rows} of length $n$, with blocks of rows
-corresponding to the \textsc{rsb} structure of the single-point complexity. For
+corresponding to the \textsc{rsb} structure of the single-point complexity.
+
+In this paper, we will focus on models with a replica symmetric complexity, but
+many of the intermediate formulae are valid for arbitrary replica symmetry
+breakings. At most {\oldstylenums1}\textsc{rsb} in the equilibrium is guaranteed if the function
+$\chi(q)=f''(q)^{-1/2}$ is convex \cite{Crisanti_1992_The}. The complexity at the ground state must
+reflect the structure of equilibrium, and therefore be replica symmetric.
+Recent work has found that the complexity of saddle points can produce
+other \textsc{rsb} order even when the ground state is replica symmetric, but the $3+4$ model has a safely replica symmetric complexity everywhere \cite{Kent-Dobias_2023_When}.
+
+For
 the scope of this paper, where we restrict ourselves to replica symmetric
 complexities, they have the following form at the saddle point:
 \begin{align} \label{eq:01.ansatz}
-- 
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