From cf4dec33e62829f40240f29b484ea2f4c7c401e7 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 26 Aug 2024 17:24:53 +0200
Subject: Lots of writing and figure work.

---
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 figs/phases_1.pdf   | Bin 11318 -> 11318 bytes
 figs/phases_2.pdf   | Bin 12012 -> 12012 bytes
 figs/phases_3.pdf   | Bin 13050 -> 13049 bytes
 topology.bib        |  14 ++
 topology.tex        | 605 ++++++++++++++++++++++++++++++----------------------
 9 files changed, 368 insertions(+), 251 deletions(-)

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diff --git a/topology.bib b/topology.bib
index 316df0f..8c17a2c 100644
--- a/topology.bib
+++ b/topology.bib
@@ -264,6 +264,20 @@
  eprinttype = {arxiv}
 }
 
+@article{Kent-Dobias_2023_How,
+ author = {Kent-Dobias, Jaron and Kurchan, Jorge},
+ title = {How to count in hierarchical landscapes: a full solution to mean-field complexity},
+ journal = {Physical Review E},
+ publisher = {American Physical Society (APS)},
+ year = {2023},
+ month = {6},
+ number = {6},
+ volume = {107},
+ pages = {064111},
+ url = {https://doi.org/10.1103/PhysRevE.107.064111},
+ doi = {10.1103/PhysRevE.107.064111}
+}
+
 @article{Kent-Dobias_2023_When,
  author = {Kent-Dobias, Jaron},
  title = {When is the average number of saddle points typical?},
diff --git a/topology.tex b/topology.tex
index 102dc6b..a7b68dc 100644
--- a/topology.tex
+++ b/topology.tex
@@ -34,7 +34,7 @@
 
 \begin{abstract}
   We consider the set of solutions to $M$ random polynomial equations with
-  independent Gaussian coefficients on the $(N-1)$-sphere. When solutions
+  independent Gaussian coefficients and a target value $V_0$ on the $(N-1)$-sphere. When solutions
   exist, they form a manifold. We compute the average Euler characteristic of
   this manifold in the limit of large $N$, and find different behavior
   depending on the ratio $\alpha=M/N$. When $\alpha<\alpha_\text{onset}$, the
@@ -110,7 +110,7 @@ tissues \cite{Fyodorov_2019_A, Fyodorov_2020_Counting,
 Montanari_2024_On, Kent-Dobias_2024_Conditioning, Kent-Dobias_2024_Algorithm-independent}. In each of these cases, the authors studied properties of
 the cost function
 \begin{equation} \label{eq:cost}
-  \mathscr C(\mathbf x)=\frac12\sum_{k=1}^MV_k(\mathbf x)^2
+  \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.
 Here we dispense with the cost function and study the set of solutions
@@ -136,9 +136,188 @@ direct sum of the Betti numbers of $\Omega$. We find that for the varied cases
 we study, these two always coincide at the largest exponential order in $N$,
 putting strong constraints on the resulting topology and geometry.
 
-\section{Results}
+\section{Methods}
 
-\subsection{Topology of solutions to many equations and the satisfiability transition}
+\subsection{The average Euler characteristic}
+
+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 $\#$
+handles. For general $d$, the Euler characteristic of the $d$-sphere is $2$ if $d$ is even and 0 if $d$ is odd. The canonical method for computing the Euler characteristic is done by
+defining a complex on the manifold in question, essentially 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 number of faces.
+
+Morse theory offers another way to compute the Euler characteristic using the
+statistics of stationary points of a function $H:\Omega\to\mathbb R$ \cite{Audin_2014_Morse}. For
+functions $H$ without any symmetries with respect to the manifold, the surfaces
+of gradient flow between adjacent stationary points form a complex. The
+alternating sum over cells to compute $\chi$ becomes an alternating sum over
+the count of stationary points of $H$ with increasing index, or
+\begin{equation}
+  \chi=\sum_{i=0}^N(-1)^i\mathcal N_H(\text{index}=i)
+\end{equation}
+Conveniently, we can express this abstract sum as an integral over the manifold
+using a small variation on the Kac--Rice formula for counting stationary
+points. Since the sign of the determinant of the Hessian matrix of $H$ at a
+stationary point is equal to its index, if we count stationary points including
+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
+\cite{Fyodorov_2004_Complexity}. Here we are correct to exclude it.
+
+We need to choose a function $H$ for our calculation. Because $\chi$ is
+a topological invariant, any choice will work so long as it does not share some
+symmetry with the underlying manifold, i.e., that it $H$ satisfies the Smale condition. Because our manifold of random
+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.
+
+We treat the integral over the implicitly defined manifold $\Omega$ using the
+method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to
+enforce the spherical constraint and $M$ multipliers $\omega_k$ to enforce the vanishing of
+each of the $V_k$, resulting in the Lagrangian
+\begin{equation} \label{eq:lagrangian}
+  L(\mathbf x,\pmb\omega)
+  =H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big)
+  +\sum_{k=1}^M\omega_k\big(V_k(\mathbf x)-V_0\big)
+\end{equation}
+The integral over $\Omega$ in \eqref{eq:kac-rice} then becomes
+\begin{equation} \label{eq:kac-rice.lagrange}
+  \chi(\Omega)=\int_{\mathbb R^N} d\mathbf x\int_{\mathbb R^{M+1}}d\pmb\omega
+  \,\delta\big(\partial L(\mathbf x,\pmb\omega)\big)
+  \det\partial\partial L(\mathbf x,\pmb\omega)
+\end{equation}
+where $\partial=[\frac\partial{\partial\mathbf x},\frac\partial{\partial\pmb\omega}]$
+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.
+
+In order for certain Gaussian integrals in the following calculation to be
+well-defined, it is necessary to treat instead the Lagrangian problem above
+with $\pmb\omega\mapsto i\pmb\omega$. This transformation does not effect the
+Dirac $\delta$ functions of the gradient, but it does change the determinant by
+a factor of $i^{N+M+1}$. We will see that the result of the rest of the
+calculation neglecting this factor is real. Since the Euler characteristic is
+also necessarily real, this indicates an inconsistency with this transformation
+when $N+M+1$ is odd. In fact, the Euler characteristic is always zero for
+odd-dimensional manifolds. This is the signature of it in this problem.
+
+\subsubsection{Calculation of the average Euler characteristic}
+
+To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
+\begin{align}
+  \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
+  &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
+  e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)}
+  \\
+  \det\partial\partial L(\mathbf x,\pmb\omega)
+  &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
+  e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]}
+\end{align}
+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
+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}.
+The result is the reduction of the average Euler characteristic to an expression of the
+form
+\begin{equation}
+  \overline{\chi(\Omega)}
+  =\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\Omega(R,D,m,\hat m)}
+\end{equation}
+where $g$ is a prefactor subexponential in $N$, and $\mathcal S_\Omega$ is an effective action defined by
+\begin{equation}
+  \begin{aligned}
+    \mathcal S_\Omega(R,D,m,\hat m\mid\alpha,V_0)
+    &=\hat m-\frac\alpha2\left[
+      \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
+      +\frac{V_0^2}{f(1)}\left(1+\frac{f'(1)R^2}{f(1)D}\right)^{-1}
+    \right] \\
+    &\hspace{7em}+\frac12\log\left(
+      1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
+    \right)
+  \end{aligned}
+\end{equation}
+The remaining order parameters defined by the scalar products
+\begin{align}
+  R=-i\frac1N\mathbf x\cdot\hat{\mathbf x}
+  &&
+  D=\frac1N\hat{\mathbf x}\cdot\hat{\mathbf x}
+  &&
+  m=\frac1N\mathbf x\cdot\mathbf x_0
+  &&
+  \hat m=-i\frac1N\hat {\mathbf x}\cdot\mathbf x_0
+\end{align}
+
+This integral can be evaluated 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^*(m)}{1-m^2} \qquad \hat m=0
+\end{equation}
+\begin{equation}
+  \begin{aligned}
+    R^*(m)
+    &=\frac{-m(1-m^2)}{2[f(1)-(1-m^2)f'(1)]^2}
+    \Bigg[
+      \alpha V_0^2f'(1)
+      +(2-\alpha)f(1)\left(\frac{f(1)}{1-m^2}-f'(1)\right) \\
+    &\quad+\operatorname{sgn}(m)\alpha\sqrt{
+      \tfrac{4V_0^2}\alpha f(1)f'(1)\left[\tfrac{f(1)}{1-m^2}-f'(1)\right]
+      +\left[\tfrac{f(1)^2}{1-m^2}-\big(V_0^2+f(1)\big)f'(1)\right]^2
+    }
+    \Bigg]
+  \end{aligned}
+\end{equation}
+\begin{equation}
+  \mathcal S_\Omega(m)
+  =-\frac\alpha2\bigg[
+    \log\left(
+      1-\frac{f(1)}{f'(1)}\frac{1+m/R^*}{1-m^2}
+    \right)
+    +\frac{V_0^2}{f(1)}\left(
+      1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+m/R^*}
+    \right)^{-1}
+  \bigg]
+  +\frac12\log\left(-\frac m{R^*}\right)
+\end{equation}
+To finish evaluating the integral, this expression should be maximized with
+respect to $m$. 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
+understood as the latitude on the sphere where most of the contribution to the
+Euler characteristic is made.
+
+The action $\mathcal S_\Omega$ is extremized with respect to $m$ at $m^*=0$ or $m^*=-R^*$.
+In the latter case, $m^*$ takes the value
+\begin{equation}
+  m^*=\pm\sqrt{1-\frac{\alpha}{f'(1)}\big(V_0^2+f(1)\big)}
+\end{equation}
+and $\mathcal S_\Omega(m^*)=0$. However, when
+\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 the solution is no longer valid. Likewise, when
+\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. Examples of
+$\mathcal S_\Omega$ in difference regimes are shown in Fig.~\ref{fig:action}.
+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}
 
 \begin{figure}
   \includegraphics{figs/action_1.pdf}
@@ -146,13 +325,13 @@ putting strong constraints on the resulting topology and geometry.
   \includegraphics{figs/action_3.pdf}
 
   \caption{
-    The effective action governing the as a function of the overlap
+    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 $V_0$. In both
+    different homogeneous polynomial functions and a variety of target values $V_0$. 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{Left:} With nonlinear
+    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
@@ -160,9 +339,133 @@ putting strong constraints on the resulting topology and geometry.
     $V_0=V_\text{sh}\simeq1.394$, $m^*$ 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.
-  }
+  } \label{fig:action}
 \end{figure}
 
+
+
+
+\subsubsection{How to interpret the average Euler characteristic}
+
+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.
+
+The many positive values the Euler characteristic can take is one problem, but
+a larger one is 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 $\mathcal S_\Omega(m)$ 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 subsection 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 $\mathcal
+S_\Omega(m)$ represents breaks down due to the vanishingly small probability of
+finding any stationary points.
+
+\subsection{Complexity of a linear test function}
+
+One way to definitely narrow possible interpretations of the average Euler
+characteristic is to compute a complementary average. The Euler characteristic
+is the alternating sum of numbers of critical points of different index. If
+instead we make the direct sum
+\begin{equation}
+  \mathcal N_H(\Omega)=\sum_{i=0}^N\mathcal N_H(\text{index}=i)
+  =\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)
+  \,\big|\det\operatorname{Hess}H(\mathbf x)\big|
+\end{equation}
+we find the total number of stationary points. The formula is exactly the same
+as that for the average Euler characteristic except for an absolute value sign
+around the determinant of the Hessian.
+
+Understanding the number of stationary points as a function of latitude $m$
+will clarify the meaning of our effective action for the average Euler
+characteristic. This is because the average number of stationary points is a
+nonnegative number. If the region of complex $\mathcal S_\Omega$ has a
+well-defined number of stationary points, it indicates that we are looking at a
+situation with a negative average Euler characteristic. On the other hand, if
+the average number of stationary points yields a complex value at some latitude
+$m$, it must be because it is either to large or small in $N$ to be captured by
+the calculation, e.g., that it behaves like $e^{N^2\Sigma}$ or
+$e^{-N^2\Sigma}$. The following calculation indicates this second situation:
+the region of complex action is due to a lack of stationary points to
+contribute to the Euler characteristic at those latitudes.
+
+To compute the complexity, we follow a similar procedure to the Euler
+characteristic. The main difference lies in how we treat the absolute value
+function around the determinant. Following \cite{Fyodorov_2004_Complexity}, we
+make use of the identity
+\begin{equation}
+  \begin{aligned}
+  |\det A|
+  &=\lim_{\epsilon\to0}\frac{(\det A)^2}{\sqrt{\det(A+i\epsilon I)}\sqrt{\det(A-i\epsilon I)}}
+  \\
+  &=\frac1{(2\pi)^N}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,d\mathbf a\,d\mathbf b\,
+  e^{
+    -\bar{\pmb\eta}_1^TA\pmb\eta_1-\bar{\pmb\eta}_2^TA\pmb\eta_2
+    -\frac12\mathbf a^T(A+i\epsilon I)\mathbf a-\frac12\mathbf b^T(A-i\epsilon I)\mathbf b
+  }
+  \end{aligned}
+\end{equation}
+for an $N\times N$ matrix $A$. Here $\bar{\pmb\eta}_1$, $\pmb\eta_1$,
+$\bar{\pmb\eta}_2$, and $\pmb\eta_2$ are Grassmann vectors and $\mathbf a$ and
+$\mathbf b$ are regular vectors. This introduces many new order parameters into
+the problem, but this is a difficulty of scale rather than principle. With this
+identity substituted for the usual determinant one, the problem can be solved
+much as before. The details of this solution are relegated to
+Appendix~\ref{sec:complexity.details}. The result is that, to largest order in
+$N$, the logarithm of the average number of stationary points is the same as
+the logarithm of the average Euler characteristic.
+
+\section{Results}
+
+\subsection{Topology of solutions to many equations and the satisfiability transition}
+
+The results of the previous sections indicate the following picture for the
+topology of the solution manifold.
+
 \begin{figure}
   \includegraphics[width=0.245\textwidth]{figs/connected.pdf}
   \includegraphics[width=0.245\textwidth]{figs/coexist.pdf}
@@ -208,6 +511,30 @@ putting strong constraints on the resulting topology and geometry.
 
 \subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold}
 
+As indicated earlier, for $M=1$ 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 scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields
+\begin{equation}
+  E_\text{onset}=\pm\sqrt{2f(1)}
+\end{equation}
+\begin{equation}
+  E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
+\end{equation}
+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 $f(q)=\frac12q^p$,
+$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
+models. This is expected, 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.
+
+However, for general mixed models 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 generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
+
 \begin{figure}
   \includegraphics{figs/dynamics_2.pdf}
   \hspace{-0.5em}
@@ -226,87 +553,22 @@ putting strong constraints on the resulting topology and geometry.
   }
 \end{figure}
 
-\section{The average Euler characteristic}
 
-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 $\#$
-handles. For general $d$, the Euler characteristic of the $d$-sphere is $2$ if $d$ is even and 0 if $d$ is odd. The canonical method for computing the Euler characteristic is done by
-defining a complex on the manifold in question, essentially 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 number of faces.
+\section{Conclusions}
 
-Morse theory offers another way to compute the Euler characteristic using the
-statistics of stationary points of a function $H:\Omega\to\mathbb R$ \cite{Audin_2014_Morse}. For
-functions $H$ without any symmetries with respect to the manifold, the surfaces
-of gradient flow between adjacent stationary points form a complex. The
-alternating sum over cells to compute $\chi$ becomes an alternating sum over
-the count of stationary points of $H$ with increasing index, or
-\begin{equation}
-  \chi=\sum_{i=0}^N(-1)^i\mathcal N_H(\text{index}=i)
-\end{equation}
-Conveniently, we can express this abstract sum as an integral over the manifold
-using a small variation on the Kac--Rice formula for counting stationary
-points. Since the sign of the determinant of the Hessian matrix of $H$ at a
-stationary point is equal to its index, if we count stationary points including
-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
-\cite{Fyodorov_2004_Complexity}. Here we are correct to exclude it.
 
-We need to choose a function $H$ for our calculation. Because $\chi$ is
-a topological invariant, any choice will work so long as it does not share some
-symmetry with the underlying manifold, i.e., that it $H$ satisfies the Smale condition. Because our manifold of random
-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.
+\cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
 
-We treat the integral over the implicitly defined manifold $\Omega$ using the
-method of Lagrange multipliers. We introduce one multiplier $\omega_0$ to
-enforce the spherical constraint and $M$ multipliers $\omega_k$ to enforce the vanishing of
-each of the $V_k$, resulting in the Lagrangian
-\begin{equation}
-  L(\mathbf x,\pmb\omega)
-  =H(\mathbf x)+\frac12\omega_0\big(\|\mathbf x\|^2-N\big)
-  +\sum_{k=1}^M\omega_k\big(V_k(\mathbf x)-V_0\big)
-\end{equation}
-The integral over $\Omega$ in \eqref{eq:kac-rice} then becomes
-\begin{equation} \label{eq:kac-rice.lagrange}
-  \chi(\Omega)=\int_{\mathbb R^N} d\mathbf x\int_{\mathbb R^{M+1}}d\pmb\omega
-  \,\delta\big(\partial L(\mathbf x,\pmb\omega)\big)
-  \det\partial\partial L(\mathbf x,\pmb\omega)
-\end{equation}
-where $\partial=[\frac\partial{\partial\mathbf x},\frac\partial{\partial\pmb\omega}]$
-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.
+\paragraph{Acknowledgements}
+The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
 
-In order for certain Gaussian integrals in the following calculation to be
-well-defined, it is necessary to treat instead the Lagrangian problem above
-with $\pmb\omega\mapsto i\pmb\omega$. This transformation does not effect the
-Dirac $\delta$ functions of the gradient, but it does change the determinant by
-a factor of $i^{N+M+1}$. We will see that the result of the rest of the
-calculation neglecting this factor is real. Since the Euler characteristic is
-also necessarily real, this indicates an inconsistency with this transformation
-when $N+M+1$ is odd. In fact, the Euler characteristic is always zero for
-odd-dimensional manifolds. This is the signature of it in this problem.
+\paragraph{Funding information}
+JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
 
-To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with
-\begin{align}
-  \delta\big(\partial L(\mathbf x,\pmb\omega)\big)
-  &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}
-  e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)}
-  \\
-  \det\partial\partial L(\mathbf x,\pmb\omega)
-  &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\,
-  e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]}
-\end{align}
+\appendix
+
+\section{Details of the calculation of the average Euler characteristic}
+\label{sec:euler}
 
 To make the calculation compact, we introduce superspace coordinates. Define the supervectors
 \begin{equation}
@@ -317,8 +579,8 @@ To make the calculation compact, we introduce superspace coordinates. Define the
 The Euler characteristic can be expressed using these supervectors as
 \begin{equation}
   \begin{aligned}
-    \chi(\Omega)
-    &=\int d\pmb\phi\,d\pmb\sigma\,e^{\int d1\,L\big(\pmb\phi(1),\pmb\sigma(1)\big)} \\
+    &\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\{
       \int d1\left[
         H\big(\pmb\phi(1)\big)
@@ -461,163 +723,17 @@ as setting everything depending on $\bar H$ and $H$ to zero. With these solution
   =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1}
   +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H
 \end{equation}
-\begin{equation}
-  D=-\frac{m+R}{1-m^2} \qquad \hat m=0
-\end{equation}
-\begin{equation}
-  \mathcal S(R,D,m,\hat m\mid\alpha,V_0)
-  =\hat m-\frac\alpha2\left[
-    \log\left(1+\frac{f(1)D}{f'(1)R^2}\right)
-    +V_0^2\left(f(1)+\frac{f'(1)R^2}{D}\right)^{-1}
-  \right]
-  +\frac12\log\left(
-    1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2}
-  \right)
-\end{equation}
-When $\alpha<\alpha_\text{onset}$ this potential has maxima at $\pm m^*$ with
-$m^*=-R^*$ where its value is zero.
-\begin{equation}
-  \alpha_\text{onset}=1-\left(\frac{V_0^2}{V_0^2+f(1)}\right)^2
-\end{equation}
-\begin{equation}
-  \alpha_\text{shatter}=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{equation}
-
-\section{Complexity of the height function}
-
-\section{Implications for the spherical spin glasses}
-
-As indicated earlier, for $M=1$ 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 scaling $V_0=\sqrt\alpha E$. With a little algebra this procedure yields
-\begin{equation}
-  E_\text{onset}=\pm\sqrt{2f(1)}
-\end{equation}
-\begin{equation}
-  E_\text{shatter}=\pm\sqrt{4f(1)\left(1-\frac{f(1)}{f'(1)}\right)}
-\end{equation}
-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 $f(q)=\frac12q^p$,
-$E_\text{shatter}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
-models. This is expected, 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.
-
-However, for general mixed models 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 generally satisfies $|E_\text{shatter}|\leq|E_\text{th}|$.
-
-\subsection{What does the average Euler characteristic tell us?}
-
-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.
-
 
+\section{Calculation of the prefactor of the average Euler characteristic}
+\label{sec:prefactor}
 
-\cite{Franz_2016_The, Franz_2017_Universality, Franz_2019_Critical, Annesi_2023_Star-shaped, Baldassi_2023_Typical}
-
-\section{Average number of stationary points of a test function}
-
-\subsection{Behavior with extensively many constraints}
-
-\subsection{Behavior with finitely many constraints}
-
-\begin{equation}
-  Mv_0^2=\frac4{f''(1)}\left[
-    f'(1)-f(1)-2\frac{f(1)}{f'(1)}f''(1)
-  \right]\left[
-    \frac{f(1)}{f'(1)}f''(1)-2\big(f'(1)-f(1)\big)
-  \right]
-\end{equation}
-
-\begin{equation}
-  Mv_0^2=\frac{f'(1)^2}{f''(1)}
-\end{equation}
+\section{Details of the calculation of the average number of stationary points}
+\label{sec:complexity.details}
 
-\section{Interpretation of our results}
+\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
+\label{sec:1frsb}
 
-\[
-  E=\frac1{\hat\omega_1}\frac12\left[
-    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1R-\omega_1^2D)f'(C)+\omega_1^2(R^2-G^2)f''(C)+\log\det\frac{CD+R^2}{G^2}
-  \right]
-\]
-$D=\beta R$, $R=r_dI$, $G=-r_dI$
-\[
-  E=\frac1{\hat\omega_1}\frac12\left[
-    \hat\omega_1^2f(C)+(2\omega_1\hat\omega_1-\omega_1^2\beta)r_df'(1)+\log\det(\beta r_d^{-1}C+I)
-  \right]
-\]
-$\beta=\hat\omega_1/\omega_1$
-\[
-  E=\frac1{\hat\omega_1}\frac12\left[
-    \hat\omega_1^2f(C)+\omega_1\hat\omega_1r_df'(1)+\log\det(\hat\omega_1\omega_1^{-1}r_d^{-1}C+I)
-  \right]
-\]
-$z=r_d\omega_1$
-\[
-  E=\frac1{\hat\omega_1}\frac12\left[
-    \hat\omega_1^2f(C)+\hat \omega_1 zf'(1)+\log\det(\hat\omega_1C/z+I)
-  \right]
-\]
-\[
-  \log\chi
-  =-\hat\omega_1 E
-  +\frac12[2\omega_1\hat\omega_1r_d-\omega_1^2d_d]f'(1)
-  +\frac12\int_0^1dq\,\left[
-    \hat\omega_1^2f''(q)\chi(q)
-    +\frac1{\chi(q)+r_d^2/d_d}
-  \right]
-  -\frac12\log r_d^2
-\]
-\[
-  0=\frac12\hat\omega_1^2f''(q)-\frac12\frac1{[\chi(q)+r_d^2/d_d]^2}
-\]
+\cite{Kent-DObias_2023_How}
 \[
   \chi_0(q)=\frac1{\hat\omega_1}f''(q)^{-1/2}-\frac{r_d^2}{d_d}
 \]
@@ -651,19 +767,6 @@ $z=r_d\omega_1$
   d_d=-\frac{1+r_d}{\int dq\,\chi(q)}r_d
 \]
 
-\paragraph{Acknowledgements}
-The authors thank Pierfrancesco Urbani for helpful conversations on these topics.
-
-\paragraph{Funding information}
-JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN.
-
-\appendix
-
-\section{Calculation of the prefactor of the average Euler characteristic}
-\label{sec:prefactor}
-
-\section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
-\label{sec:1frsb}
 
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