From 5834e535839406e6200f6ebdf464a3b0584f081d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 10 Mar 2025 17:00:23 -0300
Subject: Changes addressing report #3, comment #3.2

Added a footnote discussing the intuition for interpreting a large Euler
characteristic as a union manifold rather than a product manifold.
---
 topology.tex | 30 ++++++++++++++++++++++++++++--
 1 file changed, 28 insertions(+), 2 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 32f1a6a..f89a6ca 100644
--- a/topology.tex
+++ b/topology.tex
@@ -322,7 +322,7 @@ and the number of variables, and $R_m$ is a function of $m$ given by
   \Bigg]
   \end{aligned}
 \end{equation}
-This function is plotted in Fig.~\ref{fig:action} for a selection of
+The effective action \eqref{eq:S.m} is plotted in Fig.~\ref{fig:action} for a selection of
 parameters. To finish evaluating the integral by the saddle-point
 approximation, the action should be maximized with respect to $m$. If $m_*$ is
 such a maximum, then the resulting average Euler characteristic is
@@ -354,6 +354,30 @@ parameters are varied.
   } \label{fig:action}
 \end{figure}
 
+The order parameter $m$ may appear similar to the magnetization that appears in
+problems that have a signal or spike, where it gives the overlap of a
+configuration with the hidden signal. Here $\textbf x_0$ is no signal, but a
+direction chosen uniformly at random and with no significance to the set of solutions. If in this problem a feature of the
+action is present at some value $m$, it should be interpreted as indicating
+that, with overwhelming probability, the feature has a proximity to a typical
+point in configuration space given by the overlap $m$. For instance, for $m$
+sufficiently close to one $\mathcal S_\chi(m)$ is always negative, which is a
+result of the absence of any stationary points contributing to the Euler
+characteristic at those overlaps. Given a random height axis $\mathbf x_0$, the
+nearest point to $\mathbf x_0$ on the solution manifold will be the absolute
+maximum of the height function, and therefore contribute to the Euler
+characteristic. Therefore, we can interpret the region of negative action in
+the vicinity of $m=1$ to mean that there is a typical distance between the
+solution manifold and a randomly drawn point in configuration space, and that
+it is vanishingly unlikely to draw a point in configuration space uniformly at
+random and find it any closer to the solution manifold than this.\footnote{
+  Other properties of the set of solutions could be studied by drawing $\mathbf
+  x_0$ from an alternative distribution, like the Boltzmann distribution of the
+  cost function, from the set of its stationary points, or from the solution
+  manifold itself. While the value of the Euler characteristic would not
+  change, the dependence of the effective action on $m$ would change.
+}
+
 \subsection{Features of the effective action}
 
 The order parameter $m$ is the overlap of the
@@ -546,7 +570,9 @@ than the shattering value $V_\text{sh}$ and $\mathcal S(0)>0$. Above the
 shattering transition the effective action is real everywhere, and its value at
 the equator is the dominant contribution. Large connected components of the
 manifold may or may not exist, but small disconnected components outnumber
-holes.
+holes.\footnote{
+  We interpret the large Euler characteristic to indicate a manifold with many (topologically) spherical disconnected components because the manifold is formed by the process of repeatedly taking non-self-intersecting slices of the previous manifold, starting with a sphere. Therefore, an outcome consisting mostly of (topological) spheres seems most plausible. However, a large Euler characteristic is also consistent with a variety of connected product manifolds, among other exotic possibilities. Definitely ruling out such scenarios is not within the scope of this paper.
+}
 
 \paragraph{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.}
 
-- 
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