Small wording tweaks in footnote.

1 files changed, 4 insertions, 4 deletions

diff --git a/topology.tex b/topology.tex
index a55a5c4..cb95e01 100644
--- a/topology.tex
+++ b/topology.tex
@@ -365,15 +365,15 @@ 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. Here, if a feature of the
 action is present at some value $m$, it should be interpreted as indicating
-that, with overwhelming probability, the feature is found in proximity to a typical
-point in configuration space given by the overlap $m$. For instance, for $m$
+that, with overwhelming probability, typical configurations contributing to that feature have an overlap $m$ with a typical
+point in configuration space. For instance, for $m$
 sufficiently close to 1, $\mathcal S_\chi(m)$ 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
+maximum of the height function, and therefore will contribute to the Euler
 characteristic. Hence the region of negative action in
-the vicinity of $m=1$ implies there is a typical distance between the
+the vicinity of $m=1$ implies there is a typical minimum 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.