diff --git a/uniform_trace_reconstruction_v2.txt b/uniform_trace_reconstruction_v2.txt
index 521b377..d7a3aed 100644
--- a/uniform_trace_reconstruction_v2.txt
+++ b/uniform_trace_reconstruction_v2.txt
@@ -410,7 +410,11 @@ exponentially damped Fourier sum
 \end{equation}
 Every monomial in $f_j$ uses positions at most $j$, so
 $\Delta[f_j]=0$ for $j<d$; hence, despite the exponent being centred at
-$d$, only $j\ge d$ contribute. The whole multiscale argument (the next
+$d$, only $j\ge d$ contribute. For the base case $k=1$, where $f_j=X_j$,
+this reads $\mathcal S_f(s,0)=\sum_{j\ge d}\Delta[X_j]\,e^{-s(j-d)}$, whose
+first term is $\Delta[X_d]=\pm1$; that single term is what makes the
+starting signal about $1/2$ at decay $s=4$, as made precise in
+\eqref{eq:initial-signal}. The whole multiscale argument (the next
 two sections) has one goal: to preserve a detectable signal
 $\mathcal S_f(s,\omega)$ while shrinking both $s$ and $|\omega|$ to
 $O(1/n)$, at the cost of raising the order $k$ by at most a factor of