diff --git a/book/src/design/circuit/gadgets/ecc.md b/book/src/design/circuit/gadgets/ecc.md
index 491fc791..61fb19e1 100644
--- a/book/src/design/circuit/gadgets/ecc.md
+++ b/book/src/design/circuit/gadgets/ecc.md
@@ -11,7 +11,8 @@ Formulae:
 
 Substituting for $\lambda$, we get the constraints:
 - $(x_r + x_q + x_p) \cdot (x_p - x_q)^2 - (y_p - y_q)^2 = 0$
-  - Note that this constraint is unsatisfiable for $P + (-P)$, and so cannot be used with arbitrary inputs.
+  - Note that this constraint is unsatisfiable for $P \;⸭\; (-P)$ (when $P \neq \mathcal{O}$),
+    and so cannot be used with arbitrary inputs.
 - $(y_r + y_q)(x_p - x_q) - (y_p - y_q)(x_q - x_r) = 0$
 
 
@@ -76,3 +77,8 @@ A \cdot \left(2y_p \cdot \lambda - 3{x_p}^2\right) &=& 0 & A \wedge y_p \neq 0 &
 $
 
 Max degree: $4$
+
+Note: It is the cross-interaction of the two $B$ constraints that fully constrain
+the implications. For example, the contrapositive of the first constraint's implication
+$x_p = 0 \implies B$ is $¬B \implies x_p \neq 0$, which is the other half of the
+second constraint's implication. The same applies to $C$.
diff --git a/book/src/design/circuit/gadgets/ecc/fixed-base-scalar-mul.md b/book/src/design/circuit/gadgets/ecc/fixed-base-scalar-mul.md
index 71a3394c..8d328f01 100644
--- a/book/src/design/circuit/gadgets/ecc/fixed-base-scalar-mul.md
+++ b/book/src/design/circuit/gadgets/ecc/fixed-base-scalar-mul.md
@@ -67,4 +67,5 @@ $$
 
 Then compute $P = [m] \mathcal{V}$, and conditionally negate $P$ using $(x, y) \mapsto (x, s \cdot y)$.
 
-We can reuse the window table from full-width fixed-base scalar multiplication, but with only $\mathsf{ceil}(64 / 3) = 22$ windows.
+We compute the window table in a similar way to full-width fixed-base scalar multiplication,
+but with only $\mathsf{ceil}(64 / 3) = 22$ windows.