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Source statement and correction. Mathialagan, published 2021 PDF, pp. 9--10, defines the energy using nonzero distances but states $D(P,Q)\geq m^2n^2/|E(P,Q)|$. That exact inequality needs correction when $P$ and $Q$ overlap. Here $|P|=m$, $|Q|=n$, $2\leq m\leq n$, and $s=|P\cap Q|$. Let $D_+$ count positive cross-distances, while $D$ also counts zero if it occurs. The corrected statement is

$$E=\{(p_1,q_1,p_2,q_2)\in P\times Q\times P\times Q: |p_1-q_1|=|p_2-q_2|>0\}, \qquad D\geq D_+\geq\frac{(mn-s)^2}{|E|} \geq\frac{m^2n^2}{4|E|}.$$

Proof. For each positive distance $\delta$ let $e_\delta$ count its ordered pairs in $P\times Q$. Exactly $s$ of the $mn$ pairs have zero distance, so $\sum_\delta e_\delta=mn-s>0$. Two ordered pairs of the same positive distance specify exactly one member of $E$, giving $|E|=\sum_\delta e_\delta^2$. Cauchy--Schwarz gives $(mn-s)^2\leq D_+|E|$. Finally $s\leq m$ and $n\geq2$ imply $mn-s\geq mn/2$. This proves all the displayed inequalities.

For example, if $P=Q$ consists of two points, then $D=2$ and $|E|=4$; the uncorrected right side is $4$. The correction changes only an absolute constant in the later asymptotic deduction.

Dependencies and use. Only the finite Cauchy--Schwarz inequality is used. This is the last energy-to-distance step in Theorem 3.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review; this compilation-supplied overlap correction and its application belong to the living record on Theorem 3.

Bears on. Problem 661.