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Statement. In the notation of Proposition 19, let $E^{\rm tr}$ consist of the quadruples whose motion in Proposition 20 is a translation. Then $|E^{\rm tr}|\leq m^2n$. The printed statement is $|E^{\rm tr}(P,Q)|=O(m^2n)$; its proof gives the explicit bound stated here.

Source and proof. Mathialagan, published 2021 PDF, p. 10, Proposition 21. Choose $p_1,p_2\in P$ and $q_1\in Q$. The only translation sending $q_1$ to $p_2$ has vector $p_2-q_1$. It must send $p_1$ to $q_2=p_1+p_2-q_1$. Thus there is at most one admissible $q_2\in Q$, and some choices do not give a positive-energy quadruple. There are $m^2n$ choices of the first three entries, proving the bound.

Use. Theorem 3 handles the remaining rotation energy by line incidences.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review; included in the living Theorem 3 record. No external theorem is needed.

Bears on. Problem 661.