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Statement. Every spatial point belongs to at most $2m$ distinct lines of $L$, and every affine plane contains at most $2m$ such lines.

Source. Mathialagan, published 2021 PDF, pp. 12, 15, Lemma 25.

Proof. The families $L_p^1=\{\ell_{p,q}:q\in Q\}$ and $L_p^2=\{\ell_{q,p}:q\in Q\}$, for $p\in P$, cover $L$. Within any one family the lines are pairwise skew by Proposition 27. A point cannot belong to two skew lines, and a plane cannot contain two skew lines. Each of the $2m$ families therefore contributes at most one line to either count. Coincidences between families only decrease the number of distinct lines. This proves both assertions.

Use and verification. Verified within the independently reviewed Theorem 3 chain, retained in the final review; the caps are applied to both external incidence bounds in Theorem 3 and belong to its living record.

Bears on. Problem 661.