---

Source and boundary. These are external premises for Mathialagan's Theorem 3, whose published proof cites Guth--Katz on p. 11 as Theorems 23 and 24. We use the precise statements in Guth and Katz, On the Erdős distinct distances problem in the plane, Annals of Mathematics 181 (2015), 155--190, DOI 10.4007/annals.2015.181.1.2, in the published alternate PDF, identified on its source card. Its proof is not reconstructed here. That source card's own annotations were read from arXiv v3.

External premise A: published Theorem 1.2, printed p. 156, physical p. 2. Let $\mathcal L$ consist of $N^2$ distinct lines in $\mathbb R^3$. Suppose each plane and each regulus contains at most $K N$ of them, for a fixed constant $K$. For $2\leq k\leq N$, the number of points incident to at least $k$ lines is $O_K(N^3k^{-2})$. This is the source's $\lesssim N$ cap and $\lesssim N^3k^{-2}$ conclusion with the fixed cap constant made explicit.

External premise B: published Theorem 4.5, printed p. 176, physical p. 22. For $T$ distinct lines in $\mathbb R^3$ with at most $B$ in any plane, and every integer $k\geq3$, the number $M_k$ of points incident to at least $k$ lines satisfies

$$M_k\leq C\left(T^{3/2}k^{-2}+TBk^{-3}+Tk^{-1}\right), \tag{1}$$

with an absolute constant $C$. There is no regulus cap or upper restriction on $k$ in this premise.

Local normalization of premise A. Suppose $T\geq4$ lines have at most $K\sqrt T$ in every plane and regulus. Put $N=\lceil\sqrt T\rceil$. Add $N^2-T$ arbitrary distinct lines not already present. Such lines exist since the family of all affine lines is infinite. The number added is less than $2N$, so every plane or regulus contains at most $K\sqrt T+2N\leq(K+2)N$ lines of the enlarged set. No general-position condition is required. Every original two-rich point remains two-rich. Since $2\leq N$ and $N\leq\sqrt T+1\leq(3/2)\sqrt T$, premise A gives

$$M_2=O_K(T^{3/2}). \tag{2}$$

This is a proved specialization of the stated premise, not a claim that the theorem was literally stated with an arbitrary nonsquare line count.

Application and normalization issue. The actual family in Theorem 3 has $mn\leq T\leq2mn$, plane and point caps $2m$, and, in its small-distance branch, a regulus cap $8\sqrt{mn}$. Thus the plane cap is at most $2\sqrt T$ and the regulus cap at most $8\sqrt T$. Use (2) for two-rich points and (1) with $B=2m$ for every $k\geq3$. This avoids assuming the literal stricter cap $\sqrt T$ or limiting the higher-richness summation to $\sqrt T$ when it actually runs to $2m$.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review. Exact statement/version fidelity, the padding argument and parameter substitution belong to the living [[distance_problems/mathialagan_2021_bipartite_distinct_distances_plane/theorem_3|Theorem 3]] record. Neither Guth--Katz proof is claimed as compiled or independently reviewed by this result page.

Bears on. Problem 661.