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Source statement and compiled scope. Mathialagan, published 2021 PDF, pp. 16--17, Lemma 34, states that if $|A|\geq2$ and $B$ consists of $x$ points of a one-dimensional algebraic variety of degree $D$, then $D(A,B)=\Omega_D(x)$. The source proves this with its preceding algebraic geometry inputs.

Only a line or a circle is needed by Theorem 3. We compile these cases completely, with the explicit bound

$$|B|\leq 2D(A,B). \tag{1}$$

The unused general algebraic-variety assertion and its external Bézout/component-counting dependencies are not claimed as compiled here.

Proof for a line. Choose any $a\in A$. A circle centered at $a$ of positive radius intersects a line in at most two points: parameterizing the line and imposing the squared radius gives a nonconstant quadratic equation with positive leading coefficient. Radius zero contributes at most one point. Thus each distance from $a$ accounts for at most two points of $B$, which proves (1).

Proof for a circle. Let the containing circle have center $q$ and positive radius $r$. Since $A$ contains two distinct points, choose $a\in A\setminus\{q\}$. A circle centered at $a$ and the circle centered at $q$ have at most two common points: subtracting their squared equations gives a genuine line because their centers differ, and the preceding quadratic argument applies. Again distance zero contributes at most one point. Every distance from $a$ accounts for at most two elements of $B$.

Application. With $A=P$ and $B=Q\cap\gamma$, where $\gamma$ is any line or circle, (1) gives $|Q\cap\gamma|\leq2D(P,Q)$. This includes overlapping $P,Q$. A circle of radius zero contains at most one point and is never needed in the regulus constructions.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review; the exact line/circle specialization and its use in Lemma 26 belong to the living Theorem 3 record.

Bears on. Problem 661.