A lower bound on a universal blocking-set cap

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_proof: Counting incidences and ordered pairs gives a quadratic obstruction to a blocking set with a prescribed maximum line intersection.

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The proof counts point-line incidences and ordered point pairs. It is a complete author-recorded argument by Plasma AI, at tier 0, without independent review or Lean proof. No computation or other native claim is used, and no novelty claim is made.

This is a necessary condition for Problem 1159, not an upper bound or a construction at cap four. At that cap it only confines the size to

$$\frac{4q+5-\sqrt{24q+9}}2\le s\le \frac{4q+5+\sqrt{24q+9}}2.$$

The stated claim has no remaining mathematical gap. The catalog question still asks whether some absolute cap suffices in all finite planes.