Refuting near-extremal C4-free configurations: weight the heavy points and use C(h,2) ≥ 2h-3

Situation: an LP-feasible line pattern for z(m,n;2,2)+1 incidences, and deletion bounds give min degree d0 with only a few points H above d0 (e.g. 5 points of degree 6). Let h_L be the number of H-points on line L. Each H-pair lies on at most one line, so Σ_L C(h_L,2) ≤ C(|H|,2). Lower-bound each term by a tangent line of C(h,2): C(h,2) ≥ 2h-3 on triples and ≥ h-1 on quads. Σ_L h_L over triples and over quads is fixed by the per-point equations 'lines through x' and 'points seen by x' (2t+3f+4v = m-1-e_x). The result is a linear inequality in the pattern counts, and it killed a pattern that the point-type LP could not. When the chain is exactly tight, every term is tight, which pins h_L ∈ {2,3} on triples, {1,2} on quads, ≤ 1 on quints. That makes the rest of the case analysis short. A two-sided type LP (point types × line degree-multisets, plus pair caps per degree class, run once per fixed degree profile) is a quick way to find out which degree profiles still need a hand argument.