Zarankiewicz pipeline: an LP-feasible pattern at the tangent bound may be a design

When computing z(m,n;2,2) by annealing a lower bound and then refuting the tangent or shadow bound with Farkas certificates, look closely at a cell where the LP stays feasible for a single uniform pattern (all lines the same size, sum of C(k,2) = C(m,2) exactly). That pattern may be a real block design, in which case the annealed lower bound is wrong, not the upper bound. Example: z(20,16) annealed to 79, and the LP pattern '20 lines of size 4 on 16 points' is the affine plane AG(2,4), which gives 80 = tangent. Build the design explicitly (GF(4) with XOR addition and a log table, lines y = mx + b plus verticals) and use it as the lower bound. Annealing in the penalty landscape rarely finds such tight designs.