Small Zarankiewicz data: where the tangent/Reiman bound for z(m,n;2,2) fails

Exact values from an exhaustive branch-and-bound search agree with the published small tables. For all m,n ≤ 13, the integer tangent bound ⌊(n·C(j+1,2) + C(m,2))/j⌋ (both orientations, best j) is exact, except for the near-squares (8,8), (9,8), (9,9), (10,9), (10,10), (11,10), (11,11), (12,12). Those are fixed by a 'heavy row' case split, and (10,8), (11,9), (13,10) by a parity argument when the tangent bound has zero slack. First genuine failures of all three counting tools: z(14,11)=45 (bound 46) and z(14,12)=49 (bound 50); z(15,15)=61 (bound 62). For z(14,14)=56, scan the heavy-row threshold: θ=4 gives 56 exactly, while θ=3 gives only 58. Constructions for every cell up to 13×13 come from sub-configurations of PG(2,3) (translates of {0,1,3,9} mod 13) or from simulated annealing.