Seven Exact Finite Zarankiewicz Numbers from a Single 13 x 18 Core

Abstract

We present a unified proof and certificate package establishing seven exact finite Zarankiewicz values for the forbidden graph $K_{3,3}$: $z(12,18;3)=108$, $z(13,17;3)=110$, $z(13,18;3)=116$, $z(14,17;3)=118$, $z(14,18;3)=124$, $z(15,17;3)=126$, and $z(15,18;3)=132$. The witnesses form a connected family generated by an explicit $13 \times 18$ matrix with 116 ones. Deleting one row or one column and adding either of exactly two admissible weight-eight rows for this fixed labeled core produces the remaining witnesses. The exceptional upper-bound closure, $z(12,18;3)\leq108$, combines the published uniqueness of the extremal $12 \times 17$ graph with 103 edges and an exhaustive rejection of all $\binom{12}{6}=924$ possible degree-six column extensions. The supplement contains all witnesses, a portable verifier, a machine-readable report, and integrity hashes. Prior numerical ingredients and the role of the present work are separated cell by cell.