来自单个13×18核心的七个精确有限Zarankiewicz数
Seven Exact Finite Zarankiewicz Numbers from a Single 13 x 18 Core
AI总结：
该研究提出统一证明与证书包，确立了禁图K_{3,3}的七个精确有限Zarankiewicz数，以含116个1的13×18矩阵为核心生成见证，补充材料含验证工具等内容。
AI中文摘要：
我们提出一个统一的证明与证书包，确立了禁图K_{3,3}的七个精确有限Zarankiewicz值：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和z(15,18;3)=132。这些见证构成一个连通族，由一个含116个1的显式13×18矩阵生成。删除该固定标记核心的一行或一列，并添加恰好两个可容许的权8行中的任意一个，即可得到其余见证。例外的上界闭包z(12,18;3)≤108，结合了已发表的含103条边的极值12×17图的唯一性，以及对所有binom(12,6)=924个可能的度6列扩展的穷举拒绝。补充材料包含所有见证、便携式验证器、机器可读报告和完整性哈希。我们还逐格区分了先前的数值成分与本工作的作用。
英文摘要：
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.
