四列递归线Zarankiewicz数
Recursive-Line Zarankiewicz Numbers with Four Columns
- School of Mathematical Sciences, South China Normal University（华南师范大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本研究通过AI辅助与证书验证，确定了四列递归线Zarankiewicz数的值，并证明了m≥15时的精确公式，揭示了极值配置的洞结构。
中文摘要 AI 辅助
递归线Zarankiewicz数最大化由极值无$C_4$二分图增广编码的结构化不可约平方和表示中的平方数。我们在Löfberg和Qi于2026年9月9日的手稿中强化的递归定义下确定其四列行为。结合AI辅助发现与精确证书验证及有限排除计算，我们确定了$2\le m\le20$中十九个值中的十八个，并将唯一未解决情形限定为$37\le\zr(14,4)\le38$。更重要的是，我们证明了四列情形下首个最终精确公式：\\[ \zr(m,4)=\floor{\frac{5m+6}{2}}\qquad(m\ge15). \\] 上界由经典恒等式$z(m,4)=m+6$和精确单元计数得出。对于匹配的下界，我们从显式$20\times4$种子构造两行扩展链，并通过固定删除推导奇数阶。解析传播连同两个独立审计的符号证书表证明了该构造对任意链长成立，而不仅限于有限计算范围。因此，当$m$为偶数时每个极值配置无洞，当$m$为奇数时恰有一个洞，且相同精确公式对二阶数$z_2(m,4)$也成立。
英文摘要
The recursive-line Zarankiewicz number maximizes the number of squares in a structured irreducible sum-of-squares representation encoded by an augmentation of an extremal $C_4$-free bipartite graph. We determine its four-column behavior under the strengthened recursive definition in the manuscript of Löfberg and Qi dated 9 September 2026. Combining AI-assisted discovery with exact certificate verification and finite exclusion computations, we determine eighteen of the nineteen values for $2\le m\le20$ and isolate the only unresolved case to $37\le\zr(14,4)\le38$. More significantly, we prove the first eventual exact formula in the four-column setting: \[ \zr(m,4)=\floor{\frac{5m+6}{2}}\qquad(m\ge15). \] The upper bound follows from the classical identity $z(m,4)=m+6$ and a sharp cell count. For the matching lower bound, we construct a two-row extension chain from an explicit $20\times4$ seed and derive the odd orders by a fixed deletion. Analytic propagation, together with two independently audited symbolic certificate tables, proves the construction for arbitrary chain length rather than merely for a finite computational range. Thus every extremal configuration has no holes when $m$ is even and exactly one hole when $m$ is odd, and the same exact formula holds for the second-order number $z_2(m,4)$.
