Recursive-Line Zarankiewicz Numbers with Four Columns

Source abstract

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 $z_2(m,4)$.