Second Order Zarankiewicz Number

Source abstract

We introduce the \emph{second order Zarankiewicz number} $z_2(m,n)$, defined as the maximum SOS rank of an $m \times n$ irreducible double simple biquadratic form. We also introduce the recursive-line parameter $z_{RL}(m,n)$, obtained by keeping $(S)$ and excluding $(W1),(W2),(W2')$, and requiring the strengthened recursive rectangle certificate $(RW3^+)$. The unconditional hierarchy is $$\operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n).$$ We exhibit an explicit $5\times 4$ irreducible form with SOS rank $13$, whereas $z_{wL}(5,4)=12$, proving $z_2(5,4)>z_{wL}(5,4)$. The construction violates $(W3)$ yet remains irreducible through recursive rectangle identities. We formalize this mechanism as $(RW3)$, then strengthen it by adjoining complementary-pair checking to obtain $(RW3^+)$, which is strictly weaker than literal $(W3)$ on the common weak ambient class. Exact search gives $z_{RL}(5,4)=13>12=z_{wL}(5,4)$. The three-column equality pattern $z_{RL}(m,3)=z_{wL}(m,3)$ is refuted by a certified construction at $(10,3)$, giving $z_{RL}(10,3)\ge 20>19=z_{wL}(10,3)$. On the $7\times 7$ benchmark, $(RW3)$ accepts the weak optimum $28$, certifies seven one-edge extensions to total $29$, and a greedy search reaches $31$, so $z_{RL}(7,7)\ge 31>28=z_{wL}(7,7)$. Finally, along the complete-graph incidence family, we prove an unconditional cubic asymptotic separation: $$z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3.$$ Thus the gap between the second-order and weak frameworks is not only a finite-dimensional phenomenon but persists asymptotically.