Algebraic Obstructions and the Collapse of Elementary Structure in the Kronecker Problem

While Kronecker coefficients $g(\lambda ,\mu ,\nu )$ with bounded rows are polynomial-time computable via lattice-point methods, no explicit closed-form formulas have been obtained for genuinely three-row cases in the 87 years since Murnaghan’s foundational work. This paper provides such formulas for the first time and identifies a universal structural boundary at parameter value 5 where elementary combinatorial patterns collapse. We analyze two independent families of genuinely three-row coefficients and establish that for $k\le 4$, the formulas exhibit elementary structure: oscillation bounds follow the triangular-Hogben pattern, and polynomial expressions factor completely over $\mathbb{Z}$. At the critical threshold $k=5$, this structure collapses: the triangular pattern fails, and algebraic obstructions – irreducible quadratic factors with negative discriminant – emerge. We develop integer forcing, a proof technique exploiting the tension between continuous asymptotics and discrete integrality. As concrete results, we prove that $g((n,n,1{)}^3)=2-(n\mathrm{m}\mathrm{o}\mathrm{d}2)$ for all $n\ge 3$ – the first explicit formula for a genuinely three-row Kronecker coefficient – derive five explicit polynomial formulas for staircase-hook coefficients, and verify Saxl’s conjecture for 132 three-row partitions.