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

Abstract

While Kronecker coefficients $g(λ,μ,ν)$ 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 $k \leq 4$, 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 $g((n,n,1)^3) = 2 - (n \mod 2)$ for all $n \geq 3$ -- 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.