The Type III realisation conjecture of Kirkland and \v{S}migoc

Abstract

Kirkland and \v{S}migoc constructed a family of stochastic matrices realising the Type III boundary polynomials in the Karpelevi\v{c} region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range $0<\alpha\le1$, for genuine Type III reduced Ito polynomials of order $n$, $f_\alpha(x)=x^y(x^q-(1-\alpha))^d-\alpha^d$, where $n=qd+y$. For $0<\alpha<1$, the proof first reduces every realisation to a two-shift cyclic normal form using the Dmitriev--Dynkin boundary theorem. The remaining argument is finite and combinatorial: Coates' coefficient formula and the equality case of a weighted Tur\'an theorem force the $q$-cycles associated with the backward edges to split into $d$ complete multipartite classes of equal total weight. A circular-arc telescoping argument then converts this additive equality into the product condition required by Kirkland and \v{S}migoc. The endpoint $\alpha=1$ is treated separately. We also explain why the closed endpoint $\alpha=0$ is degenerate: the literal extension to this endpoint fails, because reducible realisations with closed $q$-cycles and transient states need not contain the global $n$-cycle.