The 2-Adic Valuation of the Order of the All-Ones Class in the Sandpile Group of a Square

Place one grain at every nonsink vertex of the wired $n\times n$ square, and let $L(n)$ be the order of this operation in the sandpile group. Thus $L(n)$ is the least positive $q$ for which $q$ uniform grain layers form an integral combination of toppling moves. We prove that, for every $n\ge 1$, $${\nu }_2(L(n))=\{\begin{array}{ll}2,& n=1,\\ 1,& n\ge 2\text{ even},\\ {\nu }_2(n+1)+2,& n\ge 3\text{ odd}\mathrm{.}\end{array}$$ For even squares, this follows from the domino–sandpile results of Florescu, Morar, Perkinson, Salter, and Xu, completed by a short parity observation. For odd squares, a unimodular cyclic basis identifies the folded cokernel with a quotient by two shifted Chebyshev polynomials and sends the all-ones class to $1$. Its order is determined by the constant part of this polynomial ideal, not just by a determinant. Two normalized Euclidean remainders reduce to consecutive Fibonacci polynomials over ${\mathbb{F}}_2$, giving the exact valuation.