Finiteness in Square Classes and Weighted Zero Density of Perfect Cuboids

Abstract

Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, $M(N)\le C^{1+r_N}$ for an absolute constant $C>1$, where $r_N$ is the rank of $y^2=x^3-N^2x$. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove \[ \lim_{X\to\infty}\frac1X \sum_{\substack{N\le X\\N\text{ positive squarefree}}}M(N)^q=0 \qquad\text{for every fixed }q>0. \] Thus the result counts cuboid classes with their full multiplicities at each $N$. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.