AI 中文总结
本文证明正有理完美长方体的相似类数量在无平方因子不变量下有限，并利用秩障碍和二次扭结果建立加权零密度极限，获得定量衰减估计。
AI 中文摘要
设 $M(N)$ 为具有有序边 $a,b,c$、空间对角线 $g$ 以及无平方因子不变量 $N=\operatorname{sf}(abcg/2)$ 的正有理完美长方体的相似类数量。我们证明对于每个正无平方因子 $N$，$M(N)$ 是有限的，且对相关联的合数椭圆曲线的秩没有任何限制。更精确地，$M(N)\le C^{1+r_N}$，其中 $C>1$ 是绝对常数，$r_N$ 是 $y^2=x^3-N^2x$ 的秩。利用已建立的 Paulsen-West 秩障碍和 Smith 关于二次扭的结果，我们随后证明 \\[ \lim_{X\to\infty}\frac1X \sum_{\substack{N\le X\N\text{ 正无平方因子}}}M(N)^q=0 \qquad\text{对每个固定的 }q>0. \\] 因此该结果以每个 $N$ 处的完全重数计数长方体类。还获得了定量衰减估计。几何步骤排除了通过相容性曲面正部分的每个正维实阿贝尔陪集；定量 Mordell-Lang 随后给出所需界限。该论证是无条件的，并使用长方体对应和秩障碍作为先前结果。
英文摘要
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.
Comments11 pages. Does not claim a proof of the nonexistence of perfect cuboids, nor the finiteness of the total number of cuboids across all parameters
