标题： 有理不平衡上的筛法和分母的首次出现 显示英文标题

标题： A Sieve on Rational Imbalances and the First Appearance of Denominators

摘要： 我们构造一个筛子，用于枚举形式为$(p-q)/(p+q)$的有理“不平衡”，其中整数$p\ge2$和$1\le q<p$按照$(p,q)$字典顺序排列。 每个不平衡都化简为最简形式，并记录它们首次出现时的不同分母序列。 我们证明每个正整数恰好作为这样的分母出现一次，且其首次出现与单位分数$1/d$相吻合。 我们随后证明，当将筛子视为从对$(p,q)$到约分分数的映射时，它在$(-1,1)$中枚举所有不重复的有理数，将其对称地扩展到整个$\mathbb{Q}$，并讨论其与双曲几何和有理数枚举理论的联系。 显示英文摘要

摘要： We construct a sieve that enumerates rational ``imbalances'' of the form $(p-q)/(p+q)$ for integers $p\ge2$ and $1\le q<p$, ordered lexicographically by $(p,q)$. Each imbalance is reduced to lowest terms, and we record the sequence of distinct denominators as they first appear. We show that every positive integer occurs exactly once as such a denominator, and that its first appearance coincides with the unit fraction $1/d$. We then prove that the sieve, when viewed as a map from pairs $(p,q)$ to reduced fractions, enumerates all rational numbers in $(-1,1)$ without repetition, extend it symmetrically to all of $\mathbb{Q}$, and discuss its connections to hyperbolic geometry and rational enumeration theory.