Simultaneous Novelty from First-Appearance Times in the Calkin-Wilf Enumeration
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arXiv
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| Format: | Preprint |
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2025
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| author | Bilokon, Paul Alexander |
| author_facet | Bilokon, Paul Alexander |
| contents | We study the first-appearance map $π:\mathbb{N}_{\ge2}\to\mathbb{N}_0$ that assigns to each denominator $d$ the earliest breadth-first index at which a reduced fraction of denominator $d$ occurs in the Calkin-Wilf enumeration of $\mathbb{Q}_{>0}$. In parallel, we consider the elementary denominator-first array $D=\big(U(2)\mid U(3)\mid U(4)\mid\cdots\big)$ with rows $U(a)=(1/a,2/a,\dots,(a-1)/a)$ and row-starts $i_0(a)=\frac{(a-2)(a-1)}{2}$. We say level $a$ locks if $π(a)=i_0(a)$. Our main theorem is purely combinatorial: for every $n\ge2$ there exists $i\in\{0,\dots,n-2\}$ such that the first appearances of denominators $n-i$ and $n+i$ align symmetrically around $i_0(n)$, i.e.\ $π(n\pm i)=i_0(n)\pm i$. We prove this pairing (or simultaneous novelty) theorem via a local-coherence analysis of $π$ around a level and a discrete intermediate-value argument. An equivalent group-theoretic restatement uses the free monoid $\langle L,R\rangle\subset SL_2(\mathbb{Z})$ underlying the Calkin-Wilf and Stern-Brocot trees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous Novelty from First-Appearance Times in the Calkin-Wilf Enumeration Bilokon, Paul Alexander General Mathematics 05 G.2.1 We study the first-appearance map $π:\mathbb{N}_{\ge2}\to\mathbb{N}_0$ that assigns to each denominator $d$ the earliest breadth-first index at which a reduced fraction of denominator $d$ occurs in the Calkin-Wilf enumeration of $\mathbb{Q}_{>0}$. In parallel, we consider the elementary denominator-first array $D=\big(U(2)\mid U(3)\mid U(4)\mid\cdots\big)$ with rows $U(a)=(1/a,2/a,\dots,(a-1)/a)$ and row-starts $i_0(a)=\frac{(a-2)(a-1)}{2}$. We say level $a$ locks if $π(a)=i_0(a)$. Our main theorem is purely combinatorial: for every $n\ge2$ there exists $i\in\{0,\dots,n-2\}$ such that the first appearances of denominators $n-i$ and $n+i$ align symmetrically around $i_0(n)$, i.e.\ $π(n\pm i)=i_0(n)\pm i$. We prove this pairing (or simultaneous novelty) theorem via a local-coherence analysis of $π$ around a level and a discrete intermediate-value argument. An equivalent group-theoretic restatement uses the free monoid $\langle L,R\rangle\subset SL_2(\mathbb{Z})$ underlying the Calkin-Wilf and Stern-Brocot trees. |
| title | Simultaneous Novelty from First-Appearance Times in the Calkin-Wilf Enumeration |
| topic | General Mathematics 05 G.2.1 |
| url | https://arxiv.org/abs/2509.10485 |