AI 中文总结
该研究针对重启的约瑟夫斯过程，证明了其递推的精确周期性、满射性，还得到相关随机变量的极限定律，提出端点优势为开放问题并完成部分验证。
AI 中文摘要
我们研究一种重启的约瑟夫斯过程，其中参与者保持线性顺序，且每次删除后计数从当前最左侧幸存者处重新开始。对于步长 $m$，令 $q=m-1$，并设 $F_n(q)$ 表示幸存者的初始位置。反向插入给出 $F_1(q)=1$，且 $F_k(q)=F_{k-1}(q)+\mathbf{1}_{\{q\bmod k<F_{k-1}(q)\}}$。记 $L_n=\operatorname{lcm}(1,\ldots,n)$，我们针对该递推中的相容剩余系统建立三个结果：第一，$F_n$ 的完整周期群恰好为 $L_n\mathbb{Z}$；第二，$F_n$ 是到 $\{1,\ldots,n\}$ 的满射映射，其证明具有构造性、无条件性且借助计算机辅助：中国剩余构造与显式素数估计将其简化为有限精确证书；第三，若 $\widetilde Q_n$ 是模 $L_n$ 的均匀分布变量，则 $(F_n(\widetilde Q_n)-1)/(n-1)$ 收敛到 $[0,1]$ 上的对称非退化定律，通过普通哈尔耦合可得到几乎处处收敛及对所有 $1\le r<\infty$ 的 $L^r$ 收敛，同时给出 $W_1$ 中 $O(n^{-1/4})$ 的界，对数边界质量估计排除了所有对称贝塔定律。此外，我们将端点优势表述为开放问题，证明对所有 $n\ge4$，其在两个最近内部位置上具有严格优势，排除素数阶作为最小反例，并通过 $n=49$ 精确验证该结论。
英文摘要
We study a restarting Josephus process in which the participants retain their linear order and counting restarts at the current leftmost survivor after every deletion. For step size $m$, put $q=m-1$, and let $F_n(q)$ denote the initial position of the survivor. Reverse insertion gives $F_1(q)=1$ and $F_k(q)=F_{k-1}(q)+\mathbf{1}_{\{q\bmod k<F_{k-1}(q)\}}$. Writing $L_n=\operatorname{lcm}(1,\ldots,n)$, we establish three results for the compatible residue system in this recurrence. First, the full period group of $F_n$ is exactly $L_n\mathbb{Z}$. Second, $F_n$ is surjective onto $\{1,\ldots,n\}$. The proof is constructive and unconditional but computer-assisted: a Chinese-remainder construction and explicit prime estimates reduce it to a finite exact certificate. Third, if $\widetilde Q_n$ is uniform modulo $L_n$, then $(F_n(\widetilde Q_n)-1)/(n-1)$ converges to a symmetric, nondegenerate law on $[0,1]$. A common Haar coupling yields almost-sure and $L^r$ convergence for every $1\le r<\infty$, together with an $O(n^{-1/4})$ bound in $W_1$. Logarithmic boundary-mass estimates rule out every symmetric beta law. We also formulate endpoint dominance as an open problem, prove strict dominance over the two nearest internal positions for every $n\ge4$, exclude prime levels as minimal counterexamples, and verify the claim exactly through $n=49$.
Comments26 pages, no figures; ancillary files contain verification code and reproduction materials; submitted to European Journal of Combinatorics
