Exact Periodicity, Surjectivity, and a Haar Limit Law for a Restarting Josephus Process

Abstract

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 $n=49$.