REVIEW
Exact Periodicity, Surjectivity, and a Haar Limit Law for a Restarting Josephus Process
Combinatorics · Submitted 10 Aug 2026 · 18 references
Pith's verdict: This is a genuinely new and mostly airtight paper on the restarting Josephus process; the exact period group, surjectivity, and Haar limit law hold up, but Theorem B's middle range rests on a finite certificate that is not yet auditable from the manuscript.
T0 · AI review
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Referee report
The paper
The paper claims
The paper proves three exact statements for the restarting Josephus process: the full period group is $L_n\mathbb{Z}$, the survivor map is surjective onto $\{1,\ldots,n\}$, and uniform inputs converge to a symmetric nondegenerate Haar…
The reading
What carries the argument
The load-bearing object is the compatible residue vector $c_n(q)=(q\bmod 2,\ldots,q\bmod n)$: it determines the whole deletion sequence, and by the general Chinese remainder theorem its admissible tuples are exactly those with $c_i\equiv c_j\pmod{\gcd(i,j)}$, with one class $q\bmod L_n$ per vector. The surjectivity proof combines this with a prime-power CRT interval construction: prescribing $q\equiv 0\pmod R$ for a prime $R>n/2$, $q\equiv -1\pmod\ell$ for chosen blocking primes $\ell$, and the common residue $P-1$ modulo every other prime power forces the trajectory to increment through the early levels and to skip exactly the blocking levels. For the limit law, the common coupling is a Haar-distributed profinite integer $Q\in\hat{\mathbb{Z}}$; the proof then uses a two-modulus interval covariance inequality, a block-discrepancy estimate for consecutive moduli, and a pathwise sandwich at the state-dependent threshold to obtain almost-sure convergence with $W_1$ error $O(n^{-1/4})$.
Extended reading notes
Core claim
On its own terms, the paper establishes that the restarting Josephus survivor map has no hidden period: $\operatorname{Per}(F_n)=L_n\mathbb{Z}$, proved by freezing trajectories and by using the least recurrence level not dividing a proposed shift as a witness. It then proves that $F_n$ is surjective onto $\{1,\ldots,n\}$: a prime-power CRT construction with blocking primes in $(n/2,n]$ produces intervals of survivor labels, a prime-interval covering criterion joins these intervals, explicit prime estimates handle the tail, and a finite exact certificate closes the remaining range. Finally, it proves a limit law: under a Haar-distributed profinite integer $Q\in\hat{\mathbb{Z}}$, the normalized survivor $X_n=(S_n-1)/(n-1)$ converges to a random variable $Y$ almost surely and in $L^r$ for every finite $r$, the law $\mu$ is symmetric about $1/2$ and nondegenerate, and $\mu([0,\varepsilon])$ and $\mu([1-\varepsilon,1])$ are each at least of order $1/\log(1/\varepsilon)$, which rules out every symmetric $\beta$ law. The paper also proves strict dominance of the endpoint fibers over the two nearest internal positions for every $n\ge 4$, excludes prime levels as minimal counterexamples to endpoint dominance, and verifies endpoint dominance exactly through $n=49$.
Load-bearing premise
The load-bearing assumption is the reliability of the finite certificate behind Theorem B: the paper's own appendix A.3 places the exact-arithmetic program on a view-only link that, as the paper notes, lacks permanent scholarly access, and if that archive does not match the stated hashes and diagnostics, surjectivity is unproved for $229\le n\le 2{,}380{,}428$.
Editorial extensions
If this is right
- Because the period group is exactly $L_n\mathbb{Z}$, any set of fewer than $L_n$ consecutive inputs necessarily misses some survivor outcomes, so the lcm bound is the true repeat length and not an artefact of the recurrence.
- Surjectivity is constructive: for every $n$ and every survivor label $j$, the CRT block construction yields an explicit $q\in\{0,\ldots,L_n-1\}$ with $F_n(q)=j$, and the finite certificate locates the only computer-assisted range.
- The Haar limit law gives $W_1(\mu_n,\mu)=O(n^{-1/4})$, all moments converge, and reflection pins the mean at $1/2$; odd centered moments vanish in the limit.
- The logarithmic boundary mass implies the limit has infinite negative moments and cannot have an $L^p$ density for any $p>1$; in particular, no symmetric beta law can be the limiting distribution.
- For endpoint dominance, the proofs show the two endpoint fibers are strictly larger than the adjacent internal fibers for every $n\ge 4$, and any possible counterexample would have to be composite with $n\ge 50$ and internal position $3\le j\le n-2$.
Reading between the lines
- Beyond the paper: the same Haar coupling could be used to test whether the $W_1$ rate $O(n^{-1/4})$ is sharp or improvable; the paper does not claim optimality.
- Beyond the paper: the deterministic equality $N_{n,1}=N_{n,n}=\varphi(L_n)$ can be read as the finite trace of the logarithmic boundary mass of $\mu$; if endpoint dominance holds, the endpoint events are the unique most probable survivor events at every finite level.
- Beyond the paper: computing $D_{n,3}$ exactly for composite $n>49$ is a natural next experiment, since the paper shows that $D_{n,3}\ge 0$ at every composite level would settle endpoint dominance.
Original 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 $n=49$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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 $n=49$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.60 | 0.44 | 0.03 | asserted 0.88 |
| s1 | For step size $m$, put $q=m-1$, and let $F_n(q)$ denote the initial position of the survivor. | method 0.45 | 0.48 | 0.03 | asserted 0.66 |
| s2 | Reverse insertion gives $F_1(q)=1$ and $F_k(q)=F_{k-1}(q)+\mathbf{1}_{\{q\bmod k<F_{k-1}(q)\}}$. | method 0.83 | 0.71 | 0.03 | asserted 1.00 |
| s3 | Writing $L_n=\operatorname{lcm}(1,\ldots,n)$, we establish three results for the compatible residue system in this recurrence. | contribution 0.89 | 0.90 | 0.07 | asserted 0.99 |
| s4 | First, the full period group of $F_n$ is exactly $L_n\mathbb{Z}$. | result 1.00 | 0.94 | 0.27 | asserted 1.00 |
| s5 | Second, $F_n$ is surjective onto $\{1,\ldots,n\}$. | result 1.00 | 0.93 | 0.20 | asserted 1.00 |
| s6 | The proof is constructive and unconditional but computer-assisted: a Chinese-remainder construction and explicit prime estimates reduce it to a finite exact certificate. | method 0.98 | 0.90 | 0.04 | asserted 1.00 |
| s7 | 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 $[0,1]$. | result 1.00 | 0.92 | 0.30 | asserted 0.99 |
| s8 | 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 $W_1$. | result 0.96 | 0.90 | 0.15 | asserted 1.00 |
| s9 | Logarithmic boundary-mass estimates rule out every symmetric beta law. | result 0.92 | 0.89 | 0.07 | asserted 1.00 |
| s10 | 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 $n=49$. | result 0.42 | 0.94 | 0.06 | asserted 0.98 |
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References
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What this paper does with the arXiv papers it cites
What it cites · 1 sentence without a confident label
Uses the method or tool of 1 paper
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The explicit estimates of [ref] handle the infinite tail, and an exact-arithmetic program certifies the remaining finite range.
Estimates of Some Functions Over Primes without R.H. · Main results · 0.82
Discussion
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