{"paper":{"title":"Exact Periodicity, Surjectivity, and a Haar Limit Law for a Restarting Josephus Process","arxiv_id":"2608.09092","primary_cat":"math.CO","submitted_at":"2026-08-10T03:43:27+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.6,"id":"s0","own":false,"role":"contribution","text":"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.","hedge":{"p":0.88,"label":"asserted"},"own_p":0.44,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.0,"background":0.39,"limitation":0.0,"future_work":0.0,"contribution":0.6,"interpretation":0.0}},{"i":1,"p":0.45,"id":"s1","own":false,"role":"method","text":"For step size $m$, put $q=m-1$, and let $F_n(q)$ denote the initial position of the survivor.","hedge":{"p":0.66,"label":"asserted"},"own_p":0.48,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.3,"role_probabilities":{"gap":0.0,"method":0.45,"result":0.0,"background":0.4,"limitation":0.0,"future_work":0.0,"contribution":0.15,"interpretation":0.0}},{"i":2,"p":0.83,"id":"s2","own":true,"role":"method","text":"Reverse insertion gives $F_1(q)=1$ and $F_k(q)=F_{k-1}(q)+\\mathbf{1}_{\\{q\\bmod k<F_{k-1}(q)\\}}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.71,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.57,"role_probabilities":{"gap":0.0,"method":0.83,"result":0.02,"background":0.08,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":3,"p":0.89,"id":"s3","own":true,"role":"contribution","text":"Writing $L_n=\\operatorname{lcm}(1,\\ldots,n)$, we establish three results for the compatible residue system in this recurrence.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.9,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.29,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.08,"background":0.02,"limitation":0.0,"future_work":0.0,"contribution":0.89,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":true,"role":"result","text":"First, the full period group of $F_n$ is exactly $L_n\\mathbb{Z}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.27,"quantitative":false,"quantitative_p":0.43,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"Second, $F_n$ is surjective onto $\\{1,\\ldots,n\\}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.2,"quantitative":false,"quantitative_p":0.33,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":0.98,"id":"s6","own":true,"role":"method","text":"The proof is constructive and unconditional but computer-assisted: a Chinese-remainder construction and explicit prime estimates reduce it to a finite exact certificate.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.0,"background":0.0,"limitation":0.01,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":7,"p":1.0,"id":"s7","own":true,"role":"result","text":"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 $[0,1]$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.92,"main_claim_p":0.3,"quantitative":false,"quantitative_p":0.41,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":8,"p":0.96,"id":"s8","own":true,"role":"result","text":"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 $W_1$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.15,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":9,"p":0.92,"id":"s9","own":true,"role":"result","text":"Logarithmic boundary-mass estimates rule out every symmetric beta law.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.92,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.06}},{"i":10,"p":0.42,"id":"s10","own":true,"role":"result","text":"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 $n=49$.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.94,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.42,"background":0.0,"limitation":0.0,"future_work":0.39,"contribution":0.19,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.65,"text":"Third, if $\\widetilde Q_n$ is uniform modulo $L_n$, then $(F_n(\\widetilde Q_n)-1)/(n-1)$ converges to a 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