Endpoint dominance for a restarting Josephus process: the frontier at n ≥ 68 and the position of the internal maximum

Abstract

The restarting Josephus process is the threshold recurrence given by F₁(q)=1 and Fₖ(q)=Fₖ₋₁(q)+1_({q mod k<Fₖ₋₁(q)}), whose value Fₙ(q) is the initial position of the survivor in a line of n people counted out with step q+1, the count restarting at the leftmost survivor after every deletion. It depends on q only through q mod Lₙ, Lₙ=lcmop(1,…,n), so the fiber sizes N_(n,j)=#{q mod Lₙ:Fₙ(q)=j} are well defined; both endpoint fibers have size exactly φ(Lₙ). Chen asks whether every internal fiber is strictly smaller — whether the two endpoints are the unique global maxima — and verifies this for 4 ≤ n ≤ 49. We compute the exact fiber-size vectors for 50 ≤ n ≤ 66, reproducing all 49 published rows digit for digit along the way, and find endpoint dominance at every level; since 67 is prime, one formally verified propagation step carries the verification to n=67, so a least counterexample must satisfy n ≥ 68. We then isolate a regularity that the computation exposes and the published account does not record: in all 64 computed rows the internal maximum sits at j=2. We prove that this positional statement alone implies endpoint dominance, with an explicit slack γₙ>0 obtained by carrying a quantitative term through the induction that proves N_(n,2)<φ(Lₙ). Finally we extend the majorization defect D_(n,k) past the published range: the sign pattern persists at every composite n ≤ 66, the record slack min D_(n,k)/Lₙ is beaten by a factor 2.146 at n=65, and D_(n,3) coincides at every level checked with the slack of the composite lift N_(n,2) ≤ ρₙN_(n-1,2), so that the route proposed in the source reduces to the same statement about where the internal maximum sits. Every statement valid for general n is machine-checked in Lean 4; the exact vectors for n ≥ 15 are the output of an independent program in C, and we say throughout which is which.

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