A Complete Classification and Reduction of the Generalized Kimberling Shuffle
Year: 2025
Type: article
Abstract: This paper introduces a two-parameter front-concatenation shuffle on the positive integers. In position n the current entry is expelled, the next m entries move to the front, and up to p predecessors move behind them. This generates an expelled sequence and a question of exhaustivity: is every integer eventually expelled? A tail-rigidity theorem reduces the infinite process to a finite core of length n+m-2. The paper solves the case m=1 for every p and proves exhaustivity for the parameters (m,p) equal to (2,0) and (2,1). For every m at least 2 and p at least 0, writing T=m+p, two arithmetic progressions in the expelled sequence give an exact linear law for its running maximum, so the asymptotic coverage of the expelled set converges to T/(2(T-1)) independently of whether the shuffle is exhaustive. A single scalar renewal map on the integers, stopped on the multiples of q=m+p-1, terminates universally exactly when the shuffle is exhaustive, and its geometric finite-time law transfers to the label streams: the number of integers up to X that are never expelled is of order X raised to the power log(q-1)/log(q). An endpoint translation shows that (m,0) is exhaustive exactly when (2,m-2) is, which settles (3,0) and identifies both (2,2) and (4,0) with a canonical open termination problem at the 4/3 endpoint studied by Dubickas and Mossinghoff. The canonical map embeds in one member of the renewal family along a single residue progression modulo 17. The paper proves this is the unique nontrivial invariant progression for q at least 3 and classifies every signed integral affine intertwiner among the renewal maps, with modulus 17 the sole nontrivial exception. The 4/3 problem is therefore affinely rigid within this family rather than one of many interchangeable encodings. At the canonical endpoint, centering the standard digit coding turns termination into the statement that a strongly separated 3-adic Cantor set meets the integers only at minus one and plus one. That set realizes every finite sign word, and its level sets carry an exact Riesz product, a full sumset and difference set, energy 6 to the power L, and a normalized Fourier coefficient of minus one half at every depth, while their prime-power Ramanujan sums vanish termwise beyond modulus 3. Consecutive orbit ratios can only be Farey neighbors of 4/3, and constant-sign runs are governed exactly by the 3-adic valuations of z-1 and z+1. These identities expose the limits of naive harmonic, continued-fraction, and bounded-memory attacks: they sharply constrain the termination problem but do not solve it.
Author: Fathi, Kevin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7090262501
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