ALGORITHMIC TRACKING AND PROBABILISTIC HEURISTICS OF THE KIMBERLING SEQUENCE

Abstract

This preprint investigates the Kimberling Expulsion Conjecture (associated with OEIS A007063), which asserts that every positive integer is eventually expelled from the main diagonal of Kimberling's expulsion array under deterministic iterative repositioning. While the underlying evolution is completely deterministic, the resulting trajectories exhibit highly complicated positional behavior. This paper formulates an explicit $O(1)$-space single-integer position-tracking representation, derives its exact normalized non-autonomous dynamics, and uses that representation as the basis for a rigorous probabilistic heuristic. By defining the normalized position $r_k = p_k/k$, the moving expulsion boundary is reformulated as a shrinking lattice-scale target. The principal unresolved mathematical issue is isolated to a specific hypothesis regarding local lattice-scale equidistribution (asymptotic uniformity). We demonstrate that, conditional on this hypothesis, the conditional expulsion hazard rate decays as $1/k$. A proof is provided showing that this harmonic divergence in the cumulative hazard rate strictly implies eventual almost-sure expulsion within the ensemble, mirroring the classical Borel-Cantelli framework. This paper clarifies the distinction between the exact time-dependent deterministic process and the statistical hypothesis required for expulsion, establishing a formal framework for analyzing the conjecture via non-autonomous dynamical systems and probabilistic heuristics.