Iterated Distance-to-Prime Function and its Universal Convergence Behaviour

The Iterated Distance-to-Prime (IDP) function d(n), defined as the minimum distance from an integer, n, to the nearest prime, exhibits a unique dynamical behaviour under iteration. This paper explores the behaviour of dk(n), where dk(n) denotes the k-th iteration of d(n) and argues that for most integers n, d2(n) converges to the attractor set {0, 1, 2}. We introduce a density-based argument, showing that the set of exceptions, {n : d2(n) > M}, has a natural density zero for any threshold M. We also identify the ‘Elite’ class of numbers where d2(n) ≥ 4, and provide insights into the role of prime gaps in the rarity of such numbers. Finally, we formalise the limit set and discuss the implications of Cramér’s conjecture [1] in explaining the exponential rarity of large d2(n) values.