The Single-Generator Theory of Cycle-Decomposed Digit-Subtraction Dynamics

For a base B≥2, digit length k, and permutation π∈ Sₖ, we study the arithmetic dynamical system F_π obtained by subtracting each local base-B numeral along the disjoint cycles of π from its cyclic rotation and scattering the resulting digits back to their original positions. We establish a complete closed-form theory for a single generator π, uniformly across all bases and cycle types. The cycle structure yields an exact direct-product decomposition of F_π into local maps gₜ, and each gₜ is semiconjugate, after identifying x~-x, to multiplication by\[c=(B-1)Bᵗ⁻¹\]on Z/(Bᵗ-1) Z. Writing\[B^t-1=N_1N_2,\]where every prime divisor of N₁ divides $B-1$ and (N₂,B-1)=1, this arithmetic reduction determines the exact fixed-point count, the criterion for total collapse, and the complete attracting-cycle structure. Each local map has (N₂+1)/2 periodic states. Their individual periods are determined by the divisor structure of N₂: for a periodic class represented by N₁r, the minimal period is the least positive integer j such that cʲ≡±1 modulo N₂/(r,N₂). The quantity P_*:=ord_( Z/N₂ Z)^×/\±1\(c) is the least common multiple of all periodic-orbit lengths. Every state reaches the periodic set after exactly a₀ or a₀+1 iterations, where a₀ is the least integer satisfying N₁ ca₀. The direct-product decomposition then yields a closed formula for the eventual rank of F_π for an arbitrary cycle type. For a two-cycle, this rank is (odd(B+1)+1)/2, giving the identification with OEIS~A003602 as a direct consequence of the underlying $2$-adic decomposition.