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

For a base $B\ge2$, digit length $k$, and permutation $\pi\in S_k$, we study the arithmetic dynamical system $F_\pi$ obtained by subtracting each local base-$B$ numeral along the disjoint cycles of $\pi$ 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 $\pi$, uniformly across all bases and cycle types. The cycle structure yields an exact direct-product decomposition of $F_\pi$ into local maps $g_t$, and each $g_t$ is semiconjugate, after identifying $x\sim-x$, to multiplication by\[c=(B-1)B^{t-1}\]on $\mathbb Z/(B^t-1)\mathbb Z$. Writing\[B^t-1=N_1N_2,\]where every prime divisor of $N_1$ divides $B-1$ and $\gcd(N_2,B-1)=1$, this arithmetic reduction determines the exact fixed-point count, the criterion for total collapse, and the complete attracting-cycle structure. Each local map has $(N_2+1)/2$ periodic states. Their individual periods are determined by the divisor structure of $N_2$: for a periodic class represented by $N_1r$, the minimal period is the least positive integer $j$ such that $c^j\equiv\pm1$ modulo $N_2/\gcd(r,N_2)$. The quantity $P_*:=\operatorname{ord}_{(\mathbb Z/N_2\mathbb Z)^\times/\{\pm1\}}(c)$ is the least common multiple of all periodic-orbit lengths. Every state reaches the periodic set after exactly $a_0$ or $a_0+1$ iterations, where $a_0$ is the least integer satisfying $N_1\mid c^{a_0}$. The direct-product decomposition then yields a closed formula for the eventual rank of $F_\pi$ for an arbitrary cycle type. For a two-cycle, this rank is $(\operatorname{odd}(B+1)+1)/2$, giving the identification with OEIS~A003602 as a direct consequence of the underlying $2$-adic decomposition.