Front–Concatenation Shuffles on N: Tail Rigidity, Finite–Core Dynamics, and Complete Classifications in Low Rank

This paper analyzes a generalized two-parameter permutation process on the positive integers, termed the Front-Concatenation (FC) (m, p) -Shuffle, inspired by the classic Kimberling problem. We first analyze the fundamental dynamics of the system, proving the Drift Dominance Theorem, which demonstrates that a strong backward drift dominates the dynamics when the re-injection parameter p>0, suggeWe study a deterministic two--parameter front--concatenation shuffle on the positive integers. At step n one expels the element in position n of the current infinite list, moves the next m elements to the absolute front, and re--injects the preceding p elements (or fewer when n p) immediately after that forward pull. We prove a tail rigidity theorem: for every (m, p) and every time n, the list agrees exactly with the shifted identity beyond position n+m-1, hence the infinite dynamics reduce to a finite recursion on a core of length n+m-2. As a main consequence we completely classify the entire m=1 family for all p 0: the expelled sequence is Eₙ=2n-1, so the process is never exhaustive and expels precisely the odd integers in increasing order. For m=2 we derive a closed core recursion and a distance--to--expulsion dynamics. This yields two universal arithmetic progressions in the expelled sequence for every p. We further give complete, fully proved classifications of the solvable cases (m, p) = (2, 0) and (m, p) = (2, 1) ; in particular, (2, 1) is exhaustive and its termination mechanism reduces to an explicit ``3x-1'' parity cascade. We isolate the remaining obstruction for general (2, p) as a one--dimensional floor--dilation map gᵣ (W) =W+ W/r with r=p+1, and prove gᵣ is injective with an explicit inverse; any hypothetical non--exhaustive case would therefore require an explicit infinite gᵣ--orbit avoiding multiples of r. Finally, Lemma~lem: digit-sensitivity yields an exact survivor tree for the wrap map: avoiding rZ up to depth T depends only on W r^T+1, and there are exactly (r-1) ^T+1 surviving residue classes. Consequently, under uniform base-r digits the hitting time of rZ is geometric, any non-terminating integers (if they exist) have density 0, and the r-adic avoidance set is a Cantor-type fractal of Hausdorff dimension (r-1) / r. sting exhaustivity across this regime. We then investigate the m=1, p>0 family and uncover a stark, parity-driven dichotomy in the structure of the chaos. We provide a rigorous proof for the underlying parity-based mechanism, showing that dynamics exhibit "rapid mixing" for even p and "structured chaos" with long time scales for odd p. Finally, we prove the Collision Lemma, a key combinatorial statement, which resolves the exhaustivity conjecture for the (1, 1) FC-shuffle.