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

We study a deterministic two--parameter \emph{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 $n\le p$) immediately after that forward pull.We prove a \emph{tail rigidity theorem}: for every $(m,p)$ and every time $n$, the list agrees \emph{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 $p\ge 0$: the expelled sequence is $E_n=2n-1$, 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 $g_r(W)=W+\lfloor W/r\rfloor$ with $r=p+1$, and prove $g_r$ is injective with an explicit inverse; any hypothetical non--exhaustive case would therefore require an explicit infinite $g_r$--orbit avoiding multiples of $r$. Finally, Lemma~\ref{lem:digit-sensitivity} yields an exact survivor tree for the wrap map: avoiding $r\mathbb{Z}$ up to depth $T$ depends only on $W\bmod r^{T+1}$, and there are exactly $(r-1)^{T+1}$ surviving residue classes. Consequently, under uniform base-$r$ digits the hitting time of $r\mathbb{Z}$ 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 $\log(r-1)/\log r$.