The e-core is independent of the rim hook removal order

Fix a $\beta$-set $X$ of a partition $\mu$ and write its elements as $h_1>\cdots >h_m$. By [F1] the hooks of length $e$ of $\mu$ correspond bijectively to the beads $b=h_i$ with $h_i\ge e$ and $h_i-e\mathrm{\notin }X$, the correspondence sending such a bead to the rim strip $R_{i,j}(\mu )$ attached to the node with $h_{i,j}(\mu )=e$, and removal of that strip replaces $X$ by $X^{\mathrm{\prime }}=(X\setminus \{h_i\})\cup \{h_i-e\}$, a $\beta$-set for $\mu \setminus R_{i,j}(\mu )$ of the same size $m$. In particular $\mu$ has no hook of length $e$ exactly when no bead of $X$ has a gap in the position $e$ labels above it on its own runner.

Compute the shift directly. For $Y=X+1$, its runner sets are $Y_0^{(e)}=(X_{e-1}^{(e)})+1$ and $Y_i^{(e)}=X_{i-1}^{(e)}$ for $1\le i<e$: the last runner wraps to runner $0$ one row lower, and the shift adds the bead at $0$. Its counts are $(c_{e-1}+1,c_0,\dots ,c_{e-2})$. Packing with these counts gives $Y^{(e)}=(X^{(e)})+1$, so $P(Y^{(e)})=P(X^{(e)})$ by [F3]. Iteration proves core independence for all beta-sets of $\lambda$. After $e$ shifts, every runner row-label set has undergone one beta-set shift by $1$, so its partition is unchanged. Normalised beta-sets differ by a multiple of $e$, proving independence of the ordered quotient and its total size.

Call a bead of $X$ movable when the position $e$ labels above it on its runner is a gap, so that by step 1.1 the movable beads are exactly the hooks of length $e$ of $\mu$; moving a bead up one row replaces $X$ by a $\beta$-set of $\mu$ minus that hook, and the counts $\mathrm{\mid }{X}_i^{(e)}\mathrm{\mid }$ of beads on each runner are unchanged. A configuration is top-packed precisely when it has no movable bead: a top-packed runner carries beads in the rows $0,1,\dots ,c_i-1$, so the row above a bead is a bead again (and row $0$ has no row above it), while if some runner is not top-packed there are a gap in row $g$ and a bead in a row $a>g$ on that runner; choosing $a$ minimal among the bead rows larger than $g$, the row $a-1$ is a gap, so the bead in row $a$ is movable.

Every sequence of moves of steps 1.1 and 2.1 is finite: the quantity $\mathrm{\Phi }:={\sum }_i{\sum }_{a\in X_i^{(e)}}a$ is the total of the row labels of all beads, and each move replaces a bead's row label $a$ by $a-1$, so $\mathrm{\Phi }$ decreases by exactly $1$ and is a nonnegative integer. A sequence is maximal exactly when it stops at a configuration with no movable bead, that is, by step 2.1, exactly when it stops at the top-packed configuration of the counts $(c_i)$; and that configuration is unique, since a runner with $c_i$ beads has exactly one top-packed state, the rows $0,\dots ,c_i-1$. Hence every maximal sequence of moves starting from $X$ ends at the same configuration, namely $X^{(e)}$, and therefore at the same terminal partition $P(X^{(e)})$.

By step 1.2 the partition $P(X^{(e)})$ does not depend on the chosen $\beta$-set $X$, so $C_e(\lambda ):=P(X^{(e)})$ is well-defined; by step 3.1 it is the common terminal partition of all maximal sequences of e-hook removals, and by step 2.1 a terminal configuration is top-packed, hence carries no movable bead, hence its partition has no hook of length $e$ by step 1.1; thus $C_e(\lambda )$ is an $e$-core. Every strip removed along such a sequence corresponds to a node hook of length exactly $e$ and so has size $e$ by [F4]. This proves all the asserted properties.