Cyclotomic multipartitions have equal residue content exactly when Jantzen-connected

Necessity. Suppose $J_{\lambda \mu }\mathrm{\ne }0$; then at least one term of its signed integer sum has nonzero valuation, and evaluating that generic foot difference at $t=1$ gives zero, so the two foot eigenvalues coincide and, since $q_H$ has order $e$, their charged foot residues agree. The two hooks have equal length and the same remainder diagram. Along a rim strip, reading from its lowest-left node to its highest-right node increases the content by one at every step, so its residue multiset consists of $h$ consecutive residues starting at its foot; hence the two removed hooks have identical residue counts and $C(\lambda )=C(\mu )$. Transitivity gives the necessity of $C(\lambda )=C(\mu )$ for every $J$-path.

Sufficiency setup: cores and lifts. Let $\kappa$ be the componentwise core of a member of the rank-$n$ content class; its size is at most $n$ and congruent to $n$ modulo $e$, and by [F2] all such cores have the same hub and are connected by the prescribed switch sequence. Every intermediate core has size at most $n$ and the same hub, so adding $(n-\text{core size})\mathrm{/}e$ $e$-hooks produces a rank-$n$ multipartition of the same content by the equal-hub/equal-size clause of [F2].

The explicit one-hook lift across a core switch. Let ${\kappa }^{\mathrm{\prime }}$ be obtained from $\kappa$ by the oriented switch in rows $i,j$ and components $a,b$ of [F2], with $\gamma \ge 2$, so $\mathrm{\mid }{\kappa }^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }\kappa \mathrm{\mid }-eD$ with $D=\gamma -2\ge 0$. Write $u=i+e{t}_{ia}$, $v=j+e(t_{ja}+1)$, $v^{\mathrm{\prime }}=j+e{t}_{jb}$, $u^{\mathrm{\prime }}=i+e(t_{ib}+1)$: the switch removes bead $u$ and inserts $v$ in component $a$, and removes $v^{\mathrm{\prime }}$ and inserts $u^{\mathrm{\prime }}$ in component $b$. Put $w=(n-\mathrm{\mid }\kappa \mathrm{\mid })\mathrm{/}e\ge 0$, form $\lambda$ from $\kappa$ by moving the lowest bead $u$ of the $i$-runner in component $a$ to $u+ew$ (adding $w$ $e$-hooks on this runner), and form $\mu$ from ${\kappa }^{\mathrm{\prime }}$ by moving its lowest bead $v$ on the $j$-runner in component $a$ to $v+e(w+D)$. Both have size $n$, the same hub, and by [F2] equal content. Their beta sets in component $a$ differ only by replacing $u+ew$ with $v+e(w+D)$, and those in component $b$ only by replacing $v^{\mathrm{\prime }}$ with $u^{\mathrm{\prime }}$. Because $(v-u)+(u^{\mathrm{\prime }}-v^{\mathrm{\prime }})=-eD$, we have $v+e(w+D)-(u+ew)=v^{\mathrm{\prime }}-u^{\mathrm{\prime }}$, so the two components exchange a single rim hook of length $\mathrm{\mid }{v}^{\mathrm{\prime }}-u^{\mathrm{\prime }}\mathrm{\mid }$, nonzero since $i\mathrm{\ne }j$ modulo $e$. If $v^{\mathrm{\prime }}-u^{\mathrm{\prime }}>0$, component $b$ unwraps and $a$ wraps a hook with foot residue $i+1$ modulo $e$; otherwise $a$ unwraps and $b$ wraps one with foot residue $j+1$. In both cases the hook feet have equal residues, and the hook pair is uniquely determined because distinct components change, so the $J$ coefficient of [F1] is a single signed valuation, a nonzero integer with no cancellation. Hence $\lambda$ and $\mu$ are Jantzen-connected by one edge; this supplies the detail abbreviated in the Lyle–Mathas tight lemma.

Termination and conclusion. The generic difference of the unique hook contribution is nonzero by [F1], and its specialization vanishes because the charged feet agree, so its positive integer valuation cannot disappear under reduction modulo $\mathrm{char}F$. The fixed-core lemma [F3] permits replacing arbitrary lifts by the chosen lifts before and after each switch, and strict descent lowers the nonnegative core size by at least $e$, so the process terminates at a minimum core; all minimum cores have equal size and are connected by $\gamma =2$ switches, whose rank-$n$ lifts are connected by the same explicit construction of step 2.1. Together with the fixed-component-size lifts of [F3], every member of the content class is Jantzen-connected to every other, and step 1.1 gives the converse. The empty rank has only one multipartition and is immediate; no affine-center or primitive-block theorem has entered the proof.