Multipartitions with fixed component cores are cyclotomic Jantzen-connected

One component at a time, at level one. Suppose first that the component sizes agree. For a single component $a$ of rank $m$, the actual level-one Hecke algebra is the finite Hecke algebra by [F3]. Its same-core Specht-block clause [F4], together with the level-one case of the Jantzen linkage equivalence [F1], connects any two rank-$m$ partitions with equal $e$-core by level-one Jantzen edges. The exact signed-conjugate bridge is $C_{\mathrm{M}}(\lambda )\cong P({\lambda }^{\mathsf{T}}{)}^{\theta }$, where $\theta (T_i)=-z{T}_i^{-1}$ and $P$ denotes the ordinary Dipper–James Specht module. If $\lambda ,\mu$ have equal $e$-cores, their transposes have the equal transposed cores, so [F4] puts $P({\lambda }^{\mathsf{T}})$ and $P({\mu }^{\mathsf{T}})$ in the same ordinary Hecke block. An algebra automorphism sends primitive central idempotents bijectively to primitive central idempotents; its exact module pullback therefore preserves the equivalence of sharing a block. Applying the SAME $\theta$ to both modules gives the required Murphy block equality and then [F1] gives the level-one Jantzen connection. No untwisted conjugate-dual identity or algebraic-closure assumption is used. The rank $m=0$ case has a single empty partition and needs no block theorem.

Equalizing the component weights with charged runners. Choose a component $a$ whose weight exceeds its target and a component $b$ whose weight is deficient. A positive-weight component has a removable rim $e$-hook by [F5]. For each component $c$ use the charged infinite beta set $B_c=\{{\lambda }_j^{(c)}+c_c-j:j\ge 1\}$. This is the ordinary beta display shifted to charge $c_c$, with all sufficiently negative positions occupied and all sufficiently positive positions vacant. A removable hook is a bead move $u\mapsto u-e$ to a gap, and its charged foot residue is $u-e+1$ modulo $e$. To see the foot formula, order the beta positions before and after the move: the affected consecutive rows form the boundary strip between the starting and ending positions, and the leftmost removed box in the last affected row has charged content one more than the ending position. The inverse move wraps a hook whose foot has charged residue one more than its starting position. Let $f$ be the charged foot residue of the hook removed from $a$. In $B_b$, choose the largest occupied integer $v\equiv f-1(\mathrm{m}\mathrm{o}\mathrm{d}e)$; it exists by the two tail conditions, and $v+e$ is vacant by maximality. Move $v$ to $v+e$ in component $b$. Its new hook has charged foot residue $v+1=f$. Equivalently, in an uncharged display for component $b$ the runner to use is $f-c_b-1$ modulo $e$, so different component charges cannot be ignored. Both moves stay on their own runners and preserve the component cores and total rank. The earlier of $a,b$ determines the strict dominance orientation: removing boxes there and adding them later decreases every intervening prefix sum; adding boxes there reverses the inequality. In that orientation the equal-remainder hook pair is unique, since the two changed components determine both strips. Its charged foot residues agree, so the foot difference evaluates to zero at $t=1$; it is nonzero over $F(t)$ by [F2], and consequently has a finite positive valuation. Thus the single signed contribution is nonzero, giving a Jantzen edge. If ${\delta }_c$ is the current component weight minus its target, then ${\delta }_a\ge 1$, ${\delta }_b\le -1$, and this transfer changes ${\sum }_c{\delta }_c^2$ by $-2{\delta }_a+2{\delta }_b+2\le -2$. Finitely many such transfers therefore equalize all component weights.

Fixed component sizes. For two full multipartitions that differ only in component $a$, every contributing equal-remainder hook pair in $J$ lies in $a$: the other components are unchanged on both sides and cancelling hooks between two unchanged components would return the same multipartition. The common residue factor $Q_a{t}^{na}$ is a unit of $\mathcal{O}$, and replacing it by the level-one factor $Q_a{t}^m$ multiplies each foot difference by a Laurent unit, so the valuation is unchanged and the full-level $J$ coefficient equals the level-one coefficient of [F2]. Inserting the level-one paths of step 1.1 in component $a$ and leaving the other components fixed connects all multipartitions with equal component sizes and the same component cores.

Conclusion. Steps 1.2 equalize component sizes along Jantzen edges from any starting multipartition, and step 2.1 connects all multipartitions of fixed component sizes with equal component cores. Composing the two finite chains connects any two multipartitions of rank $n$ with equal componentwise $e$-cores, proving that they are $J$-equivalent; the fixed-size comparison of coefficients is step 2.1.