The restricted-partition component is the basic crystal in the symmetric-group convention

The conjugation dictionary identifies the vertex sets of the two components. By [F1] transposition is a bijection from the $e$-restricted onto the $e$-regular partitions, fixing the empty partition, and by [F3] the vertex set of the empty component of the full Fock crystal is exactly the set of $e$-regular partitions. Hence the $J^{-1}$-image of the empty component is exactly the $e$-restricted set, with the empty partition corresponding to the vacuum, and by [F2] the transported modular operators on these labels are the transported normalized Fock operators ${\tilde{e}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}=J^{-1}{\tilde{e}}_{-i}^{\mathrm{Q}\mathrm{G}}J$, ${\tilde{f}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}=J^{-1}{\tilde{f}}_{-i}^{\mathrm{Q}\mathrm{G}}J$.

The transported component is connected and generated from the vacuum. The empty component of [F3] is the canonical $B({\mathrm{\Lambda }}_0)$, hence connected and generated from its highest vertex by the ${\tilde{f}}^{\mathrm{Q}\mathrm{G}}$; transporting along the bijection of step 1.1, the $e$-restricted labels form one connected component of the transported crystal, generated from the empty partition by the operators ${\tilde{f}}_i^{\mathrm{s}\mathrm{y}\mathrm{m}}$, that is, by modular cogood additions. In particular every $e$-restricted shape is reached from the empty partition by successive cogood additions, and dually by successive good deletions; for $e=p$ prime this is also the choice-free statement of [F4], whose good and cogood nodes are the modular ones by [F2].

The weight function is the transported one. By [F1] the weight of $M_{\lambda }$ is ${\mathrm{\Lambda }}_0-{\sum }_i{m}_i(\lambda ){\alpha }_i$ with $m_i(\lambda )$ the number of residue-$i$ nodes, and by [F2] it transforms under conjugation as $\mathrm{wt}({\lambda }^{\mathrm{\prime }})=\tau (\mathrm{wt}(\lambda ))$; the vacuum $\mathrm{\varnothing }$ has weight ${\mathrm{\Lambda }}_0$ since $m_i(\mathrm{\varnothing })=0$. The modular ${\epsilon }_i(\mu )$ and ${\phi }_i(\mu )$ are by construction the numbers of surviving $-$ and $+$, that is, of normal removable and conormal addable $i$-nodes in the sense of the modular signature.

This proves part 1 of the Statement: the empty component of $F_{\mathrm{s}\mathrm{y}\mathrm{m}}$ is exactly the $e$-restricted partitions (step 1.1), its operators are the modular good deletions and cogood additions (step 1.1), its weight is ${\mathrm{\Lambda }}_0-{\sum }_i{m}_i(\lambda ){\alpha }_i$ (step 2.2), $\epsilon$, $\phi$ count normal and conormal nodes (step 2.2), and it is $B({\mathrm{\Lambda }}_0)$ with every restricted shape reached from the empty partition by cogood additions (step 2.1).

For part 2, the full $F_{\mathrm{s}\mathrm{y}\mathrm{m}}$ has all partitions as vertices and its module is the transport of the modules of [F3], so its field decomposition is $F_{\mathrm{s}\mathrm{y}\mathrm{m}}\cong {⨁}_{k\ge 0}V({\mathrm{\Lambda }}_0-k\delta {)}^{\oplus p(k)}$ with $p(k)$ ordinary partition numbers; this direct sum is infinite, so $F_{\mathrm{s}\mathrm{y}\mathrm{m}}$ is reducible whenever at least two summands are nonzero (for instance the summands $V({\mathrm{\Lambda }}_0)$ and $V({\mathrm{\Lambda }}_0-\delta )$). Since the empty component is exactly the $e$-restricted shapes by step 3.1, every other vertex of the full crystal is a non-restricted shape, and conversely each non-restricted shape lies outside the empty component. The axiom of choice is inherited through [F3], both here for the field decomposition and in steps 1.1-2.1 for the canonical highest-weight-crystal identification. The transport itself introduces no choice. The vertex-set identification and reachability can also be read directly from the choice-free partition-graph supplier of [F2], and [F4] gives the prime-case reachability without choice.

Part 3, the James labels and the socle formula. Let $k$ be of characteristic $p>0$ and $e=p$; $e$-restricted $\lambda$ are exactly the labels $D(\lambda )$ of the simple $k{S}_d$-modules in the dual-Specht convention, and for $p$-regular ${\lambda }^{\mathrm{\prime }}$ the published dictionary gives $D^{{\lambda }^{\mathrm{\prime }}}\cong D(\lambda )\otimes \mathrm{sgn}$ by [F6]. Suppose ${\epsilon }_i(\lambda )>0$ and let $A$ be the good $i$-node, so that ${\tilde{e}}_i\lambda =\lambda \setminus A$ and, by [F5], $\mathrm{soc}{E}_{i}D(\lambda )\cong D(\lambda \setminus A)$. Tensoring with $\mathrm{sgn}$ negates the Jucys-Murphy action: for $v\otimes w$ in $M\otimes \mathrm{sgn}$, $L_r(v\otimes w)={\sum }_{s<r}((sr)v)\otimes ((sr)w)=-L_{r}v\otimes w$ because each $(sr)$ acts by $-1$ on $\mathrm{sgn}$ by [F6]; hence the content vector of the sign twist is the negative, and by [F5] the residue summands satisfy $E_{-i}(M\otimes \mathrm{sgn})\cong (E_{i}M)\otimes \mathrm{sgn}$. The twist is an exact invertible endofunctor (its square is the identity on isomorphism classes), so it preserves socles and simples; applying it to $\mathrm{soc}{E}_{i}D(\lambda )\cong D(\lambda \setminus A)$ gives $\mathrm{soc}(E_{-i}(D(\lambda )\otimes \mathrm{sgn}))\cong D(\lambda \setminus A)\otimes \mathrm{sgn}$, and the two published re-identifications $D(\lambda )\otimes \mathrm{sgn}\cong D^{{\lambda }^{\mathrm{\prime }}}$ and $D(\lambda \setminus A)\otimes \mathrm{sgn}\cong D^{(\lambda \setminus A{)}^{\mathrm{\prime }}}$ of [F6] yield $\mathrm{soc}{E}_{-i}{D}^{{\lambda }^{\mathrm{\prime }}}\cong D^{({\tilde{e}}_i\lambda {)}^{\mathrm{\prime }}}$; this is the stated formula, and it exhibits $E_{-i}$ as the sign twist of the residue functor $E_i$.

Part 4, the boundary and the naming caveats. At $e=0$ the colors are $\mathbb{Z}$ and, by [F7] together with the distinct-content computation recorded in the convention item The symmetric-group level-one residue and partition-label convention, each residue class contains at most one corner of a partition, so no signature contains a cancellable pair; every partition reaches the empty partition by good deletions, and the source's restricted set for level $1$ is all partitions. No finite affine datum, charge, weight formula or symmetric-group module statement is made in this boundary; and for a general finite $e\ge 2$ the crystal labels of parts 1 and 2 are Hecke labels, valid for any quantum characteristic $e$, not only for prime $e$ as in part 3.