The degenerate and nondegenerate Ariki theorems have distinct parameter ranges

The actual generic tower comparison. Use the first-strand presentations and the finite support of [F1]. On a first color $i$, the two cyclotomic polynomials become $y_1^{{\lambda }_i}$ times an invertible nilpotent-evaluated series, since $m_j-i\mathrm{\ne }0$ and ${\xi }^{m_j}-{\xi }^i\mathrm{\ne }0$ when $m_j\mathrm{\ne }i$. Thus the proved affine inverse dictionaries descend in both directions to the same homogeneous cyclotomic KLR presentation. Their inverse composites fix words, dots and crossings, with the prescribed right-side order of their series. This gives the actual maps ${\varphi }_d^{\mathrm{deg}},{\varphi }_d^{\xi }$ and hence ${\mathrm{\Theta }}_d$; no arbitrary rankwise isomorphism is selected. Their local series are independent of rank, charge order and window. The prefix projector is the sum over its appended colors, so the tower maps agree; the common last-color idempotent identifies the two induction bimodules by $a{ϵ}_i\otimes v\mapsto {\mathrm{\Theta }}_{d+1}(a){ϵ}_i\otimes v$, and identifies the restrictions. In a fixed finite rank bound every series is a finite nilpotent Taylor sum; window enlargement adds only zero unsupported words. Therefore the comparison preserves actual word projectors, residue functors and their finite compositions.

Word characters and the generic Murphy standard. Define ${\mathrm{ch}}_{\mathrm{deg}}M={\sum }_{\nu }\dim e_{\mathrm{deg}}(\nu )M[\nu ]$ and ${\mathrm{ch}}_{\xi }M={\sum }_{\nu }\dim e_{\xi }(\nu )M[\nu ]$, with finite sums in rank $d$ by [F1]. An idempotent image is an exact functor, so both characters are additive and descend to $G_0$. Infinite order makes $\nu \mapsto ({\xi }^{{\nu }_1},\dots ,{\xi }^{{\nu }_d})$ injective. Order the Murphy basis by a linear extension of dominance; [F2] gives a simultaneous flag for the commuting $X_r$, with one-dimensional factors of eigenword ${\xi }^{m_j+c-r}$. Applying the exact primary word projector to this flag counts each matching factor once and every nonmatching factor zero, even when distinct tableaux give the same word. Star fixes every $X_r$, so right-to-left conversion preserves these dimensions. Hence ${\mathrm{ch}}_{\xi }{S}_{\xi }(\mathit{\mu })={\sum }_{T\in \mathrm{Std}(\mathit{\mu })}[{\mathrm{res}}_{\mathbf{m}}(T)]$.

The actual degenerate standard character. Put $d_j=\mathrm{\mid }{\mu }^{(j)}\mathrm{\mid }$ and $W_P=S_{d_1}\times \cdots \times S_{d_l}$. The affine PBW basis of [F3], read also in reverse order through its generator-fixing anti-involution, makes the affine algebra free as a right affine Young-subalgebra module on the distinguished finite shuffles $w\in S_d\mathrm{/}{W}_P$. Thus the actual induced standard has basis $w\otimes v$, where $v$ is a tensor of ordinary Specht eigenvectors. The mixed relation moves a polynomial generator through the shuffle as $x_{r}w=w{x}_{w^{-1}(r)}+{\sum }_{\mathrm{\ell }(u)<\mathrm{\ell }(w)}{c}_{u}u$: each scalar correction deletes a crossing. Reducing $u$ to its minimal Young-coset representative cannot increase length. Therefore the shuffle-length filtration is stable under all commuting $x_r$, and its factors have the permuted component eigenwords $m_j+c-r$ of [F3]. Each component standard-tableau tuple and order-preserving shuffle specifies one standard multitableau, and conversely a multitableau recovers that tuple and shuffle by local renumbering. Applying exact word projectors to this filtration consequently gives ${\mathrm{ch}}_{\mathrm{deg}}{S}_{\mathrm{deg}}(\mathit{\mu })={\sum }_{T\in \mathrm{Std}(\mathit{\mu })}[{\mathrm{res}}_{\mathbf{m}}(T)]$, with full multiplicities. The joined Schur theorem [F3] identifies this actual affine-induced module with the Schur image; a proposed finite-W or Hecke character is not substituted for that module.

Injectivity from actual projective words. For $\nu =({\nu }_1,\dots ,{\nu }_d)$ put $P(\nu )=f_{{\nu }_d}\cdots f_{{\nu }_1}\mathbb{C}$, where the ground object is projective. The actual right-appending residue bimodules identify $P(\nu )$ with $H_d^{\mathrm{deg}}{e}_{\mathrm{deg}}(\nu )$ by repeated balanced tensor multiplication. Evaluation at the idempotent gives $\mathrm{Hom}(P(\nu ),M)\cong e_{\mathrm{deg}}(\nu )M$, so its Cartan pairing is exactly the corresponding word-character coefficient. Under the projective highest-weight theorem [F4], the classes of these words span each rationalized $K_0$ weight: the lower lattice is ground-generated under divided powers, and over $\mathbb{Q}$ each divided power is an ordinary power divided by a factorial. Therefore an integral $G_0$ class of zero character pairs to zero with every projective cover. The integral dual bases in [F5] make every simple coefficient zero, proving word-character injectivity on $G_0$, not just on its rationalization. Only the lower projective lattice is used for this spanning argument; integral ground cyclicity of the full upper lattice is not assumed.

Exact standard classes and charge order. Step 1.1 makes pullback preserve every word coefficient. Steps 1.2–2.1 give equal characters for ${\mathcal{E}}_d{S}_{\xi }(\mathit{\mu })$ and $S_{\mathrm{deg}}(\mathit{\mu })$, and step 2.2 gives their equality in integral $G_0$. A simultaneous permutation of charges and partitions bijects their standard tableaux without changing a residue word, so it preserves both character sums and, by step 2.2, the degenerate standard class. Sort these pairs into the source charge order. The entries $m_j+1-r+{\mu }_r^{(j)}$ are strictly decreasing down the column and have the ground tail; their ground differences are exactly the component partitions. The source Schur/Fock dictionary [F3, F4] therefore sends the sorted standard class to $S_{B_{\mathit{\mu }}}$. This proves the stated generic standard-class dictionary for every finite integral charge order. The calculation uses the same-shape Murphy cell and its star conversion. A convention using a contragredient dual has the same primary dimensions because star fixes the polynomial generators; a signed-conjugate convention must also carry its sign and charge/reflected-residue change. No untwisted conjugate module or unchanged signed residue word is inferred.

Transfer the full based and pairing data. An algebra isomorphism gives an exact equivalence and a Hom bijection on every pair. Thus it takes simples to simples, indecomposable projectives to indecomposable projectives and projective covers to the corresponding covers, defining precisely $D_{\xi }(A)$ and $Y_{\xi }(A)$. It preserves the Cartan pairing and the inclusion of projective classes into module classes. Composing with [F4] therefore transfers the full lower and restricted upper integral lattices, their normalized canonical/dual-canonical families at $s=1$, and the Cartan map as their lattice inclusion. The bimodule maps in step 1.1 transfer the Chevalley actions, and the integral divided-power actions transfer under these integral group isomorphisms. Combining step 3.1 with the degenerate standard multiplicities of [F4] gives $[S_{\xi }(\mathit{\mu }):D_{\xi }(A)]=d_{A,B_{\mathit{\mu }}}(1)$. This coefficient comparison is in the transported simple basis and implies neither a Specht-module isomorphism nor a same-shape cellular-head identification.

All compatible cuts and all ranks. On a content $\alpha$, the sum of word projectors with that content is preserved by ${\mathrm{\Theta }}_d$, as is the central sum over words in any supported interval $I$. Hence the same comparison restricts to the same supported finite content algebras and to their residue corners. For any selected finite list of contents and operations, choose lower/upper letter and root cuts containing all charges, content supports and intermediate supports; [F1] permits a common finite generic window, and [F4] identifies the actual degenerate finite-cut based lattices and standards by their common-ground-tail labels. The inverse dictionaries use fixed pair series, so the tower isomorphisms agree on overlaps. The unchanged intrinsic affine standards and their corner projections have the same characters and classes there. Thus their standard, simple and projective classes, pairings and integral lower/upper bases glue under the same composable tail maps, proving the full arbitrary-consecutive-$I$ assertion weight by weight. No proper supported cut becomes the whole rank algebra. At rank zero both maps are the identity on $\mathbb{C}$, with the empty word, tableau and standard. At level zero the first cyclotomic polynomial is $1$, so both positive-rank algebras and all residue operations are zero, and only the ground line remains.

The root-of-unity obstruction and characteristic boundary. For level one with charge zero and $d=2$, the degenerate presentation gives $x_1=0$, $x_2=s_1$ and $s_1^2=1$, hence $H_2^{\mathrm{deg}}\cong \mathbb{C}[S_2]\cong \mathbb{C}\oplus \mathbb{C}$. At $\xi =-1$ the nondegenerate level-one presentation gives $X_1=1$, $X_2=-T_1^2$ and $(T_1+1{)}^2=0$, hence $H_2^{-1}\cong \mathbb{C}[T_1]\mathrm{/}((T_1+1{)}^2)$. Its class $T_1+1$ is nonzero and nilpotent, whereas the product of two fields has no nonzero nilpotent. These algebras are therefore not isomorphic. Moreover, the generic local inverse series rely on ${\xi }^a-{\xi }^b\mathrm{\ne }0$ for distinct integers; these constants can vanish at a root of unity, so the proved generic inverse maps cannot simply be specialized there. The cyclic characteristic-zero and positive-characteristic assertions retain their own parameters and suppliers, and the grading/basis variable is never identified with the numerical $\xi$. This proves the stated obstruction and all comparison claims without claiming a general positive-characteristic decomposition formula. The only choice is the inherited AC in [F4]; the remaining constructions use finite data.