Weighted cell quotients and the leading finite-Weyl basic-set labels

If a coefficient transition leads from $y$ to $x$, P4 applied to its generator product gives $a(x)\ge a(y)$. This holds on both sides and along every finite preorder chain. Mutual comparison therefore makes $a$ constant on each cell of all three types. Dagger is an algebra involution and carries $C_x{C}_y={\sum }_z{h}_{x,y,z}{C}_z$ to the identical formula in $D$; hence the coefficient preorders have the same labels. The map from the dagger quotient to the native quotient sending $[D_w]$ to $[C_w]$ intertwines the action of $h$ on the former with $h^{†}$ on the latter. This proves the precise twist convention.

Fix an integer $i$ and consider the finite set of left cells with value $i$. Their preorder descends to a partial order because mutual comparison is the cell equivalence. Repeatedly take a minimal remaining cell to obtain a total listing with every smaller cell earlier. This finite construction terminates as the number of remaining cells decreases by one. In $B_i$, the spans of initial unions in that listing are left submodules: an action term either has strictly larger $a$ and vanishes in the layer, or has the same $a$ and lies in a cell no later than the input cell. Their successive quotients have exactly one cell as basis and exactly its quotient multiplication law, by [F1]. The same argument on right or two-sided cells gives the corresponding filtrations with their sidedness. These modules and their successive layers are free in their displayed bases.

Over the fraction field $F$ of [F3], $H_F\cong J_F$ and the layer $B_{i,F}\cong J_F^i$ is its regular support ideal. Its generic simple constituents are exactly the $E_F$ with support degree ${\nu }_E=i$, each with positive multiplicity: in a matrix block $M_{d_E}(F)$ the left regular module consists of $d_E$ copies of its standard simple. For each $E$, the explicit free model in [F3] has nonzero finite-dimensional residue, hence some simple factor $M$ with $d_{E,M}>0$. Reciprocity in [F3] then puts $V_E$ in $B_{{\alpha }_E,k}e$ for a projective-cover idempotent $e$ of $M$. Since $B_i$ is supported in $J$-degree $i$ by [F2], this forces ${\nu }_E={\alpha }_E$. Thus the generic simple constituents of $B_{i,F}$ are exactly the labels with ${\alpha }_E=i$. Additivity along the finite cell filtration in step 2.1 shows that every such $E_F$ occurs in at least one dagger cell quotient and that every generic constituent of each of those quotients has exponent $i$. The same statements hold over $\mathbb{Q}(v)$ because all generic blocks are split and extension to $F$ preserves their multiplicities.

For a fixed special simple $M$ work first over its prime-generated field $k_0$, as in [F3], and lift its projective-cover idempotent $e$ to $q$ over the complete local ring. If $X_R$ is any displayed free dagger cell module, then $q{X}_R$ is a direct summand of a finite free $R$-module, so it is finite projective and local-free. Its residue and generic ranks agree. Exactness of multiplication by $e$ along a composition series gives $[X_{k_0}:M]={\dim }_{k_0}e{X}_{k_0}={\dim }_{F}q{X}_F={\sum }_E[X_F:E_F]d_{E,M}$, where the last equality follows from split generic decomposition and [F3]. The field-extension part of [F3] preserves this equality over $k$. No reduction-exactness assertion for arbitrary nonfree lattices is used.

Take $E=E(M)$ and $i={\alpha }_E=a_M$. Step 3.1 supplies a dagger cell module $X$ of value $i$ with $[X_F:E_F]>0$. The sum of nonnegative integers in step 4.1 then has a positive contribution $[X_F:E_F]d_{E,M}=[X_F:E_F]$, so $M$ occurs in its specialization. By [F4], any other generic label contributing to $M$ must have exponent strictly larger than $i$. Step 3.1 says all constituents of this cell module have exponent $i$, so those other contributions vanish. Thus the selected label gives precisely the claimed leading contribution, without asserting that the cell module has only one generic constituent or one special factor.