A nonsemisimple L-good rank-one specialization has a singleton basic set

The generic algebra has two distinct roots $v,-v^{-1}$ over $\mathbb{Q}(v)$, so it is split with its two displayed one-dimensional characters. The KL basis is $C_1=1$, $C_s=T_s+v^{-1}$, and $C_s^2=(v+v^{-1})C_s$ by expansion of the quadratic relation. Therefore $a(1)=0$, $a(s)=1$, and $\tau (C_1)=1$, $\tau (C_s)=v^{-1}$ give $\mathcal{D}=\{1,s\}$ and $n_1=n_s=1$. The only nonzero $\gamma$ are ${\gamma }_{1,1,1}={\gamma }_{s,s,s}=1$.

P2, P3, P5, P6, P7 and P8 follow immediately from these two diagonal $\gamma$ entries and $s^2=1$. P4 holds for the only products $C_1{C}_w=C_w=C_w{C}_1$ and $C_s^2=(v+v^{-1})C_s$. For P15′, the condition $a(w)=a(y)$ forces $w=y$, since the two values are distinct. Its left side is then ${\delta }_{w,x^{\mathrm{\prime }}}{h}_{x,w,w}$ and its right side is ${\delta }_{y,x^{\mathrm{\prime }}}{h}_{x,w,y}$, identical because $w=y$. Thus every required P property is checked, not imported from a general cell theorem.

For the two characters write $\tau =c_{+}^{-1}{\chi }_{+}+c_{-}^{-1}{\chi }_{-}$. Evaluating on $1$ and $T_s$ gives $c_{+}^{-1}+c_{-}^{-1}=1$ and $v{c}_{+}^{-1}-v^{-1}{c}_{-}^{-1}=0$. Hence $c_{+}=1+v^2$ and $c_{-}=1+v^{-2}$. Their lowest terms give $(f_{+},{\alpha }_{+})=(1,0)$ and $(f_{-},{\alpha }_{-})=(1,1)$ as in [F2]. Both specialize to zero at $v=i$, whereas the leading integers stay nonzero.

At $v=i$ both roots equal $i$, so $H_k=\mathbb{C}[T_s]\mathrm{/}((T_s-i{)}^2)$. Put $\epsilon =T_s-i$. The element $\epsilon$ is nonzero and square-zero, since $1,T_s$ form the rank-two basis. On a simple module, the nilpotent ideal $(\epsilon )$ must act by zero: otherwise its image would be the whole simple and iteration would contradict its nilpotence. Thus the quotient $H_k\mathrm{/}(\epsilon )=\mathbb{C}$ gives the unique simple $M$. Both global free rank-one character models reduce to it, so the decomposition column is $(1,1{)}^t$. Its unique least Schur exponent is zero, with multiplicity one, selecting exactly the first label. These are also precisely the axioms and uniqueness conclusion in [F3]. This verifies concretely that full-Schur vanishing is compatible with the general nonsemisimple basic-set theorem.