Spherical and antispherical normalizations give different polynomials

For $y=e$ and $x=s_2{s}_1$ the antispherical alternating sum [F3] has the two terms $z=e$ and $z=s_1$: $n_{e,s_2{s}_1}=p_{e,s_2{s}_1}-v{p}_{s_1,s_2{s}_1}$. By [F1] the first value has length gap two, so $p_{e,s_2{s}_1}=v^2$; the second has gap one (by [F2], $s_1<s_2{s}_1$), so $p_{s_1,s_2{s}_1}=v$. Hence $n_{e,s_2{s}_1}=v^2-v\cdot v=0$.

For the same pair the spherical reduction [F4] gives $m_{e,s_2{s}_1}=p_{s_{1}e,s_1{s}_2{s}_1}=p_{s_1,w_0}$; the pair has length gap two by [F2] ($\mathrm{\ell }(w_0)-\mathrm{\ell }(s_1)=3-1=2$), so by [F1] $p_{s_1,w_0}=v^2$ and $m_{e,s_2{s}_1}=v^2$.

Specializing at $v=1$ gives $n_{e,s_2{s}_1}(1)=0\mathrm{\ne }1=m_{e,s_2{s}_1}(1)$: no unit of $\mathbb{Z}$ (indeed no unit of $\mathbb{C}$) multiplied by $0$ gives $1$, so the antispherical value is not a unit multiple of the spherical one; and since the two elements $0$ and $v^2$ of $\mathbb{Z}[v]$ are distinct, they differ as polynomials and not only after specialization.

Consequently the two normalizations cannot be substituted for one another in a multiplicity statement: a singular-block or parabolic multiplicity theorem must fix the side of the parabolic and the corresponding module. In particular the singular-block multiplicity theorem of this page is stated in the spherical normalization, whereas a genuine regular parabolic locally finite category uses the antispherical family; replacing the spherical value $m_{e,s_2{s}_1}(1)=1$ by the antispherical value $n_{e,s_2{s}_1}(1)=0$ produces the false multiplicity $0$ for this pair. The computation refutes the normalization-blind claim, not the parabolic basis theorem or the alternating-sum identity that produced the two values.