Cellularity does not make every cell module simple

The abstract instance: in $A=F[x]\mathrm{/}(x^2)$ with $c_{11}=1$, $c_{22}=x$ and $2⊳1$, the product $c_{22}{c}_{22}=x^2=0$ is congruent to $0\cdot c_{22}$ modulo $A^{⊳2}=0$, so the axiom holds with zero form value; the products $c_{11}(a+bx)=a+bx\equiv a{c}_{11}(\mathrm{m}\mathrm{o}\mathrm{d}(x))$ and $c_{22}(a+bx)=ax=a{c}_{22}$ confirm the triangularity. Hence $C^2$ has identically zero form, $\mathrm{r}\mathrm{a}\mathrm{d}{C}^2=C^2$ and $D^2=0$; the module $C^2$ itself is one-dimensional with $a+bx$ acting as $a$, hence simple with zero Jacobson radical and head $C^2\mathrm{\ne }0$. Moreover $a+bx$ is a unit exactly when $a\mathrm{\ne }0$: in that case its inverse is $a^{-1}-b{a}^{-2}x$, while every element of $(x)$ squares to zero. Thus $(x)$ is the unique maximal ideal, so $\mathrm{rad}A=(x)\mathrm{\ne }0$. The regular module is not semisimple: its submodule $(x)$ has no complementary ideal, since any nonzero ideal meeting $A\setminus (x)$ contains a unit and equals $A$.

The Hecke instance directly. At $q=1$, put $x={\sum }_{w\in S_3}{T}_w$. Pairing terms gives $x{T}_i=x$, and multiplication gives $x^2=6x=0$ in characteristic $2$. The top Murphy cell is the free rank-one row-character module generated by $x$, and there is no higher shape than $(3)$; hence its cell form has value $6=0$ and its labelled quotient is zero. The signed-conjugate bridge in [F2] identifies the ordinary module $S^{(1,1,1)}$ and its form radical with this Murphy cell and its radical, through a unit factor. Since the sign twist is trivial in characteristic $2$, this is the nonzero trivial module, while $D^{(1,1,1)}=0$. Thus this witness does not require the general finite-quantum-characteristic classifier.

The reducible Weyl module. In the precise lift of [F3], the sum of the positive filtration-submodule classes is $[L^{(3,1^2)}]+2[L^{(2,2,1)}]\mathrm{\ne }0$. At least one of these submodules is nonzero. Since all are contained in the first reduced submodule $W_F(1)={\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}{W}_F$, that radical is nonzero. The Schur cell form has nonzero simple head, so its form radical is its unique maximal proper submodule. Consequently $W_F^{(3,2)}$ is reducible, despite having simple head $L^{(3,2)}$. The reduced form radical is not identified with $F{\otimes }_R$ the generic radical: specialization need not commute with this kernel, and the lifted generic form here is nondegenerate.

Conclusion: the three instances refute the claims that cellularity forces semisimplicity, forces every cell module to be simple, or forces every labelled cell-form quotient to be nonzero; each is an application of the classification of cell-form quotients [F1] together with the explicit degenerate data.