Complex-root standard radicals, simple heads and reflection linkage

The form is nonzero unless $\delta =0$ and the positive-rank defect number is zero. Indeed, when $\delta \mathrm{\ne }0$ a state's pairing with itself is ${\delta }^p\mathrm{\ne }0$. When $\delta =0$ and $t=n-2p>0$, take $x$ with arcs $(1,2),(3,4),\dots ,(2p-1,2p)$ and $y$ with arcs $(2,3),(4,5),\dots ,(2p,2p+1)$; all other sites are defects. Their overlay on sites $1,\dots ,2p+1$ is one open alternating path, with no cycle, and all remaining defect paths match. Thus $\varphi (x,y)=1$. If $p=0$ the form is $(1)$. For $t=0,n>0$, every overlay has a closed component and hence value zero at $\delta =0$. Consequently [F1] applies to exactly the stated nonexceptional labels.

First observe the rank-one identity: form the algebra diagram whose two halves are $x,y$. Stacking $z$ against its $y$ half gives $\varphi (y,z)$ times the identity on the defect strands; if fewer strands propagate, it is zero in the standard quotient. Therefore this diagram sends $z$ to $\varphi (y,z)x$, by exactly the matching rule in [F5]. More strongly, any submodule not contained in the form radical is the entire standard. If $z$ lies outside the radical, choose $y$ with $\varphi (y,z)=1$. For any vector $x$, the diagram with halves $x,y$ sends $z$ to $x$ by [F5]; hence $z$ generates the standard. Also, if $p>p^{\mathrm{\prime }}$ and the source form is nonzero, every map between any quotients of $V_{n,p}$ and $V_{n,p^{\mathrm{\prime }}}$ is zero. Choose $y,z$ in the source with $\varphi (y,z)=1$. A diagram with halves $x,y$ has $n-2p<n-2p^{\mathrm{\prime }}$ propagating strands and therefore acts as zero on the target and its quotients. Applying a proposed homomorphism to its action on the class of $z$ gives zero on the class of every $x$, hence the map is zero.

Consider an adjacent reflected pair $a<p$ and a nonzero $f:V_{n,a}\to V_{n,p}$ from [F2]. Its kernel is proper and therefore lies in $R_{n,a}$ by step 2.1. Its image also lies in $R_{n,p}$: otherwise step 2.1 makes $f$ surjective, and its inverse isomorphism $V_{n,p}\to V_{n,a}\mathrm{/}\ker f$ contradicts the quotient-vanishing assertion in step 2.1. The source is never the zero-form exceptional label, since $a<p$. If the target is the exceptional label, its simplicity in [F4] instead gives $\mathrm{im}f=V_{n,p}=R_{n,p}$ directly. Thus in every case $\ker f\subseteq R_{n,a}$ and $\mathrm{im}f\subseteq R_{n,p}$.

Rank-nullity and [F3] now force equality in both inclusions: $\dim V_{n,a}=\dim \ker f+\dim \mathrm{im}f\le \dim R_{n,a}+\dim R_{n,p}=\dim R_{n,a}+\dim L_{n,a}=\dim V_{n,a}$. Therefore $\ker f=R_{n,a}$ and $\mathrm{im}f=R_{n,p}\cong L_{n,a}$, which is simple by [F1]. For an absent reflected label [F3] gives $R_{n,p}=0$. At a critical label [F4] gives the same.

For a nonexceptional standard, steps 2.1 and 4.1 show every proper submodule is contained in a radical which is zero or simple. Thus its only submodules are $0,R,V$ and every quotient is indecomposable. When the radical is nonzero, the exact radical/head sequence cannot split: a complementary submodule would be proper and not contained in the radical, contradicting step 2.1. It has exactly the two displayed simple factors. The critical $\kappa =1$ branch is [F4]. If $\kappa =\mathrm{\infty }$, no quantum integer in the determinant vanishes, so the determinant and hence the form are nondegenerate; this is [F6]. Exhaustion and distinction of the nonzero heads are [F1].

At $\delta =0$ and $n=2p>0$, $\kappa =2$ and the reflected arc label is $p-1$. Step 1.1 identifies the form radical as all of $V_{2p,p}$, while step 4.1 identifies it with $L_{2p,p-1}$. This agrees with the separately proved actual simplicity [F4]; its zero form quotient introduces no extra simple label. The remaining standards have nonzero forms and hence the complete simple labels are $0,\dots ,p-1$. Finally, each noncritical standard shares factors only with its adjacent reflection neighbors by step 4.1, and every present neighbor occurs. Taking the equivalence generated by these edges gives precisely the stated reflection chains. A critical head occurs only in its own simple standard, giving singleton classes.