Root fusion rules and completeness of the simple objects

If $r=2$, then $p=4$, $\delta =0$, and the sole proposed retained label is $0$. The boundary projector is $p_1=1_1$ and has zero trace. The tensor ideal it generates contains $1_n=1_1\otimes 1_{n-1}$ for every $n\ge 1$. Therefore every morphism with a positive source or target is zero in the quotient, because it factors through the corresponding identity. The endomorphism space of object $0$ is $\mathbb{C}$, and its trace is the identity, so it is non-negligible. The quotient's additive idempotent completion is thus finite-dimensional vector spaces with one self-dual simple $0$. Its fusion is $0\otimes 0=0$, its unit twist is $1$, and its Hopf-link matrix is the $1\times 1$ matrix $(1)$, which is invertible.

Suppose $r\ge 3$ and first take the canonical root $A_0=e^{\pi i\mathrm{/}(2r)}$, so $q_0=A_0^2=e^{\pi i\mathrm{/}r}$ and the BHMV loop is $-(q_0+q_0^{-1})$. By [F3], the diagram quotient with its parity-corrected pivotal structure is ribbon-equivalent to the positive-pivotal quotient of [F4]–[F5]. Thus the quotient has exactly the labels $0,\dots ,r-2$, the displayed truncated fusion multiplicities, self-dual simples, and a nondegenerate Hopf-link matrix for that corrected pivotal structure.

On the semisimple quotient define ${\epsilon }_{p_a}=(-1{)}^a{1}_{p_a}$. This is a monoidal natural automorphism: each retained summand $p_c$ of $p_a\otimes p_b$ has $c\equiv a+b(\mathrm{m}\mathrm{o}\mathrm{d}2)$ by [F4], so ${\epsilon }_{p_c}={\epsilon }_{p_a}{\epsilon }_{p_b}$; it is dual-compatible because every $p_a$ is self-dual. The planar pivotal structure is $\epsilon$ times the parity-corrected one of [F3]. Multiplying a pivotal structure $j$ by a dual-compatible monoidal automorphism $\epsilon$ changes the twist $u^{-1}j$ to $\epsilon \theta$, preserves balancing by monoidality, and preserves dual compatibility; hence the planar convention is also ribbon. Its dimensions are $d_a=(-1{)}^a{d}_a^{+}$, which equal the BHMV recurrence value $[a+1]$: the sequence $v_j=(-1{)}^{j-1}[j{]}_{q_0}$ has $v_0=0$, $v_1=1$ and satisfies $v_{j+1}=-(q_0+q_0^{-1})v_j-v_{j-1}$, so uniqueness of the recurrence gives $v_j=[j]$ for the BHMV loop. Its simple twists are ${\theta }_a=(-1{)}^a{\theta }_a^{+}=(-1{)}^a{A}_0^{a(a+2)}$. Its unnormalized Hopf-link entries are $s_{ab}={\epsilon }_a{\epsilon }_b{s}_{ab}^{+}$, since the pivotal trace on $p_a\otimes p_b$ is multiplied by ${\epsilon }_a{\epsilon }_b$. Therefore its Hopf-link matrix is obtained from the invertible matrix in [F5] by multiplying row $a$ and column $b$ by $(-1{)}^a$ and $(-1{)}^b$, so it remains invertible. The criterion in [F6] proves modularity.

For any primitive $A$, write $A=A_0^k$ with $\gcd (k,4r)=1$. The automorphism of the cyclotomic field $\mathbb{Q}(A_0)$ sending $A_0$ to $A$ transports the finite diagram coefficients, the recursively defined projectors, the braiding and twist matrices, and the Hopf-link matrix. By [F7], every fixed Hom space is finite-dimensional, so this transport preserves all ranks and nonzero determinants. The fusion multiplicities are dimensions of these finite Hom spaces and so are unchanged under the field automorphism and extension to $\mathbb{C}$. Thus the canonical-root result applies to every primitive $2p$-th root with the exact stated BHMV convention; in particular ${\theta }_a=(-1{)}^a{A}^{a(a+2)}$.