The root negligible quotient is semisimple with truncated fusion

Write $m=\mathrm{\ell }-1$ and $p=p_m$. It is a nonzero idempotent with one-dimensional corner and trace $[\mathrm{\ell }]=0$ by [F3,F4]. For every $x$, cyclic closure gives $\mathrm{Tr}(px)=\mathrm{Tr}(pxp)$; the latter is a scalar multiple of $\mathrm{Tr}(p)=0$. Thus $p$ is negligible. By [F4], the categorical tensor ideal it generates is negligible, and [F2] identifies its endomorphism component as exactly $I_n$. Consequently $I_n\subseteq \mathrm{Rad}({\mathrm{Tr}}_n)$ at every rank. The scalar ${\delta }^{-n}$ exists and does not change this radical.

In the quotient by $I_n$, [F1] gives matrix units $E_{a,b}$ for paths ending at each retained $c$, with trace $\mathrm{Tr}(E_{a,b})={\delta }_{a,b}[c+1]$. If an element has a nonzero coefficient at $E_{a,b}$, multiplying by $E_{b,a}$ detects it in this nonzero weighted matrix-trace pairing. The quotient trace pairing is therefore nondegenerate. Together with step 1.1 this proves $I_n=\mathrm{Rad}({\tau }_n)$ and the full matrix-block description. Inserting the path decompositions on both ends describes every unequal-degree Hom space as the corresponding rectangular matrix blocks; its pairing with the reverse Hom space is also nondegenerate. Thus the categorical negligible ideal is precisely the boundary tensor ideal, not a larger ideal known only on endomorphisms. The endpoint corners are $\mathbb{C}{p}_c$, so its additive Karoubi quotient is semisimple with exactly the stated simple labels.

Sandwich a basis matching from $c$ to $a+b$ between $p_c$ and $p_a\otimes p_b$. Any arc joining two points of the same labelled arm is killed by its innermost adjacent turnback. A remaining matching must have the three bundle sizes $(a+b-c)\mathrm{/}2$, $(b+c-a)\mathrm{/}2$, $(c+a-b)\mathrm{/}2$, so it exists only for a triangle/parity triple and is unique. Thus this projected Hom space is spanned by the vertex of [F5], and its reverse by the reflected vertex. Their trace pairing is the exact theta scalar. When the labels are retained, [F5] makes it positive exactly for $a+b+c\le 2\mathrm{\ell }-4$ and zero above that level. Step 2.1 identifies quotient Hom with this pairing modulo its radical, so its dimension is respectively one or zero. In a semisimple category these are precisely the simple multiplicities in $p_a\otimes p_b$. The triangle, parity and level inequalities yield the displayed truncated rule. No theta value is divided at a forbidden level.

Finally work at rank $m=\mathrm{\ell }-1$ before quotienting. The same up/down path proof as [F1] applies without deleting its final up endpoint $p_m$: a length-$m$ path cannot exceed height $m$, and every split before its final step uses only $[1],\dots ,[m]$, which are nonzero. All endpoint projectors up to $p_m$ are nonzero by their identity coefficient. The matching-turnback corner calculation gives full matrix units just as before, now including the endpoint $m$; their completeness is the unreduced JW split. Hence ${\mathrm{T}\mathrm{L}}_m$ is semisimple. The endpoint-$m$ block has the unique all-up path and is $\mathbb{C}{p}_m$. Its trace weight is $[m+1]=0$, whereas every other endpoint weight is positive. Thus precisely this one-dimensional block is the trace radical, while semisimplicity makes the Jacobson radical zero. This proves the boundary assertion directly, without assuming a later ordinary semisimplicity criterion.