= Dimension-weighted fusion resolution
{title2=$\sum_b d_bU_c^bV_c^b=d_c1_{c\otimes a}$}
In the reduced Jones-Wenzl theory, bend a fixed self-dual color a in the mutually inverse inclusion/projection maps of $c\subset b\otimes a$, obtaining $U_c^b:b\to c\otimes a$ and $V_c^b:c\otimes a\to b$. Cyclic closure trace and straightening cups and caps give $V_c^bU_c^b=(d_c/d_b)1_b$. Completeness of the normalized maps for all b then proves the weighted resolution. The dimensions are nonzero in the reduced theory. This local identity gives the precise weight preservation in the <Kirby color handle-slide identity>.
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