= Semisimplicity of the reduced Temperley-Lieb category
The <Jones-Wenzl idempotent> recursion decomposes $f_j\otimes1$ into the orthogonal $f_{j+1}$ and $f_{j-1}$ channels. The <Jones-Wenzl partial trace> supplies the nonzero scalar normalizing the cap-cup channel. At the highest surviving color the $f_{r-1}$ channel is zero. Induction therefore decomposes every tensor power into the surviving colors. A planar matching between different projected colors has a turnback on the larger side and vanishes; between equal colors only the identity matching remains. Their <endomorphism> spaces are consequently one-dimensional, and the resulting finite <matrix> blocks split. Taking direct sums and splitting these <idempotents> produces a <semisimple category>, supplying the complete resolutions needed in <fusion of Jones-Wenzl colors> and the <Kirby color handle-slide identity>.
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