Finite configurations and nuclear quotients of Thompson's group F

Let F be Thompson’s group, let P = F′, and let Ug denote the canonical prefix-table unitaries in the Cuntz algebra O₂. We prove that the finite ordinary tensor images Bn = C*(Uf⊗n : f ∈ F) and Cn = C*(Up⊗n : p ∈ P) are nuclear. A representation has finite separated derived capacity when products of sufficiently many displacement operators with separated interior supports vanish. Capacity at most d gives an actual marked quotient of B2d+2, or of C2d for a nontrivial P-representation. Infinite capacity is equivalent to admitting the regular representation as an actual quotient. Consequently, if F is nonamenable, every nonzero nuclear quotient of its full group C*-algebra, and likewise of C*(P), is a quotient of one of these finite canonical images. The finite nuclearity and norm estimates are unconditional; nonamenability enters only in passing from nuclearity to finite capacity. The result gives a countable cofinal family of nuclear quotients, with a degree depending on the quotient. GitHub repository: nuclear-quotients-thompsons-group-f