A simple nuclear C*-algebra not isomorphic to its opposite in ZFC

We construct in ZFC a unital simple nuclear complex C*-algebra of norm density ℵ₁ which is not isomorphic to its opposite. The algebra is approximately matricial and has a unique tracial state. This answers the ZFC existence question recorded by Farah and Hirshberg in 2016. We use the norm completion of the ordered cubic parafermion algebra, whose canonical unitaries satisfy u_αu_β = ζu_βu_α for α < β < ω₁, where ζ = e2πi/3. We prove that its unitary group contains no ω₁-indexed family satisfying the inverse-phase relation with ζ². The tracial von Neumann closure is a hyperfinite type II₁ factor with predual density ℵ₁, and there is no nonzero complex-linear *-antihomomorphism from the algebra into this factor. Moreover, a club of separable subalgebras are copies of the UHF algebra M3^∞, giving a ZFC failure of reflection of nonselfoppositeness. The proof combines finite-support Δ-systems with a uniform tracial estimate that allows arbitrary noncommuting matrix coefficients. GitHub repository: eulogica/simple-nuclear-nonselfopposite-zfc