An Exact Classical–Quantum Coexistence Trade-off Through One Counted Qubit, With Covariant Attainability

We study a minimal coexistence task: a single qubit of memory must be used, through one unassisted encode–decode factorization, both to report a classical binary label (which of two orthogonal "sectors" a carrier occupied) and to preserve quantum information within one sector, in the presence of a Z₂ symmetry. We quantify the first by a label accuracy L — which we show is exactly a two-state discrimination success probability for a decoder-induced measurement — and the second by the unconditional same-sector entanglement fidelity F_same. Our main result is the exact optimal trade-off: for every required accuracy τ in [1/2, 1], the maximum attainable F_same is a single closed-form curve Φ(τ), and this maximum is the same whether or not the factorization is required to be covariant under the symmetry (at the fixed memory representation V = Z). An explicit two-parameter covariant CPTP family attains Φ at every threshold, so covariance imposes no optimal-fidelity penalty anywhere on the trade-off. The frontier has a structural transition at τ = 5/8, above which every optimal code-restricted encoder is necessarily nonunitary, and the attainable score region is nonconvex. A directly applicable prior hybrid-memory inequality (Kuperberg) specializes here to F_same ≤ (5 − 2τ)/4, which is valid but strictly loose for τ > 1/2 and tight at τ = 1/2. Whether the exact frontier, the transition, or covariant optimality follow from earlier results remains open. The results are proved analytically, with selected exact symbolic identities, explicit constructions, and regression tests. This record includes the manuscript and a verification supplement of seven standalone SymPy scripts, a README, and a runner (run_all.sh) that reproduce the exact checks (each exits 0 on success; tested with Python 3.11.15, SymPy 1.14.0, NumPy 2.4.4). The results are mathematical; any interpretive applications are developed separately and are not part of this work.