Covariant Perfect Anticorrelation and Modal Simplicity: A Characterization of the Singlet Cosine

We characterize, within quantum theory, the two-party correlation E(θ_A, θ_B) = −cos(θ_A − θ_B) that saturates the Tsirelson bound |S| = 2√2. Let E be the correlation kernel of a U(1)-covariant, perfectly anticorrelated bipartite quantum realization (a jointly invariant state with covariant local dichotomic observables). We prove that −E is then automatically a positive-definite function on U(1), because it equals a diagonal matrix coefficient of an associated unitary representation. We prove a converse: every normalized nonnegative harmonic spectrum is realized by an explicit bipartite quantum model with genuinely local observables. In the real-even category we prove that spectral extremality of the kernel is equivalent to irreducibility of its real cyclic (GNS) representation, and we separate this from minimality of the cyclic realization: a reducible multi-harmonic kernel still has a minimal cyclic realization. Adding one axiom — Primitive Modal Simplicity, that the response occupies a single irreducible real sector — together with faithfulness of that active carrier, singles out −cos Δ. We prove by an explicit, reproducible counterexample (a spin-3/2 singlet with the dichotomic observable sign(S_x), giving the kernel −(7/8)cos Δ − (1/8)cos 3Δ) that modal simplicity is independent of purity, covariance, perfect anticorrelation, one reference dichotomic observable per party, absence of classical randomness, and minimality of the cyclic realization. We state carefully what is and is not established: this is a characterization of a scalar correlation-function family, not a reconstruction of the quantum state/observable framework, the Born rule, or a new empirical prediction. Code: the accompanying verification suite (routeB_verification_suite.tar.gz) is licensed MIT; this manuscript is licensed CC BY 4.0.