Classically Forbidden Signatures of Quantum Coherence in the Mesoscopic Lipkin-Meshkov-Glick Model

derive strict quantitative conditions under which a collective quantum system of N ∼ 370 spins exhibits classically forbidden temporal correlations in a spinor Bose–Einstein condensate (BEC). The Lipkin–Meshkov–Glick (LMG) model near its Z2-breaking quantum critical point supports a mesoscopic superposition α|P⟩+β|R⟩ of two macroscopic ordered phases (|P⟩: mz ≈ +m∗; |R⟩: mz ≈ −m∗) at the Goldilocks crossover N ≈ Nc, where the tunnel splitting equals kBT and macroscopic susceptibility χ ∼ N coexists with finite quantum coherence. We establish two quantumdiscriminating predictions. P4 (Landau–Zener crossover, proposed discriminator): quantum tunnelling drives Perror → 0 exponentially with quench time, while the classically non-ergodic system remains kinetically frozen at Perror → 1 — a parametrically large, computable separation that rests on a specific kinetic foil. P5 (LeggettGarg inequality violation, strictly model-independent): K3 > 1 is forbidden by macrorealism for all classical models satisfying non-invasive measurability. A fivelevel Lindblad simulation yields K3 ≈ 1.32 and dephasing threshold γϕ ≲ 0.289 s−1 at the BEC target (γϕ = 0.05 s−1, N = 370, Γ/J = 0.95) — a margin of 8.8× above the optimal measurement interval, well within current experimental reach. The threshold has a precise physical origin in the emergent collective Z2 symmetry: exact parity eliminates the dominant dephasing cross-term and renders the LGI correlator immune to T1 population mixing without requiring ground-state preparation (2.35× improvement over the naive mean-field estimate); constructive dynamical phase alignment of higher odd-parity states at the benchmark parameters contributes a further 2.47×. All results are reproducible from the provided self-tested Python code.