Suppression of Coherent Mode Formation in Central-Spin Systems Coupled to Spin Baths: A Quantitative Dephasing Crossover Criterion

We derive a quantitative, non-circular criterion for coherent-mode formation in a central spin coupled to a spin bath. The criterion compares the pre-decoherence rate to the quantum speed limit rate from the Margolus-Levitin bound, defining Gamma = gamma_pre * t_ML; the crossover occurs exactly at Gamma = 1. The characteristic energy scale is E_char = 2 pi hbar v_LR / L in the finite-size regime and E_char = Delta in the gap-dominated regime, with correlation length xi = 2 pi hbar v_LR / Delta. The two regimes match continuously at L = xi. For the spin-boson model with a hard gap, the polaron transformation gives exactly gamma_pre = pi J_0 / 2 (independent of the gap) and a coherence plateau exp(-J_0/Delta). The criterion becomes Gamma = pi^2 J_0 / (4 Delta), with crossover at J_0 = 0.405 Delta; the dephasing-dominated regime is a half-axis with no upper bound in the domain of validity. For the NV/P1 system, using the measured T2 we find Gamma ~ 2e-3 at 10 ppm, deep in the coherent regime for all accessible concentrations (0.1-300 ppm). We validate the analytical results by exact diagonalization of central-spin-bath systems with up to three bath spins. For non-interacting bath spins, the numerical coherence reproduces the exact analytical solution to within the Monte-Carlo statistical error. For interacting bath spins (Ising coupling J), we find a non-monotonic dependence of the central-spin coherence on J, with a maximum at J of order the bath precession frequency Omega --- a collective motional narrowing effect that is absent for the bosonic bath. We formulate eight falsifiable predictions. The derivation relies only on established theorems (Lieb-Robinson, Margolus-Levitin), exact solutions (polaron transformation, spin-bath dephasing), and exact diagonalization; it contains no adjustable parameters and no fitting.