Which Coherence Decoheres? Basis-Dependent Decoherence Rates in Symmetry-Broken Collective Spin Systems

In the ordered phase of a mathbb{Z}₂-symmetric collective spin system, two natural bases - localised pointer states \{|Prangle,|Rrangle\} and energy eigenstates \{|E_0rangle,|E_1rangle\} - yield Lindblad dephasing rates that differ by a factor approaching 2 as N→infty and reaching 2.42 near the quantum-critical crossover. The discrepancy has a single algebraic origin: parity forces langle E_i|hat{J}_z|E_irangle=0 exactly, eliminating the cross-term that doubles the localised-state rate. Two distinct protection factors are identified: η_{rm MF}=\(Nm_*\)²/\(2G₀₁\)approx2.42, where m_* is the order parameter and G₀₁=frac{1}{2}\(langle E₀|hat{J}_z²|E_0rangle+langle E₁|hat{J}_z²|E_1rangle\) (advantage over the classical mean-field estimate), and η_{rm exact}=\(G₀₁+J₀₁²\)/G₀₁approx1.86, where J₀₁=langle E₀|hat{J}_z|E_1rangle (exact physical ratio of pointer-state to eigenstate decay rate). In the thermodynamic limit the secular approximation fails, the doublet degenerates, and both rates converge. The three-regime structure is demonstrated in the Lipkin-Meshkov-Glick model via exact diagonalisation, and the algebraic origin of the discrepancy is established via the mathbb{Z}₂ parity of the Lindblad jump operator.