The T₂ₒ state observed in the decay Dₒ₁ (2460) ^+ Dₛ^+π^+π^- provides direct evidence for an isovector open-charm tetraquark state with strangeness. We develop a unified theoretical framework that consistently incorporates triangle loops and DK-Dₛπ rescattering. Especially, DK-Dₛπ coupled-channel interactions through off-diagonal potential terms provide a novel perspective on the origin of the T₂ₒ pole. Based on the systematic description of the Dₒ₁ mass spectrum, the two-peak structure in Dₒ₁ (2460) decay is perfectly reproduced, and explained by the interference of the f₀ resonance and rescattering diagrams. In contrast, only one-peak structure is predicted in Dₒ₁ (2536) decay, since it is dominated by rescattering only. This difference originates from the S- and D-wave dominance for Dₒ₁ (2460) and Dₒ₁ (2536) coupling with D^*K channel, respectively, which reflects the internal structures of the two Dₒ₁ states. This unified approach demonstrates how decay and production mechanisms encode different aspects of dynamics, offering an opportunity to disentangle the nature of exotic hadrons.
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