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We analyze hidden-state dynamics in trained long short-term memory (LSTM) networks and show that small networks near their optimal training epochs (steps) exhibit scale-free avalanche statistics and branching parameters close to unity, indicative of near-critical dynamics, while larger models remain subcritical. To explain the coexistence of subcritical branching with robust $1/f^{\\beta}$ noise, we introduce a mixture branching process framework that links heterogeneous branching dynamics to long-range temporal correlations. 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To explain the coexistence of subcritical branching with robust $1/f^{\\beta}$ noise, we introduce a mixture branching process framework that links heterogeneous branching dynamics to long-range temporal correlations. 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