<![CDATA[This study addresses the limitations of conventional nonlinear autoregressive models, which struggle to maintain stability and generalization in high-dimensional, non-stationary forecasting environments. The research aims to develop a mathematical framework that integrates deterministic dynamics with probabilistic uncertainty through the proposed Quantum-Entropy NARX (Q-ENARX) model. The methodology combines nonlinear autoregressive modeling, entropy-based trust-region optimization, and quantum information theory to establish a unified formulation for dynamic forecasting. The model embeds NARX states into a quantum Hilbert space, introduces an entropy-regularized loss function to balance accuracy and uncertainty, and employs a quantum Fisher Information Matrix for curvature-aware optimization. Analytical derivations reveal that Q-ENARX achieves enhanced stability, improved generalization, and robust convergence by leveraging quantum state dynamics, entropy-energy duality, and fractional learning operators. The results shows that the integration of entropy and quantum principles transforms traditional NARX forecasting into a probabilistically interpretable and physically grounded framework capable of capturing complex temporal correlations with high mathematical precision.]]>
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