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This paper presents a comprehensive performance analysis of a downlink non-orthogonal multiple access (NOMA) network assisted by an active reconfigurable intelligent surface (ARIS) in a cognitive spectrum-sharing scenario. Unlike conventional passive RIS (PRIS), the ARIS can both adjust phase shifts and amplify incident signals, thereby mitigating inter-user interference and overcoming multiplicative fading. We consider a two-user secondary network coexisting with a primary user, where the base station communicates with the secondary users via the ARIS. Closed-form expressions for the outage probability (OP), throughput, energy efficiency (EE), and an approximation for the ergodic data rate (EDR) are derived under Nakagami-m fading, along with asymptotic OP analysis to reveal the achievable diversity order. We also formulate and solve an optimization problem for the NOMA power allocation coefficient to minimize OP. The main contributions are: (i) proposing an ARIS-assisted NOMA architecture for spectrum-sharing networks and comparing it with PRIS and orthogonal multiple access (OMA) schemes; (ii) deriving exact OP, throughput, and EE expressions, and an approximate EDR expression, validated by Monte Carlo simulations; (iii) providing asymptotic OP analysis to characterize the diversity order; and (iv) optimizing NOMA power allocation to minimize OP. Numerical results confirm that ARIS significantly outperforms PRIS and OMA in terms of OP, throughput, EC, and EE, demonstrating its potential to enhance spectral efficiency, reliability, and coverage in next-generation spectrum-sharing NOMA networks.