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Q-learning is one of the most fundamental algorithms in reinforcement learning. We analyze constant-stepsize Q-learning through a direct stochastic switching system representation. The key observation is that the Bellman maximization error can be represented exactly by a stochastic policy. Therefore, the Q-learning error admits a switched linear conditional-mean recursion with martingale-difference noise. The intrinsic drift rate is the joint spectral radius (JSR) of the direct switching family, which can be strictly smaller than the standard row-sum rate. Using this representation, we derive a finite-time final-iterate bound via a JSR-induced Lyapunov function and then give a computable quadratic-certificate version.