Algorithms for solving nonlinear fixed-point equations---such as average-reward Q-learning and TD-learning---often involve semi-norm contractions. Achieving parameter-free optimal convergence rates for these methods via Polyak–Ruppert averaging has remained elusive, largely due to the non-monotonicity of such semi-norms. We close this gap by (i.) recasting the averaged error as a linear recursion involving a nonlinear perturbation, and (ii.) taming the nonlinearity by coupling the semi-norm's contraction with the monotonicity of a suitably induced norm. Our main result yields the first parameter-free ~O(1/√t) optimal rates for Q-learning in both average-reward and exponentially discounted settings, where t denotes the iteration index. The result applies within a broad framework that accommodates both synchronous and asynchronous updates, single-agent and distributed deployments, and data streams obtained from either simulators or along Markovian trajectories.