Stable training is often treated as evidence that learning is succeeding, but stability characterizes optimization behavior rather than correctness relative to an external objective. We study what happens when the signal being optimized remains persistently biased. We define Stable but Wrong (SBW) as a learning state in which optimization remains active or stabilizes while the learned outcome stays systematically displaced from an independently defined objective. A minimal theoretical model shows that persistent systematic bias can separate a stable convergence point from the true objective even under standard conditions for stable convergence. We then introduce static and dynamic bias in reinforcement learning, supervised learning, and continual fine-tuning of large language models. Across these settings, optimization of the current learning signal continues or stabilizes while correctness relative to the external objective declines or remains displaced. Finally, two controlled challenge-correction experiments with external ground truth test whether a prespecified error can be exposed and changed. In controlled scientific discovery and CIFAR-10, targeted evidence corrects or reduces established errors more effectively than equal-budget general evidence, with no evident harm to paired correct states under the tested protocols. These experiments do not provide a general method for automatically discovering unknown errors. Overall, SBW identifies a basic limit of convergence and other internal optimization metrics as evidence of correctness in systems that learn through persistent bias and feedback.

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