Murphys Laws of AI Alignment: Why the Gap Always Wins

We study reinforcement learning from human feedback under misspecification. Sometimes human feedback is systematically wrong on certain types of inputs, like a broken compass that points the wrong way in specific regions. We prove that when feedback is biased on a fraction alpha of contexts with bias strength epsilon, any learning algorithm needs exponentially many samples exp(n*alpha*epsilon^2) to distinguish between two possible "true" reward functions that differ only on these problematic contexts. However, if you can identify where feedback is unreliable (a "calibration oracle"), you can focus your limited questions there and overcome the exponential barrier with just O(1/(alpha*epsilon^2)) queries. This quantifies why alignment is hard: rare edge cases with subtly biased feedback create an exponentially hard learning problem unless you know where to look. The gap between what we optimize (proxy from human feedback) and what we want (true objective) is fundamentally limited by how common the problematic contexts are (alpha), how wrong the feedback is there (epsilon), and how much the true objectives disagree there (gamma). Murphy's Law for AI alignment: the gap always wins unless you actively route around misspecification.