Transformers Can Learn Rules They've Never Seen: Proof of Computation Beyond Interpolation

A central question in the debate over large language models is whether transformers can learn rules they have never seen, or whether they can only interpolate: predict new cases from their similarity to training examples. We test this in a controlled setting where interpolation provably fails, so success can only come from computation beyond interpolation. We train small transformers to predict the rollout of a cellular automaton whose update rule is pure XOR, and remove one entry of the rule's truth table from all direct supervision. The missing entry's output is never shown to the model; its only trace is indirect, as wrong values corrupt visible predictions at later timesteps. Because XOR parity flips whenever one input bit is changed, every one-bit neighbour of the missing entry carries the opposite label, and we prove that similarity-based predictors, including nearest-neighbour, kernel, and Gaussian-process methods, are forced to the wrong answer. A two-layer transformer can nevertheless recover the missing entry, and circuit extraction confirms it computes XOR exactly. Ablations show the recovery depends on gradient signal propagating through multi-step prediction, and a second, structurally unrelated benchmark on symbolic operator chains exhibits the same capacity under ordinary autoregressive training. Together with a constructive proof that a standard transformer block can implement exact local Boolean rules, these results provide an existence proof that transformers can learn rule structure not directly observed in training and express it explicitly. This rules out the strongest architectural form of the interpolation-only account, the claim that transformers cannot in principle discover and communicate unseen rules, while leaving open when such behaviour arises in large-scale language training.