Two Runs, One Law: An Entropy Balance for Reasoning Bias

We investigate how chain-of-thought reasoning handles biases by analysing its geometry. We show that reasoning bias is a two-run quantity, which can only be known as the difference between a cued run and its uncued twin. Taking as a mediator the vector a layer passes forward, its hidden state, we apply VanderWeele's four-way decomposition to split the bias into a natural indirect effect the state carries, a natural direct effect the later computation supplies, and their interaction. The depth at which the state becomes a near-complete mediator of the cue's effect we call the crossover, found in nine models: 7B to 70B, base and mixture-of-experts. We measure the second moment of the cued-minus-uncued differences across items and read two quantities from it, the number of directions the correction occupies and the share carried by one direction common to all items. Across nine bias datasets and six models these two quantities sort biases into those that a single fixed direction can correct and those that require a per-item correction; both published corpora of cognitive biases fall in the second class, on every model, and neither the amount of cue text nor the size of the state's response is what decides it. The crossover says where a correction belongs and why it must be per-item. Only the uncued twin supplies a per-item target, so we distill it into one pass. This counterfactual twin distillation on the uncued run's answers improves held-out balanced accuracy on cued prompts by 11 to 24 points across 30 Kahneman–Tversky families in five models, without prior knowledge of the target bias type. The two-run geometry holds for MMLU position bias, 11 BBQ categories, and six further published bias datasets.