Accuracy stops being a pass: shared errors make three observers worth one, so certify the deployment

Whether a theory-of-mind module should be deployed depends on what its errors cost together in that deployment, not on how accurately it infers. The prior practice sets a threshold on inference accuracy and trusts a module that clears it. Across task assignment, Level-Based Foraging and Hanabi, accuracy errs in both directions: it approves harmful language-model deployments in Hanabi and rejects beneficial ones in task assignment.

Errors are not independent, and their co-occurrence has a sign: positively associated errors raise the benefit of acting where their damage overlaps and lower it where damage compounds. An accuracy threshold assumes the errors of the many inferred beliefs are independent, so its failure has a predictable direction. In Hanabi, sharing each teammate's belief across all observers costs 0.17 to 0.35 of the achievable benefit with two or more lives in 16 of 16 cells and raises the threshold by 0.04 to 0.11.

Accuracy buys far less redundancy than independence predicts, because the remaining errors of accurate models are largely blind spots every copy shares. Independence predicts that three observers all misread a teammate at the cube of the single-observer error rate. On teammates the models had evidence about, per-belief error falls from 32% for GPT-5.4-mini to 11% for Opus 5.5, but joint misreading falls only from 16% to 9%, where independence predicts 3.4% to 0.15%.

Pricing error costs, pairwise interactions and measured dependence certifies a module without deploying it. Accuracy reads only the module's own inference correctness, not what a wrong action costs in the deployment. On held-out data from the cells it was developed on, the gate makes no significant false approval in 20 splits against 3.2 per split for accuracy; on modules it was not developed on, it reaches regret 0.004 on seven classifiers against 0.479 for accuracy.

The findings rest on controlled benchmarks with known hidden variables, which is what allows errors of known kinds to be injected; the human data, whose hidden states nobody defined, show the same knee and the same two directions. The gate is a second-order certificate and is no longer exact when three or more interacting beliefs are often wrong together, a regime its own statistics flag when the mean error load exceeds about four, where direct measurement can take over. It needs a simulator that accepts injected beliefs and assumes a module's errors are unrelated to the episode state that sets their cost.

The language-model tests use one prompt family, relabelled once, and most human team analyses combine judgments made independently, so richer prompting and teams that deliberate are natural next tests. The gate matches a 95% lower-bound A/B test on the cells it was developed on, but on the out-of-sample Bayesian module it still makes one significant false approval, on a cell whose accuracy sits at the bar. The preregistered prediction that a third-order correction would help was not met, since the measured interactions are small and noisy at 250 games, so the second-order gate stands.