Message capacity and claim wording set the transition points of collective truth-finding in language-model networks

Whether human or large language model (LLM), an agent in a discussion reads only a few of the others' contributions, bounded by cognition, context, or cost. LLM collectives can settle on a wrong consensus even when a majority starts out correct; we ask how far that reading bound alone decides the outcome. We model the bound with one number, the message capacity, which sets how many of the others' messages an agent reads, and generate the communication network from it. Over 31,824 randomized queries, we found that an 8-billion-parameter model's judgment of a claim effectively reduces to a logistic function of a weighted sum of its inbox, the update rule of a stochastic binary neuron with divisively normalized weights. From these weights and the network's degree statistics alone, the wrong consensus should become unreachable from any start once agents read, on average, fewer than 6.4 of their 31 sources. In 1,414 episodes with assigned starts the prediction failed: the correct side won in fewer than 50% of episodes from every start, and in only 28-45% when 75% of agents started correct. The failure traces to the field, the threshold that a claim's wording sets for the agent's answer before any message is read: the experimental claims' fields lay below the calibration mean, and with each claim's own field the same weights reproduce the outcomes. Reversing the wording showed that the threshold follows what a claim asserts, not whether it is true. On a second 8B model the pipeline predicts claim-dependent bistability; transition points appeared where computed, and an eight-claim calibration matched in 15 of 16 conditions. At 70B the assertion bias is not detected. Thus a collective's fate is largely set by two single-agent measurements: the threshold a claim's wording sets, and the message capacity that sets the transition point.