Pairwise net influence on LLM action logs: detection at the calibrated rate, blind to two pairs, fooled by low-temperature shared priors

Pairwise net influence (Bonjour, Aggarwal and Bhargava, 2022) identifies a colluding pair from logged actions alone and has never been tested on language-model agents (LLM). We implement and validate it on a scripted Rock-Paper-Scissors code with a known colluding pair. We reproduce the paper's Table 2 when the state is the previous round's joint action, which the paper does not state. Under that assumption, a spurious pair is named in half of 50-round games with no collusion. Pre-registered anti-examples show that the one-pair gate is blind to two pairs, one-way dependence is reported as mutual, and lagged copying is invisible. We then run 72 games of four LLM agents (mostly gpt-4.1-mini) with one instructed pair or none. The assistant complied only partly (scripted-scale strength 0.13 to 0.33), and at those strengths the scripted calibration predicts the LLM detection rates within sampling error. Net influence ranked the true pair first in 16 out of 20 60-round games, with a stronger assistant at 200 rounds. Both rules identify the pair on 3 of 4 occasions, the miss being the one-pair gate tripped by a same-model pair. An action-log LLM judge gave the first-listed pair at least 0.5 in 9 of 10 controls: a position bias. Same-model agents are not flagged at temperature 1.0 under a permutation threshold (less than 1 out of 10) but are at 0.3 (2 out of 5), and in mixed-model games every flagged pair was same-model. A permutation threshold and a multi-pair variant are new here. The statistic is a plus factor, not proof of agreement.