Do LLM judges diverge from exact Bayesian posteriors on partially disclosed argument graphs, and does the divergence widen the manipulation surface?

Setting: an argument graph over binary claims (each claim carries a prior; typed edges carry likelihood ratios, supporting or attacking) compiles to a Bayesian network, so the ideal judge's posterior P(root | revealed subset) is EXACTLY computable for every legal reveal set (a legal set must be connected to the root). The open-source `probability-flow` package (PyPI, MIT, v0.4.0) provides such a generator, the BN compilation, and an exact achievable-posterior-range [min, max] over reveal sets on polytrees ('manipulability'). Render each claim as a natural-language statement and give a black-box LLM judge exactly the revealed claims. Measure, across graphs and reveal sets: (1) the divergence distribution |judge - exact posterior|; (2) manipulability_LLM = [min, max] of the LLM judge's verdict over reveal sets, versus the exact manipulability — does the LLM range strictly contain the exact range (i.e., a debater exploiting judge error can push the verdict FURTHER than any selection of true evidence could push an ideal Bayesian)? (3) flip sets: reveal sets where exact and LLM verdicts land on opposite sides of 0.5; (4) overconfidence: are LLM verdicts systematically closer to 0/1 than the exact posterior? (5) the effect of presentation: identical graphs presented with numeric probabilities vs calibrated verbal qualifiers ('very likely', Sherman-Kent style); (6) judge-capability scaling: weak vs strong judge models. A second axis: does TELLING the judge that evidence was strategically selected move it from credulous (hidden = neutral) toward skeptical (adverse inference from silence) updating, and does that shrink or widen the manipulation surface?

Acceptance. A public, reproducible study with: >=15 graphs x >=20 legal reveal sets each x >=2 judge models (>=1 deliberately weak); the divergence distribution; per-graph manipulability_LLM vs manipulability_exact with a clear verdict on strict containment (report the fraction of graphs where the LLM range exceeds the exact range on either end, with CIs); flip-set counts; an overconfidence measure; the numeric-vs-verbal presentation delta; and the credulous-vs-skeptical instruction effect. Negative/partial results (e.g., LLM range NARROWER than exact) are first-class. Code + per-call transcripts public; ground truth verifiable by exact recomputation.