Web answer synthesis with Mixture-of-Agents (MoA) requires reliable factual support for every atomic claim, yet agreement among agents is not verification. Correlated proposers can repeat a shared misconception, and a correct specialist fact receives little support when the other proposers stay silent. We introduce C-MoA, which decomposes MoA outputs into atomic claims, scores inter-agent semantic support, and calibrates claim retention at the level of whole answers. Under exchangeability and a policy frozen before calibration, it provides distribution-free marginal control of the probability that an answer retains any claim unsupported by the reference labels. Because calibration only selects a cutoff on a fixed ranking and cannot repair shared errors, we also study CONTRA-MoA, an optional extension that adds blinded near-miss tournaments, leave-one-agent-out stability, and availability-aware fusion, and we derive a competence--discrimination law for its verifier. Across 300 splits of the FactScore biography benchmark, the mean rate of fully supported answers (Fact) exceeds all six nominal targets, but only $1.7\%$--$9.8\%$ of supported claims are retained and $30\%$--$75\%$ of answers are empty. At $α=0.10$, mean agreement retains $3.1\%$ of supported claims, about $3.4$ times the $0.9\%$ retained by the best single-proposer score. On the medical Golden Dataset, it retains $37.3\%$ at a mean Fact of $0.898$, slightly below the $0.90$ target, and on NQ mean Fact reaches $0.904$. On a separate balanced FactScore cohort, memory-only counterfactual scoring is close to chance (AUC $0.522$ on 182 claims), and raw maximum fusion lowers the agreement AUC from $0.687$ to $0.652$. These results separate the statistical calibration of factuality from informative claim verification and clarify the role of each in multi-agent Web synthesis.

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