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Crowdsourced fact-checking systems have been adopted by major social media companies such as X, Meta, TikTok and Google with the aim of combating misleading information at scale without relying on centralized editorial control. These systems have been developed around a common underlying concept: a bridging mechanism that identifies notes flagging misleading information when they receive support from people with different perspectives rather than simple majority support. To our knowledge the only publicly disclosed bridging algorithms deployed for fact-checking are based on matrix factorization, as deployed by both X and Meta, augmented with additional components addressing abuse, targeted manipulation, and contributor brigades. This work examines the core matrix factorization portion of these systems, presenting theoretical and empirical evaluations of the degree to which coordinated users could vote strategically by leveraging the latent representations to fabricate the appearance of synthetic consensus within the bridging mechanism. Using historic production data, we find that up to 10.7% of lower quality notes could be manipulated above consensus thresholds using less than 10 ratings. We complement these findings with a theoretical analysis, revealing counterintuitively that rating a note as "Not Helpful" can increase its helpfulness score, as well as a cost model quantifying manipulation effort. We have developed and deployed mitigations within X's Community Notes algorithm to address synthetic consensus.