Can Ideologues Agree on Quality? From Non-identifiable Latent Factors to Collective Outcomes

To surface "objective" fact-check notes, X's Community Notes uses Biased Matrix Factorization (BMF) on crowd-sourced ratings to separate note quality from reviewer ideology. We show that for any correctly specified model that decomposes ratings into quality and ideological alignment, this separation requires fixing an ideological anchor: a reference point for what counts as neutral. We give a characterization for the standard implementation: with $L_2$ regularization, BMF anchors quality at a point proportional to the mean reviewer ideology, recovering objective quality if and only if the mean reviewer is neutral. This parallels the Condorcet Jury Theorem: collective judgment recovers truth when the pool is unbiased on average. When that assumption fails, BMF's recovered ranking shifts toward maximizing aggregate reviewer satisfaction, which coincides with social welfare only when the pool is representative. This is not specific to BMF; the rating matrix admits a continuous family of gauge-equivalent decompositions, each implying a different quality ranking, so no mechanism based on ratings alone can recover quality without implicitly choosing such an anchor. What appears to be routine hyperparameter tuning is in fact a normative choice about whose preferences define quality.