Coordinates Without a Compass: Gauge Symmetry and Trust Anchors in LLM Verification

Relational evidence can recover every pairwise truth relation while leaving absolute truth unresolved. We study verification systems whose judgments are correct only up to a shared binary orientation. Flipping both latent truth and this orientation preserves every observation, so no aggregation of such evidence can certify an individual claim. We formalize this ambiguity as a gauge symmetry and show that certification is possible exactly when truth is constant across observationally indistinguishable states. A trusted anchor resolves the ambiguity only when it fixes the target mechanism's orientation. Consequently, in noiseless relational systems, trust cost is determined by the number of independently orientable components, not by the amount of evidence. Experiments support these predictions. On MMLU-Pro, DeepSeek recovers the truth partition with 98.5% accuracy, yet unanchored absolute accuracy remains 50%; one held-out anchor raises it to 97.0%. HumanEvalPlus audits show that anchors help aligned mechanisms but can hurt oppositely oriented ones. A label-free router topology predicts the same within-component advantage in two frozen cohorts. Relational coordinates become a compass only when a mechanism-compatible anchor supplies their orientation.