Multi-stakeholder recommendation systems optimize proxy objectives that may harm unobserved stakeholders. We identify a precise condition governing when this occurs: in Bradley-Terry (BT) preference learning with K stakeholders, additional training data degrades a hidden stakeholder's expected utility if and only if the cosine similarity between the learned weight vectors of the optimization target and the hidden stakeholder is negative. We validate this directional Goodhart condition on 32 data points with | cos | > 0.2 across 4 datasets (MovieLens-100K, MovieLens-1M, MIND, Amazon Kindle), 13 stakeholders, and 3 feature families (D {19, 32, 35} dimensions), with 100% accuracy (32/32) under softmax-weighted evaluation and 91% (29/32) under hard top-K selection. We also show that a standard Pareto frontier metric (Hausdorff distance) produces a false positive Goodhart signal on synthetic data where the condition correctly predicts no harm. Two precisions emerge from the cross-dataset analysis. First, the relevant cosine is between BT-trained weight vectors, not raw stakeholder weights; on high-dimensional sparse features, BT training remaps the cosine geometry (up to 3 of 10 pairs flip sign on MIND). Second, the condition governs expected utility under the learned score distribution; under hard top-K selection, it can reverse for stakeholders with high utility variance near the selection boundary (5 of 20 MIND pairs, 0 of 8 Amazon, 0 of 6 MovieLens). The reversal vanishes at softmax temperature T ≥ 1. Applied to X's (formerly Twitter) open-sourced recommendation algorithm, the Goodhart risk reduces to one observable: whether the platform treats negative signals (blocks, reports) as positive engagement. We provide an audit toolkit, a data budget analysis (median 34 preference pairs to recover 50% of hidden stakeholder harm across 16 dataset-stakeholder configurations), and practical guidance for platform transparency under the EU Digital Services Act.