Predicting the Frozen Ceiling: A Train-Free Diagnostic for Self-Supervised Retrieval

How far can frozen self-supervised features go for strict nearest-neighbor retrieval, and can that limit be known without training? We show that the retrieval ceiling collapses to a single sufficient statistic—the signal-to-spread ratio of the per-query log-distance ratio, \(J=E[\log(b/a)]/Var(\log(b/a))\)—so that a closed-form diagnostic \(R@1=\Phi(J)\), computed from just five summary statistics with no training and no labels, recovers observed strict \(R@1\) to within 2.4 points (raw) and 5.2 points (readout) and ranks backbones at zero cost (ImageNet-100 Spearman 0.943). The diagnostic reveals that frozen DINOv2 features are linearly closed: no linear post-processor improves retrieval, and only a supervised nonlinear readout opens the space (0.688\(\to\)0.739). The ceiling is governed by the spread of the log-ratio, not by margins or dependence alone; raising dependence \(\rho\) is provably insufficient as shared query difficulty can drive \(\rho\to1\) without changing retrieval. The trained readout acts through regime-dependent mechanisms—\(\rho\)-driven on COCO, margin-driven on Stanford Dogs—and the theory-derived loss \(L_J\) underperforms SupCon when trained, exposing a prediction-optimization gap we interpret as a mechanistic finding. The result is a train-free, mechanism-aware diagnostic that reveals when frozen features are closed and why supervised readouts open them.