Semantic-Frame Identifiability and Recovery in Geometric Representation Learning

Geometric representations are often used not only to predict relations, but also to infer hierarchy, abstraction, or uncertainty from an embedding’s origin and direction. However, accurate distance-based prediction does not establish that these semantic choices are identifiable. We formalize this distinction as semantic-frame identifiability: observationally equivalent representations can support different semantic targets despite identical intrinsic predictions. We establish a minimax lower bound determined by the semantic ambiguity within each observational equivalence class and characterize how asymmetric evidence resolves decision-relevant ambiguity. Building on this analysis, we introduce FramePosterior, a semantic decision layer that leaves the encoder and native predictor frozen, conditions candidate frames on sparse, noisy anchors, and aggregates task-specific decisions across rooted charts. Exact-isometry audits distinguish unchanged native predictions, altered semantic readouts, and measurable downstream loss. On ImageNet-100, a radial-depth readout loses 12.68 percentage points in within-one-level accuracy, while 32 noisy landmarks recover 89.88% of its depth-error gap. On the full WordNet noun DAG, posterior voting outperforms matched MAP inference by 1.48 percentage points with 16 anchors under 20% corruption. A topology-informed routing atlas improves delivery from 69.90% to 77.92% over a matched no-revisit control. Stable visual readouts provide an important boundary: frame ambiguity does not necessarily degrade every task. These results clarify when metric prediction is insufficient for semantic decisions and how explicit evidence enables targeted recovery without changing native predictions.