Relational Rank Geometry in Transformers: Detecting and Steering Hidden-State Relation Frames

Abstract

Transformer hidden states are often interpreted through local or low-order objects: neurons, sparse features, attention heads, residual-stream directions, or activation patches. This paper studies a complementary object: the rank-indexed geometry of relations among token tuples. I use Plucker sign entropy to test whether r-argument relations leave arity-matched orientation signatures in hidden-state space. Across Llama-family 8B, 70B, and 405B checkpoints, true relation tuples show stronger orientation-sign consistency at the expected rank k=r for r=3,...,6 than scrambled tuples under matched random-control audits. Multi-template audits show that the effects survive surface variation, with all tested 405B rows retaining positive expected-rank margins and 8B/70B retaining positive rows with constructor-specific mixed cells. I then ask whether the same relation geometry can be steered. In an edge-grid clean/corrupt intervention assay over 32 prompts, the row/column scaffold and answer format stay fixed while the YES/NO relation map changes, and the corrupt hidden-state relation frame is patched toward clean or placebo targets. In 70B and 405B, clean-targeted relation-frame paths recover clean-answer behavior and residual relation geometry, while centroid-only and equal-norm controls show negligible recovery. Site/order controls further separate marker-site importance from ordered clean-frame geometry: target clean shape and cross-prompt clean shape recover behavior and residual geometry at the marker interface, whereas corrupt-donor transfer, same-site permutation/reflection, wrong-site clean deltas, centroid-only motion, and equal-norm noise fail or remain far below clean-frame paths. The result is a controlled bridge from relation probing to relation-frame intervention: relation rank geometry can be detected, targeted, and behaviorally validated in transformer hidden states.