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

arXiv: https://arxiv.org/abs/2605.29634 · HTML: https://mlsanity.com/paper/2605.29634/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 27 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 20 of 20)

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- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Attribution Patching Outperforms Automated Circuit Discovery](https://mlsanity.com/paper/2310.10348/) (2023)
- [Linearity of Relation Decoding in Transformer Language Models](https://mlsanity.com/paper/2308.09124/) (2023)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [How do Language Models Bind Entities in Context?](https://mlsanity.com/paper/2310.17191/) (2023)
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- [Do Sparse Autoencoders Capture Concept Manifolds?](https://mlsanity.com/paper/2604.28119/) (2026)
- [Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior](https://mlsanity.com/paper/2605.05115/) (2026)
- [A polar coordinate system represents syntax in large language models](https://mlsanity.com/paper/2412.05571/) (2024)
- [Interpretability without actionability: mechanistic methods cannot correct language model errors despite near-perfect internal representations](https://mlsanity.com/paper/2603.18353/) (2026)
- [The Shape of Beliefs: Geometry, Dynamics, and Interventions along Representation Manifolds of Language Models' Posteriors](https://mlsanity.com/paper/2602.02315/) (2026)
- [Arithmetic in the Wild: Llama uses Base-10 Addition to Reason About Cyclic Concepts](https://mlsanity.com/paper/2605.01148/) (2026)
- [Hypothesis-Driven Feature Manifold Analysis in LLMs via Supervised Multi-Dimensional Scaling](https://mlsanity.com/paper/2510.01025/) (2025)
- [Cell-Based Representation of Relational Binding in Language Models](https://mlsanity.com/paper/2604.19052/) (2026)
- [Evaluating Relational Reasoning in LLMs with REL](https://mlsanity.com/paper/2604.12176/) (2026)

## Abstract (arXiv)

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.