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

Mazen Kobrosly

cs.LGMay 28, 2026

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

Authors: Mazen Kobrosly

Organizations: Independent Researcher

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.

Explore similar work

May 31, 2026cs.AI

Emergent Ordinal Geometry in Transformers Trained on Local Comparisons

Transitive inference is the challenge of inferring that A < C from knowing only adjacent relations (A < B, B < C). It is solved by humans and animals not through logical chaining but via an analogue mental number line, whose signature is the symbolic distance effect: distant comparisons are easier than nearby ones. We ask whether Transformers acquire the same primitive, training small models exclusively on adjacent comparisons from a hidden total order and evaluating generalization to unseen distant pairs. We find that out-of-distribution generalization emerges alongside a striking geometric reorganization: entity embeddings collapse onto a one-dimensional manifold whose principal axis recovers the hidden rank order with near-perfect fidelity, and this structure is sensitive to optimization in ways that produce grokking-like transient dynamics. Critically, even when accuracy is at ceiling, decision confidence and geometric separation both scale monotonically with rank distance, directly mirroring the symbolic distance effect observed across decades of behavioural experiments on humans, primates, and rodents. We further show the same rank-aligned geometry in a pretrained large language model, where it tracks the topology of each ordinal relation: linear for sizes and digits, cyclic for months. These results ground a 50-year-old behavioural regularity in the geometry of learned representations, offering a mechanistic account of transitive inference that bridges cognitive science and modern neural networks.

Nishit Singh

Transformer Architectures Distance Cognitive Science Comparison

Ask PDF

Apr 26, 2026cs.LG

Rank, Head-Channel Non-Identifiability, and Symmetry Breaking: A Precise Analysis of Representational Collapse in Transformers

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

Giansalvo Cirrincione

Transformer Architectures Batch Normalization Feed-Forward Network Symmetry Identifiability Ranking

Ask PDF

May 12, 2026cs.CL

Geometric Factual Recall in Transformers

How do transformer language models memorize factual associations? A common view casts internal weight matrices as associative memories over pairs of embeddings, requiring parameter counts that scale linearly with the number of facts. We develop a theoretical and empirical account of an alternative, \emph{geometric} form of memorization in which learned embeddings encode relational structure directly, and the MLP plays a qualitatively different role. In a controlled setting where a single-layer transformer must memorize random bijections from subjects to a shared attribute set, we prove that a logarithmic embedding dimension suffices: subject embeddings encode \emph{linear superpositions} of their associated attribute vectors, and a small MLP acts as a relation-conditioned selector that extracts the relevant attribute via ReLU gating, and not as an associative key-value mapping. We extend these results to the multi-hop setting -- chains of relational queries such as ``Who is the mother of the wife of

x

?'' -- providing constructions with and without chain-of-thought that exhibit a provable capacity-depth tradeoff, complemented by a matching information-theoretic lower bound. Empirically, gradient descent discovers solutions with precisely the predicted structure. Once trained, the MLP transfers zero-shot to entirely new bijections when subject embeddings are appropriately re-initialized, revealing that it has learned a generic selection mechanism rather than memorized any particular set of facts.

Shauli Ravfogel, Gilad Yehudai, Joan Bruna +1

Transformer Architectures Factual Recall Relational

Ask PDF