Off-Axis, On Purpose: Where a Transformer Computes Concepts and Why it Does So

2026-08-10

AI Summary

This study investigates why Transformer intermediate layers encode conceptual information in subspaces nearly orthogonal to the output reading axis—termed “off-axis representations”—and demonstrates that such geometry is functional rather than redundant. Through geometric analysis, subspace projection, controlled rotational interventions, and ablation studies—augmented with early-exit training and custom loss functions—the authors find that off-axis subspaces are critical for cross-token information mixing. While forcing alignment with the reading axis preserves standard task performance, it substantially compresses the effective representational dimensionality. Notably, models can effectively utilize predefined random bases without performance degradation, and sparse rotation strategies enhance training stability. This work provides the first evidence of the functional role of off-axis structure and proposes a method to guide models toward stably exploiting this geometric property via predefined rotations.

Abstract

A transformer's answer lives on one axis: the direction its unembedding reads. Its intermediate states largely do not, and that off-axis position is usually treated as an obstacle to interpretation. We show it is functional. A 12-layer model computes in two phases. Through the first, every sublayer writes into a subspace held near-orthogonal to the read-out, attention 75 to 96 degrees off it at every depth. Moving attention's values onto the read-out is 64 to 84 times more damaging than a matched random rotation, and the damage is entirely in cross-token mixing: the subspace insulates composition from the vocabulary. Beneath it the frame itself turns rigidly with depth. In the second phase the answer arrives on-axis, late, and by addition rather than by turning accumulated content onto the read-out. Pressing every layer onto the read-out instead, as training for early exit does, matches the baseline on perplexity, LAMBADA and BLiMP while cutting the concept-phase workspace from about twenty-five effective dimensions to fourteen, a change none of those benchmarks register. The geometry can also be imposed, though not by asking for it. Prescribing it through the loss is a lottery: six of eight seeds collapse, because a model told to null its read-out projection obeys most cheaply by discarding dimensions. Inserting one fixed rotation at the phase boundary lands it instead, at baseline quality. A sparse rotation the surrounding weights can absorb converges on all nine seeds, against five of nine for ordinary training. Which rotation is immaterial: twenty-five runs across thirteen distinct ones reach the same quality, and two baselines from different seeds hold their concepts in near-orthogonal frames while agreeing on their read-outs. That freedom is usable: a basis drawn at random and prescribed before training is adopted across the concept phase, with quality unchanged.

Problem

• Transformer
• off-axis representation
• concept computation
• geometric structure
• interpretability

Innovation

• off-axis representation
• orthogonal subspace
• concept composition
• geometric inductive bias
• fixed rotation injection