Structural Entanglement in the Informative Subspace: Eight Experiments on Why Every Direction Carries Every Concept

Executive Summary

This paper uses factorial direction decomposition — multi-output ridge regression over three simultaneously varied concepts, followed by SVD — and eight experiments across four transformer models (GPT-2 124M, Qwen-0.5B, Qwen-7B, and Qwen-7B-Instruct). The V-matrix shows each direction loading primarily on one concept, but the damage matrix shows every informative direction causing 39–88% accuracy drops across all three concepts. Random Gaussian projections to 448 or more dimensions reproduce the learned entanglement intensity (EI = 1.50 in Qwen-7B), while projections to the 7-dimensional informative rank show near-baseline EI (0.18 in Qwen-7B). PCA reduces it below baseline. Stratified bootstrap confidence intervals (2,000 iterations) place three of the four models entirely above EI = 1.0; the fourth (Qwen-7B) has a lower bound of 0.965.

Superlinear amplification is confirmed: when three concepts are probed simultaneously, the triple EI exceeds the mean pairwise EI by 2x (GPT-2: 1.87x, Qwen-7B: 2.15x). This is not a measurement artifact — nesting two concepts into one reduces EI below the pairwise baseline, confirming that independent concept axes drive the superlinearity. Concept-type independence is validated by replacing linguistic concepts with software engineering concepts, producing comparable EI (mean SE/Original ratio = 0.97).

RLHF increases entanglement and accelerates its crystallization: the instruction-tuned Qwen-7B crosses EI = 1.0 at depth fraction 0.222 versus 0.296 for the base model, and reaches a higher peak EI (1.620 vs. 1.599). The paper's conclusion is methodological: any interpretability method claiming to isolate a concept in activation space should demonstrate isolation in the damage matrix, not just in classifier weights.

Key Findings

• Random projection reproduces entanglement: Gaussian projections to 448d match learned EI (1.50 in Qwen-7B); projections to 7d yield near-baseline EI (0.18)
• PCA reduces entanglement: PCA to 28 dimensions gives EI below 0.10 in all three models, versus 0.35-0.41 for random projection to the same dimension
• Superlinear amplification: Triple/pairwise EI ratio 1.87x (GPT-2) to 2.15x (Qwen-7B)
• Concept-type independence: SE concepts produce comparable EI (mean ratio 0.97)
• RLHF accelerates crystallization: The Instruct model crosses EI = 1.0 earlier in the forward pass
• Cross-model consistency: All four models show EI greater than 1.0 at terminal layers (1.39-1.53); bootstrap 95% CIs lie entirely above 1.0 for three of the four

Key References

• McEntire (2026) — Entangled Directions (AI-25): INLP blind spots and asymmetric feature coupling
• Ravfogel et al. (2020) — Null It Out: Guarding Protected Attributes by Iterative Nullspace Projection (ACL)
• Elhage et al. (2022) — Toy Models of Superposition (Transformer Circuits Thread)
• Park, Choe, and Veitch (2024) — The Linear Representation Hypothesis and the Geometry of Large Language Models (ICML)