Executive Summary
Earlier papers in the series established INLP (Iterative Nullspace Projection) as the default basis for domain-selective noise injection. But INLP is classification-optimized: it finds directions that best separate domains in activation space, not necessarily the directions through which domain-specific computation actually flows. Perhaps causal methods — activation patching, contrastive activations, gradient-aligned directions — could discover directions more aligned with the model's actual domain processing. Each basis is extracted at the selectivity-peak layer (layer 10, Qwen 2.5-7B) and evaluated with the 4×4 entropy response matrix from Paper IX.
The hypothesis is refuted. INLP remains the only basis with positive selectivity (+0.618, diagonal dominance +1.818), while all three causal bases are anti-selective: patching (−0.630), contrastive (−0.777), and gradient (−0.116). The overlaps between INLP and each causal basis are near-zero (0.011–0.016 mean cosine, at or below the random baseline of about 0.017 in R3584), so the four bases occupy essentially orthogonal subspaces. This is the classification-intervention dissociation: the directions that best classify domains also produce the best interventions, while causally-derived directions are actively worse than random in this geometry.
The paper explains the result through the isotropic forward pass (Paper X): when all directions are amplified roughly equally, there is no causal "highway" to exploit, and only the classification boundary provides domain-specific targeting. The one exception is code, where the gradient basis (+0.71) beats INLP (−0.14) — a hint that code processing may follow a less isotropic pathway, left for future work.
Key Findings
- INLP is uniquely selective: INLP is the only basis with positive selectivity (+0.618); all causal bases (patching, contrastive, gradient) produce anti-selective effects
- Near-zero subspace overlap: Pairwise overlaps between INLP and causal bases range from 0.011 to 0.016, confirming they occupy almost entirely disjoint subspaces
- Classification-intervention dissociation: The classification-optimized basis gives the best interventions; causal relevance is not domain-selective relevance
- Causal bases are anti-selective: Noise along patching, contrastive, or gradient directions produces more entropy change in non-target domains than in the target domain
- Code exception: The gradient basis outperforms INLP for code (+0.71 vs. −0.14), the one domain where INLP is weakest
Key References
- McEntire (2026) — The Concentration Barrier: dimensionality bounds on selectivity (Paper XI)
- McEntire (2026) — Shaped Noise Injection: terminal measurement limit (Paper VIII)
- McEntire (2026) — Layer-Resolved Response Tensor: selectivity profile (Paper IX)
- McEntire (2026) — Spectral Geometry of the Forward Pass: the isotropic Jacobian (Paper X)
- McEntire (2026) — INLP domain directions (Paper IV)