The Geometry of Failure: Why Direction-Space Interventions Cannot Achieve Domain Selectivity in Neural Networks

Executive Summary

This consolidated paper unifies six previously separate works tracing the failure of direction-space intervention in neural networks. The arc begins with a practical tool that works (shaped noise breaks 100% of repetition loops) and ends with a theorem explaining why it cannot be made selective: any k directions capture at most a fraction k/d_eff of activation variance (the concentration barrier). The collected finding is a dissociation: classification accuracy does not imply intervention precision.

Shaped Noise Injection (former AI-8)

Shaped noise along INLP domain directions breaks 100% of repetition loops at 3B and 7B with near-perfect token uniqueness (0.99+). Domain-specific entropy reductions reach 6.1% for legal at 7B. But cross-domain selectivity fails: targeting medical reduces legal entropy by 10.7% while reducing medical by only 1.9%. All three correction attempts fail -- subspace decomposition, scalar cancellation, and optimal linear correction via matrix inversion. The response matrix is invertible (determinant -84.5, condition 21.2), the linear algebra is clean, and the optimal weights are computable. They simply do not produce the predicted effects. Root cause: the terminal measurement problem -- the response matrix characterizes the system's input-output mapping but cannot invert nonlinear transformations at intermediate layers.

Layer-Resolved Response Tensor (former AI-9)

Maps the selectivity profile at nine sampled transformer layers in Qwen-2.5 7B (28 layers total). Mean selectivity peaks modestly at layer 10 (s̄ = 0.57) and declines toward both input and output; no layer achieves mean selectivity above 1.0. Strong domain asymmetry: code and science are positively selective at most layers (7 and 6 of 9), while medical and legal are positive at only one layer each.

Spectral Geometry (former AI-10)

Measures amplification spectrum of INLP domain directions through the layer Jacobian using finite-difference Jacobian-vector products. INLP directions are not preferentially amplified (INLP/random ratio 0.991, range 0.897-1.046 across injection layers). PCA-INLP alignment is near-random at intermediate layers but increases toward terminal layers where selectivity is lowest. The forward pass is an isotropic amplifier treating domain directions as generic directions in activation space.

The Concentration Barrier (former AI-11)

Proves that any k directions capture at most a fraction k/d_eff of activation variance; the link from this variance bound to behavioral selectivity is empirically consistent but not formally derived. Measures effective dimensionality at every transformer layer: d_eff ranges 4.7-26.0 (mean 19.2) for last-token extraction. Mean-pooled d_eff collapses to 1.0 at layers 3-25, revealing previously undocumented representation anisotropy invisible to position-aware measurements. INLP variance fraction increases from 1.3% at early layers to 12.5% at terminal, yet the concentration barrier holds at all 28 layers.

SR Channel Capacity (former AI-12)

Quantifies information-theoretic content of domain-selective noise injection via KL divergence across a sigma sweep at optimal injection layer (layer 10). All four domains exhibit inverted-U KL profiles confirming stochastic resonance in information space. Total KL peaks at 14-15 bits but domain-specific component is small: +1.3 bits (medical), +1.9 bits (legal), -1.0 bits (code), -1.5 bits (science). Consistent with a heuristic bound C <= log2(1 + k^2/d_eff) = 2.24 bits.

The Activation Geometry Program (former AI-13)

The four measurements converge. The barrier is geometric -- a property of high-dimensional computation -- not an engineering limitation of any particular intervention method. Classification exploits any separable signal, including shared substrate that happens to correlate with the label; intervention requires causal signal for the target concept specifically. The legal INLP direction is the sharpest case: it classifies legal text correctly, yet the R^-1 optimization assigns it zero weight for the legal target. The barrier constrains direction-space interventions along linear discriminative directions (INLP, DAS, LEACE) and likely extends to activation addition, representation engineering, and ROME; nonlinear, multi-layer, or non-discriminative methods may achieve selectivity that linear single-layer injection cannot.

Key Findings

• Repetition loop breaking: 100% escape rate at 3B and 7B with near-perfect token uniqueness (0.99+)
• Cross-domain selectivity fails: Targeting one domain affects non-target domains comparably or more strongly; all corrections fail
• Terminal measurement limit: Response matrix characterizes but cannot invert nonlinear mixing at intermediate layers
• Selectivity peaks at intermediate layers: Layer 10 shows the highest mean selectivity (0.57), declining toward input and output; no layer exceeds 1.0
• Isotropic amplification: Forward pass treats INLP directions identically to random directions (ratio 0.991)
• Concentration barrier theorem: Variance fraction captured by any k directions bounded by k/d_eff -- proved analytically, verified at all 28 layers
• Information bound: Domain-specific output perturbation of 1-2 bits against ~15 bits of domain-agnostic perturbation, consistent with a heuristic 2.24-bit bound
• Classification-intervention dissociation: Classification accuracy does not imply intervention precision

Superseded Papers

This paper consolidates and supersedes:

• AI-8: Shaped Noise Injection -- domain precision, loop breaking, and the terminal measurement limit
• AI-9: Layer-Resolved Response Tensor -- where domain selectivity lives in the forward pass
• AI-10: Spectral Geometry of the Forward Pass -- how INLP directions interact with layer Jacobians
• AI-11: The Concentration Barrier -- effective dimensionality bounds domain selectivity
• AI-12: Channel Capacity of Domain-Specific Stochastic Resonance
• AI-13: The Activation Geometry Program -- twelve papers on the mathematical structure of neural network representations

Key References