Spectral Geometry of the Forward Pass: How INLP Directions Interact with Layer Jacobians

Executive Summary

Paper IX showed that domain selectivity peaks at intermediate layers (7-10) and declines at both input and terminal layers. Paper XI showed that effective dimensionality bounds achievable selectivity. This paper asks why selectivity peaks where it does by measuring the amplification spectrum of INLP domain directions through the layer Jacobian.

Using finite-difference Jacobian-vector products, we compute how perturbations along INLP directions propagate through consecutive layer pairs and compare to random directions and PCA structure. Both hypotheses are cleanly rejected.

Key Findings

• No preferential amplification: INLP/random amplification ratio is 0.99 +/- 0.05 — the forward pass does not treat domain directions differently from arbitrary directions
• PCA-INLP alignment is near-random at intermediate layers but increases toward terminal layers where selectivity is lowest — the opposite of what would be needed for selective intervention
• The forward pass is an isotropic amplifier: It treats INLP directions as generic directions in activation space, consistent with the concentration barrier as the sole constraint

Significance

The result closes the spectral route around the concentration barrier. The forward pass does not know which directions are INLP directions and which are random; it amplifies and mixes all of them roughly equally (about 1.06x per layer, with a terminal spike at layers 22-27 that explains why terminal injection produces large but unselective effects). Without preferential amplification, the only leverage for selectivity is the geometric overlap k/d_eff that Paper XI measures.

Key References

• McEntire (2026) — Layer-Resolved Response Tensor: selectivity profile across layers (Paper IX)
• McEntire (2026) — The Concentration Barrier: dimensionality bounds (Paper XI)
• McEntire (2026) — Shaped Noise Injection: the terminal measurement limit (Paper VIII)