This paper has been superseded
This paper has been consolidated into The Geometry of Failure (DOI: 10.5281/zenodo.19432168). The content below is retained for reference.
Executive Summary
Papers VIII-IX measured domain selectivity as entropy change ratios. Paper XI established that geometric structure bounds achievable selectivity. This paper completes the measurement chain by quantifying the information-theoretic content of domain-selective noise injection.
We measure the KL divergence between noisy and clean output distributions across a sigma sweep at the optimal injection layer (layer 10). The results confirm stochastic resonance in information space and show that almost all of the information the noise injects is domain-agnostic.
Key Findings
- Stochastic resonance confirmed: All four domains exhibit inverted-U KL profiles — the signature of stochastic resonance in information space
- Total KL peaks at 14-15 bits: Noise injection produces substantial total information change at optimal sigma
- Domain-specific component is small: +1.3 bits (medical), +1.9 bits (legal), -1.0 bits (code), -1.5 bits (science)
- Information anti-selectivity: Medical and legal show weak information selectivity (IS = 1.09-1.13); code and science are information-anti-selective (IS = 0.91-0.93) — noise shaped for them is more informative about other domains' outputs
- Consistent with approximate bound: Domain-specific differential consistent with C ≤ log2(1 + k2/d_eff) = 2.24 bits, an approximate (not rigorously derived) bound motivated by the concentration barrier
Significance
This paper completes the measurement chain: Paper VIII showed noise injection changes output distributions; Paper IX showed where selectivity peaks; Paper XI proved the geometric bound; Paper X showed the Jacobian provides no spectral shortcut. Here the actual information content is quantified: about 15 bits of total perturbation, of which at most about 2 bits per token are domain-specific. That is barely sufficient for a 4-way domain distinction and far too little for fine-grained domain-selective control.
Key References
- McEntire (2026) — The Concentration Barrier: geometric bounds (Paper XI)
- McEntire (2026) — Spectral Geometry: Jacobian amplification (Paper X)
- McEntire (2026) — The Generative Lossy Channel: five sufficient conditions for SR (Paper VI)
- McEntire (2026) — Layer-Resolved Response Tensor: source of the optimal injection layer (Paper IX)