Per-Layer Gamma Slope Dynamics: Breaking the Amplification Ceiling in Sparse Model Editing

1. The Uniform Gamma Ceiling Problem

Sparse knowledge editing techniques — such as Sparse Knowledge Patches (SKP / SKIT [1]) — apply parameter updates $\Delta W$ scaled by a global amplification scalar $\gamma$. When $\gamma$ is applied uniformly across all transformer layers, model performance exhibits distinct phenomenological regimes:

1.2 The Oscillation Phenomenon & Zombie Comebacks

Between $\gamma = 42$ and $\gamma = 65$, we observe a novel non-monotonic oscillation pattern: after primary pathways fail completely at $\gamma = 42$, specific higher gamma values ($\gamma = 60\text{--}65$) temporarily recover correct answers before final degradation. This demonstrates that neural networks contain redundant representational pathways encoding the same knowledge at different parameter depths; extreme perturbation disrupts primary circuits before accidentally activating secondary pathways.

1.3 Why Uniform Gamma Fails

Uniform scaling fails because transformer layers perform asymmetric functional roles:

• Early Layers (0–8): Token embeddings, syntactic construction, and core model identity.
• Middle Layers (9–18): Abstract reasoning patterns and compositional semantics.
• Late Layers (19–27): Specialized domain knowledge, factual recall, and vocabulary output projection.

At uniform $\gamma = 42$, early layers suffer a $\approx 3\%$ weight shift — sufficient to break syntax generation before late layers can project their amplified domain knowledge.

2. Layer-Level Distribution of Learned Corrections

2.1 Correction Concentration

Analyzing the gradient-importance-weighted corrections of a medical SKP (182,589 parameters across 28 layers of Qwen 2.5 7B) reveals extreme depth asymmetry:

2.2 Hotspot Layers: Layer 19 & Layer 27

Two individual layers account for over 37.3% of all learned knowledge positions:

• Layer 27 (Output Projection): 37,713 positions (20.7% of total) with max correction $0.0153$.
• Layer 19 (Knowledge Transition): 30,268 positions (16.6% of total) with max correction $0.0077$.

3. The Per-Layer Gamma Slope Function

3.1 Mathematical Definition

We define the Per-Layer Gamma Slope Modulation Function $\gamma(l)$ as:

where:

• $l \in \{0, 1, \dots, L-1\}$ is the zero-indexed layer index.
• $\gamma_{\text{base}}$ is the baseline anchor factor (optimized via binary search on held-out validation data).
• $l_{\text{pivot}} = \lfloor L / 2 \rfloor$ is the central pivot layer ($l_{\text{pivot}} = 14$ for a 28-layer model).
• $\text{slope} \ge 0$ is the amplification gradient scalar ($\text{slope} = 0$ corresponds to uniform scaling).
• $\max(\cdot, 1.0)$ provides automatic self-clamping, ensuring early layers never receive negative or disruptive scaling.

3.2 Effective Layer-Wise Curves

4. Empirical Validation

4.1 Coherence Preservation on Foundation Benchmarks

We evaluated baseline factual integrity and conversational coherence across slope values:

4.4 The Magnesium Clinical Pearl: Surfacing Deep Latent Knowledge

We designed a diagnostic stress-test based on a multi-domain attending-level clinical trap:

"A 68-year-old patient presents with severe hyperkalemia ($K^+ = 7.2\text{ mEq/L}$) with peaked T waves. The medical team administers standard emergency therapy (calcium gluconate, insulin + dextrose, albuterol, and sodium polystyrene sulfonate). Follow-up potassium improves to $6.8\text{ mEq/L}$, but the patient abruptly develops polymorphic ventricular tachycardia (torsades de pointes). What went wrong?"

The Underlying Medical Mechanism: Unrecognized and uncorrected hypomagnesemia. Low serum magnesium prolongs the myocardial QT interval, creating the substrate for torsades de pointes when combined with rapid potassium shifts.

5. Theoretical Interpretation

5.2 Neural Superposition & Interference Suppression

Neural networks store multiple semantic concepts in overlapping parameter coordinates (superposition [2]). Uniform high amplification scales all features sharing a coordinate equally — including unrelated representations — inducing severe destructive interference in early layers.

The gamma slope confines extreme scaling exclusively to late layers where representation spaces are highly specialized, thereby preserving general syntax and identity in early layers.

5.3 The Knowledge Depth Continuum

We propose that parameterized knowledge exists along a continuous depth spectrum:

• Surface Knowledge ($\gamma \in [1, 10]$): High-frequency pretraining associations (e.g., "Aspirin treats pain").
• Mid-Depth Knowledge ($\gamma \in [10, 35]$): Specialized domain procedures (e.g., "Digoxin-furosemide interaction cascade").
• Deep Latent Knowledge ($\gamma \ge 50$): Rare cross-domain clinical pearls requiring late-layer graduated amplification.
• Unreachable (Not in Weights): Concepts absent from training corpora.

6. Cross-Architecture Validation & Boundary Conditions

6.1 Validation on Mistral 7B & Qwen 14B

We observe an empirical correlation between architecture sensitivity and safe slope:

6.3 Refinement vs. New-Knowledge Boundary Conditions

The gamma slope acts as a refinement amplifier for existing latent representations. When tested on an aviation exam where the base model possessed zero pretraining representation (baseline accuracy $16\%$), applying $\text{slope} = 4.0$ amplified late-layer noise, reducing accuracy to $8\%$.

6.4 Quantization Survival for 4GB Edge Deployment

Slope-amplified SKPs survive post-training 4-bit NF4/AWQ quantization. Because sparse parameter modifications operate at $\Delta \approx 0.001\text{--}0.01$ (exceeding the NF4 quantization noise threshold of $\approx 0.005$), patched models reload identically into 4GB VRAM with zero runtime overhead:

7. Conclusion

We have demonstrated that the uniform amplification ceiling in sparse model editing is an artifact of layer-asymmetric representational fragility. By applying a per-layer gamma slope that protects early syntactic foundation while maximally amplifying late domain representations, models break the uniform $\gamma=42$ ceiling, achieving stable operation 50% beyond previous limits and surfacing complex latent clinical associations unreachable by standard scaling.

References

1. [1] Martin, J. (2026). "Architecture-Compatible Sparse Knowledge Transfer for Large Language Models." Cross Domain Reasoning Technical Report CDR-TR-2026-02.
2. [2] Elhage, N., et al. (2022). "Toy Models of Superposition." Anthropic Research.
3. [3] Hu, E. J., et al. (2022). "LoRA: Low-Rank Adaptation of Large Language Models." ICLR 2022.
4. [4] Li, Y., et al. (2016). "Convergent Learning: Do Different Neural Networks Learn the Same Representations?" ICLR 2016.

Citation

@techreport{martin2026gammaslope,
  title={Per-Layer Gamma Slope Dynamics: Breaking the Amplification Ceiling in Sparse Model Editing},
  author={Martin, J.},
  institution={Cross Domain Reasoning},
  year={2026},
  month={August},
  number={CDR-TR-2026-03},
  doi={10.5281/cdr.2026.03},
  url={https://crossdomainreasoning.com/papers/gamma-slope-dynamics/}
}