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lim_{z→∞} F'(z)/F(z) = 1/e— The Gradient Axiom
| Constant | Value | Role |
|---|---|---|
| C₁ (Attractor) | -0.007994021805953 | Zero-point / quantization |
| C₂ (Tripwire) | 0.000200056042968 | Security / curriculum |
| 1/e (Axiom) | 0.367879441171442 | Gradient scaling limit |
F(z) = Σ zⁿ/nⁿ (Sophomore's Dream, Bernoulli 1697)
lim_{z→∞} F'(z)/F(z) = 1/e ≈ 0.367879441171442| Prior | Form | Preference |
|---|---|---|
| Gaussian | exp(-n²) | Medium depth |
| Laplace | exp(-n) | Shallow |
| Sentinel | n⁻ⁿ | Very shallow, wide |
1@misc{abdel-aal2026sentinel,
2 title={The Sentinel Manifold: A Unified Mathematical Framework for Machine Learning},
3 author={Abdel-Aal, Romain},
4 year={2026},
5 url={https://huggingface.co/5dimension/sentinel-manifold-discoveries}
6}