>[!warning] >This content has not been peer reviewed. # BSD solver — calculation This note documents the **RST BSD conceptual solver** for [[BSD Conjecture (RST)]]. The script illustrates how **fidelity** (capacity for rational points) responds when **noise** (a proxy for the $L$-function near the critical point $s=1$) vanishes. --- ## Purpose - **BSD conjecture:** The rank $r$ of an elliptic curve equals the order of the zero of $L(E,s)$ at $s=1$. - **RST reading:** The $L$-function is the substrate “noise report” across primes. A zero at $s=1$ means noise is vacated; by conservation of workload, **signal** (rational points) fills that space. So **fidelity** (ability to render full-resolution points) peaks where the noise proxy vanishes. --- ## Engine - **Noise proxy:** $L_{\text{noise}}(s) = (s-1)^r$ — mimics a zero of order $r$ at $s=1$ (here $r=2$ for the plot). - **Fidelity from noise:** When noise dominates, fidelity is low; when noise $\to 0$, fidelity $\to 1$. Using the same $L^n$-norm structure: $\text{fidelity} = \frac{1}{(1 + L_{\text{noise}}^n)^{1/n}}$ So zeros in $L_{\text{noise}}$ produce peaks in fidelity (rank presence). --- ## Embedded code ```python import numpy as np import matplotlib.pyplot as plt n_sharpness = 1.25 def bsd_fidelity_from_noise(L_noise, n=n_sharpness): """Fidelity = 1 / (1 + L_noise^n)^(1/n); peaks where L_noise = 0.""" L_safe = np.maximum(np.abs(L_noise), 1e-30) return 1.0 / (1.0 + L_safe**n)**(1.0/n) # Around critical point s=1; zero of order 2 (rank r=2) s = np.linspace(0.5, 1.5, 500) L_noise = (s - 1.0)**2 rational_density = bsd_fidelity_from_noise(L_noise) # Plot: fidelity (rational-point capacity) vs L_noise vs s; vertical line at s=1 # Output: bsd_solver.png ``` --- ## Parameters | Parameter | Value | Meaning | |:---|:---|:---| | $n$ | 1.25 | Substrate sharpness (same as core RST). | | Zero order | 2 | $(s-1)^2$ — mock rank $r=2$ for illustration. | --- ## Output - **Figure:** `bsd_solver.png` — fidelity (blue) and L-noise proxy (red dashed) vs $s$; vertical line at $s=1$. **Results:** [[BSD Solver Results]]. --- ## Links - **Application:** [[BSD Conjecture (RST)]] - **Fidelity:** [[expanded theory/Fidelity]], [[expanded theory/Fidelity Derivation]] - **Resource Triangle:** [[expanded theory/Resource Triangle]]