>[!warning] >This content has not been peer reviewed. # Substrate Hardware — Code Documentation ## Script `rst_hardware.py` — derives physical constants ($\alpha$, $m_p/m_e$, $\alpha_G$) from $L^n$ ball geometry. ## Dependencies - `numpy`, `matplotlib`, `scipy` (Gamma function, Brent root-finding) ## Core functions ### `volume_ln_ball(d, n)` Volume of the $d$-dimensional $L^n$ unit ball: $V_d(n) = \frac{(2\,\Gamma(1+1/n))^d}{\Gamma(1+d/n)}$ Works for any real $d > 0$ via the analytic continuation of the Gamma function. ### `lacunarity_d(d, n)` Lacunarity (enclosing hypercube to ball volume ratio): $\mathcal{L}_d(n) = 2^d / V_d(n)$. ### `perimeter_ln_2d(n)` Euclidean arclength of the 2D $L^n$ unit ball boundary. Uses the polar parameterisation $r(\theta) = (|\cos\theta|^n + |\sin\theta|^n)^{-1/n}$ and numerically integrates $P = \int_0^{2\pi} \sqrt{r^2 + (dr/d\theta)^2}\,d\theta$. ### `surface_area_ln_3d(n)` Euclidean surface area of the 3D $L^n$ unit ball boundary. Parameterises the boundary in spherical coordinates and computes the cross-product magnitude $|\partial\mathbf{r}/\partial\phi \times \partial\mathbf{r}/\partial\theta|$ via finite differences. ### `find_d_star(n_target, n_ref, alpha_target)` Root-finding (Brent's method) for the embedding dimension $d^*$ where $V_d(n_\text{target})/V_d(n_\text{ref}) = \alpha_\text{target}$. ## Modes ### `--alpha` 1. Continuous sweep: $V_d(1.25)/V_d(2)$ for $d \in [0.5, 25]$. Finds $d^*$ via root-finding. 2. Integer scan at $d = 3, \ldots, 12$: volume ratio, lacunarity ratio, and cross-dimensional combinations (SA/V, P/V mixed ratios). 3. Reports the closest integer match to $\alpha^{-1} = 137.036$. 4. Outputs: `hardware_alpha_sweep.png`, `hardware_alpha_integer_scan.png`. ### `--mass` 1. Computes geometric primitives at $n = 1.25$: $V_3$, $\text{SA}_3$, $P_2$ (both L^{1.25} and L^2). 2. Tests candidate topology factors: $\text{SA}_3/P_2$, $d^* \times \text{SA}_3/P_2$, cross-ratios. 3. Derives $m_p/m_e = \alpha^{-1} \times (\text{Topology Factor})$ and ranks by match quality. 4. Output: `hardware_mass_ratio.png`. ### `--gravity` 1. Computes $\alpha_G$ (measured) and the hierarchy ratio $\alpha/\alpha_G$. 2. Finds the power $k$ where $\alpha^k = \alpha_G$ (logarithmic ratio). 3. Computes $V_{d^*}(n)/V_{d^*}(2)$ for all Dimensional Ladder sectors. 4. Output: `hardware_gravity_hierarchy.png`. ### `--discovery` Runs all three modes sequentially and produces a master summary table plus combined figure (`hardware_discovery.png`). ## Usage ```bash python "expanded theory applied/further applications/Substrate Hardware/rst_hardware.py" --discovery ``` Individual modes: ```bash python rst_hardware.py --alpha python rst_hardware.py --mass python rst_hardware.py --gravity ``` ## Output files | File | Content | |:---|:---| | `hardware_alpha_sweep.png` | Volume ratio vs embedding dimension (continuous) | | `hardware_alpha_integer_scan.png` | Volume ratio at integer dimensions (bar chart) | | `hardware_mass_ratio.png` | Topology factor candidates for $m_p/m_e$ | | `hardware_gravity_hierarchy.png` | Coupling hierarchy ($\alpha^k$) and ladder ratios | | `hardware_discovery.png` | Combined three-panel discovery summary |