>[!warning] >This content has not been peer reviewed. # RST Equilibrium Proof — Calculation Solver for the **Main Equation** ([[RST Baseline 1.0]]) and the **Kinematic Shift** (Gate 1). The identity $g_N = g^2/(g^n + a_0^n)^{1/n}$ is solved for $g_{\text{obs}}$; orbital velocity $v = \sqrt{g_{\text{obs}} \cdot r}$ is plotted for $z=0$ and $z=2$. **Results:** [[RST Equilibrium Proof Results]]. --- ## Logic - **Main Equation:** $g_N = g_{\text{obs}}^2 / (g_{\text{obs}}^n + a_0^n)^{1/n}$; solve for $g_{\text{obs}}$ by root-finding (bracket $[g_{\text{lo}}, g_{\text{hi}}]$). - **Parameters:** $r$ (log-spaced), $GM=1$, $a_0^{\text{local}}=0.05$, $a_0^{z=2}=0.15$ (3× higher noise floor). - **Output:** Plot of $\sqrt{g \cdot r}$ vs $r$: local (blue), $z=2$ (magenta), Newtonian (red dashed). Gate 1: asymptotic velocity ~32% higher at $z=2$. ## Interpretation - **Blue:** $z=0$ — local $a_0$; flat rotation curve at asymptotic velocity $\propto (GM\,a_0)^{1/4}$. - **Magenta:** $z=2$ — $a_0(z=2) \approx 3\times a_0(z=0)$; asymptotic velocity **$\sim 32\%$ higher** for same mass (Gate 1). - **Red dashed:** Newtonian (no noise); no flat regime. **Reference:** [[RST Baseline 1.0]]; [[Relational Substrate Theory (RST)]].