Goodbye to (ϕ) - Replacing the Galvani Potential with a Boltzmann Geometric State Equation for Next-Gen Battery Operation and Reduced-Gravity Molten Salt Electrolysis (Lean 4 and Python)

Classical electrochemistry relies on the macroscopic Galvani potential ($\phi$) to account for inner-phase electrical states, treating it as an unmeasurable primitive or operational convention. Here, we demonstrate that $\phi$ is an emergent property rather than an independent variable. By formulating the electrostatic potential from first principles via the Principle of Superposition and transforming discrete coordination shell interactions into a bulk convolution integral, we eliminate the need for extrathermodynamic assumptions. The resulting framework yields a closed-form Geometric State Equation, proving that ion stability and concentration are governed by a Boltzmann distribution scaled by a local structural density and a Geometric Coupling Constant ($\Gamma$).$$c_i(\mathbf{r}) = C_0 \exp\left( - \frac{\Gamma \langle \rho_s \rangle_{shell}}{k_B T} \right)$$