The Zeta-Minimizer Theorem as a Deductive Variational Foundation for HOR and ORR Kinetics in Proton Exchange Membrane Fuel Cells

The Zeta-Minimizer Theorem provides a fully deductive variational foundation for the hydrogen oxidation reaction (HOR) and oxygen reduction reaction (ORR) in proton exchange membrane fuel cells. Starting from three primitive thermodynamic axioms and the helical geometry of the phase functional, a multi-extent dynamical system is constructed that simultaneously treats the electrochemical reaction coordinates and the adsorption extents of the participating species at the solid–electrolyte interface. The combined Hessian of the phase functional yields a complete spectrum of relaxation rates whose eigenvalues and eigenvectors emerge directly from the solid blackbox constants Ck and the helical partition functions of the reactive species. Adiabatic elimination of the fast surface modes produces an effective single-extent description in which voltage (or overpotential) appears as the conjugate variable, exactly analogous to the role of pressure in the corresponding gas-phase ammonia synthesis framework. The resulting nonlinear rate law is thermodynamically consistent at all conditions, recovers the Butler–Volmer and Tafel forms as well-defined limiting cases, and incorporates the effects of temperature, dilution, and catalyst-specific interface constants without empirical activation energies or adjustable reaction orders. The framework therefore unifies equilibrium, kinetics, and modal dynamics of HOR and ORR within a single variational structure, offering a parameter-light, first-principles alternative to classical empirical electrocatalytic rate expressions while preserving transparent contact with established limiting laws.