$k_d\left(E\right)$	Water dissociation rate constant vs. field; unit s−1.
$\left(0\right)$	Thermal WD rate constant (no field); s−1.
$E$	Electric field in the BPM junction; V·m−1.
$U_j$	Junction voltage drop across active region; V.
$\delta$	Junction thickness; m (e.g., 1 nm benchmark).
$J$	Current density at the junction; mA·cm−2.
$K$	Prefactor in junction law $J=K U_j^2$; mA·cm−2·V−2 (when $J$ in mA·cm−2, $U_j$ in V).
$f$	Water volume fraction in the junction (hydration); dimensionless $\left[0,1\right]$.
$q$	Elementary charge; C (e.g., $1.6\times {10}^{-19}$).
${\mu }_{H_3{O}^{+}},{\mu }_{O{H}^{-}}$	Ionic mobilities; m2·V−1·s−1 (e.g., $3.62\times {10}^{-7}$, $2.05\times {10}^{-7}$ at 25 °C).
$c_{\mathrm{ions}}$	Concentration of autoprotolysis ions $H_3{O}^{+}/O{H}^{-}$ in water; mol·L−1 (≈${10}^{-7}$ at 25 °C).
$c_{{\mathrm{H}}_2\mathrm{O},\mathrm{l}}$	Concentration of free water; mol·L−1 (≈55.5 at 25 °C).
$c_{{\mathrm{H}}_2\mathrm{O},\mathrm{j}}$	Concentration of water in the junction; mol·L−1
$E_d$	Threshold dissociation energy per water molecule; J (≈$1.33\times {10}^{-19}$ at 25 °C).
$J_{\lim }$	Limiting current density (plateau); mA·cm−2 (≈2.19 measured here).
$E^{\backslash *}$	Crossover field where $k_d=k_0$; V·m−1 (≈$1.43\times {10}^5$).
$E_{cell}$	External cell voltage applied; V.
$E_{cell}^{rev}$	Reversible (equilibrium) voltage of the cell; V (see Equation (49)).
$R_s$	Series resistance (two-compartment cell, area-normalized); Ω·cm2 (EIS-derived).
${\eta }_{\mathrm{HER}},{\eta }_{\mathrm{OER}}$	Overpotentials for HER/OER; V.
${\mathrm{\Lambda }}^{\circ }$	Limiting molar conductivity at infinite dilution; S·m2·mol−1 (used to infer $\mu$).
$F$	Faraday constant; C·mol−1 (96,485).
$\eta$	Viscosity (in Stokes–Einstein context for $\mu$); Pa·s.
$b$	Tafel slope; mV·dec−1.
$j_0$	Exchange current density; A·cm−2.
$v_d$	Drift velocity of ions under the field; m·s−1. (Relation: $v_d=\mu E$.)
$P$	Per-ion field-supplied power dissipated by drift; (J·s−1). (Model: $P=q \mu E^2$.)
$P_{{\mathrm{H}}_3{\mathrm{O}}^{+}},P_{{\mathrm{OH}}^{-}}$	Per-ion power for ${\mathrm{H}}_3{\mathrm{O}}^{+}$ and ${\mathrm{OH}}^{-}$; (J·s−1).
$n_{{\mathrm{H}}_3{\mathrm{O}}^{+}},n_{{\mathrm{OH}}^{-}}$	Number density of autoprotolysis ions; m−3.
$V_j$	Junction volume (membrane area × active thickness $\delta$); m3.
$P_{total}$	Total power dissipated in the junction by all ions; (J·s−1).
$w$	Power density dissipated by ${\mathrm{H}}_3{\mathrm{O}}^{+}/{\mathrm{OH}}^{-}$ in the junction; J·s−1·m−3.
$\Delta G_{\mathrm{d}}$	Molar Gibbs free energy of water dissociation; (kJ·mol−1). (Used to compute the per-molecule threshold $E_d=\Delta G_{\mathrm{d}}/N_A$).
$N_A$	Avogadro constant; mol−1.
$\kappa$	Thermal conductivity of water used in the heating estimate; W·m−1·K−1.
$\Delta T$	Estimated junction temperature rise from dissipation; K.
$A$	Membrane area used in estimates; m2 (our benchmark: 1 cm2).
${\epsilon }_r$	Relative permittivity; dimensionless.
$T$	Absolute temperature; K (reference 298 K/25 °C).
R	Universal gas constant (8.314 J·mol−1·K−1).
$K_w$	Water equilibrium constant (${10}^{-14}$ at 25 °C).