Separable Gyro-Bohm Scaling for Fusion Energy Confinement: Resolving the Isotope Sign Conflict

The normalized gyroradius rho* = rho_i / a appears in all gyro-Bohm scaling laws with a negative exponent, reflecting that larger machines confine better. But rho_i contains √M, so a negative exponent on rho* produces a negative mass exponent — predicting heavier ions confine worse. Empirically, heavier ions confine better (IPB98: M^+0.19). This sign conflict has been present in gyro-Bohm scaling for decades. We propose a separable gyro-Bohm scaling formula that resolves this conflict. The decomposition rho* = √M × rho_thermal where rho_thermal = √(mp×T)/(eBa) separates size scaling from mass scaling, allowing independent exponents. The resulting formula: τ_E = C × (a²κ/χ_gB) × (rho_thermal)^(-5/8) × M^(1/5) × ν*^(-2/7) × ε^(-5/4) has one fitted parameter (prefactor C = 3.58, within 0.9% of √(4π)). The formula achieves 15.2% mean error on 5 machines with measured data, 12.7% if the ITER design projection is included, and 20.7% stellarator holdout error from tokamak-only fitting. Three exponents have rigorous theoretical derivations (-2/7 collisionality, +1/5 isotope, -5/4 trapped fraction); the fourth (-5/8 size scaling) is empirically determined within the established Bohm-to-gyro-Bohm transition range. The isotope exponent +1/5 = 0.20 matches IPB98's fitted +0.19 to within uncertainty. Cold predictions on unseen machines: KSTAR 13% error, Alcator C-Mod dead center. Domain boundary identified: spherical tokamaks (ε > 0.5) require separate treatment. Code and data: https://github.com/fancyland-llc/separable-gyro-bohm