The cubic invariant of charged-fermion masses

Three positive mass amplitudes with fixed mean and quadratic spread retain one independent symmetric quantity: their product. For a threefold charged-sector operator this product is a cubic function of its complex hop and fixes the spectrum up to scale and permutation. A family of symmetric reciprocal costs, including costs of amplitudes, masses, and their pairwise ratios, decreases strictly with that product. Its minimum therefore has two equal generations. The measured muon–electron ratio gives a separate precision constraint. Exact Koide balance combined with an exact ring phase of 2/3 predicts a ratio 442 standard uncertainties above the CODATA 2022 value, sharpening a discrepancy identified by Brannen. The exact allowed curve between balance and phase is derived. Relative weak-current frames and the gravitationally normalized mass scale are independent data beyond these spectral invariants.