Minimal Current Algebra

We devise an algebra of currents and their first time derivatives designed to damp at high momentum the asymptotic behavior of lepton-pair scattering amplitudes from hadrons consequent from the local current algebra of Gell-Mann. Given certain criteria, the algebra we find is unique, and the commutators are expressed linearly in terms of the currents themselves. The Jacobi identity, however, is formally violated for this algebra; we argue that this does not invalidate it. A possible realization of this "minimal" algebra is found in terms of the formal limit of a massive Yang-Mills theory as $g_0$, $m_0\to 0$; $\frac{g_0}{{m_0}^2}\to \mathrm{const}\ne 0$. With this algebra, all electromagnetic masses of hadrons are finite. Experimental consequences, the strongest of which occurs in inelastic lepton-hadron scattering, are outlined.