We investigate the restrictions that analyticity, unitarity (positivity), and current conservation place on the asymptotic behavior in $q_0$ at fixed q of the connected, covariant amplitude ${{\tilde{M}}_{\mu \nu }}^c(q_0,\mathbf{q})$, where ${{\tilde{M}}_{\mu \nu }}^c$ is the connected part of the forward amplitude for the scattering of a charged conserved current on a proton target. Our main tool is a theorem of Jin and Martin which was used to give lower bounds for forward amplitudes. Our main result is to establish a rigorous lower bound on the contribution of the class-I intermediate states to ${{\tilde{M}}_{00}}^c$. The bound gives $|{M_{00}}^I+\frac{p_0}{m{q}_0}|>C{\left|q_0\right|}^{-2\mathrm{}}$, $C>\mathrm{}0\mathrm{}$, for large complex $q_0$. This is larger by a factor (${q_0}^{I+\epsilon }$) than the asymptotic behavior of ${{\tilde{M}}_{00}}^c$ needed in order to have the Bjorken-limit definition of the space-space equal-time commutator agree with the naive quark commutator. This bound has the following consequences: (a) The existence of operator Schwinger terms is shown to be equivalent to the existence of a frame of reference for which the contribution of class-II intermediate states does not exactly cancel the $O\left({q_0}^{-2\mathrm{}}\right)$ contribution of the class-I states. (b) The existence of the Bjorken-limit definition of the space-space commutator depends on yet another cancellation between the $O\left({q_0}^{-3\mathrm{}}\right)$ contributions to ${{\tilde{M}}_{00}}^c$ of the class-I and class-II states, and the cancellation also of all terms between $O\left({q_0}^{-2\mathrm{}}\right)$ and $O\left({q_0}^{-3\mathrm{}}\right)$. The sum rule whose validity is needed to guarantee this cancellation is known to diverge in perturbation theory. It is also shown to be divergent when the $\mathbf{P}\to \infty$ limit is allowed. (c) The convergence of the Cottingham formula can only be guaranteed if in all frames of reference all the contributions between $O\left({q_0}^{-2\mathrm{}}\right)$ and $O\left({q_0}^{-4\mathrm{}}\right)$ of the class-I states are exactly canceled by the class-II contributions, and if in addition the remaining contribution at $O\left({q_0}^{-4\mathrm{}}\right)$ of the class-I states is also exactly canceled out. The separation of the absorptive part of the amplitude into class-I and class-II contributions is carried out explicity through a fully reduced Lehmann-Symanzik-Zimmermann formula. The class-I states are the usual $s$-channel states and the so-called $Z$-graph states. The class-II states are those that couple directly to the current, contribute only for timelike $q$, and are physically quite distinct from the class-I states.

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