  The Maxwell-Klein-Gordon equation $ \partial^{\alpha} F_{\alpha \beta} =
-Im(\Phi \overline{D_{\beta} \Phi}) $ , $ D^{\mu}D_{\mu} \Phi = m^2 \Phi $ ,
where $F_{\alpha \beta} = \partial_{\alpha} A_{\beta} - \partial_{\beta}
A_{\alpha}$, $D_{\mu} = \partial_{\mu} - iA_{\mu} $, in the (3+1)-dimensional
case is known to be unconditionally well-posed in energy space, i.e. well-posed
in the natural solution space. This was proven by Klainerman-Machedon and
Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge.
The main purpose of the present paper is to establish that for both gauges this
also holds true for data $\Phi(0)$ in Sobolev spaces $H^s$ with less
regularity, i.e. $s < 1$, but $s$ sufficently close to $1$. This improves the
(conditional) well-posedness results in both cases, i.e. uniqueness in smaller
solution spaces of Bourgain-Klainerman-Machedon type, which were essentially
known by Cuccagna, Selberg and the author for $s > \frac{3}{4}$ , and which in
Coulomb gauge is also contained in the present paper. In fact, the proof
consists in demonstrating that any solution in the natural solution space for
some $s > s_0$ belongs to a Bourgain-Klainerman-Machedon space where uniqueness
is known. Here $s_0 \approx 0.914$ in Coulomb gauge and $s_0 \approx 0.907$ in
Lorenz gauge.
