= Solution
For <causal reachability on a spatial circle>, put $P=2\pi R$ and use a representative $\bar x\in[-P/2,P/2)$ of the spatial circle. The lower-left panel of part <1/c> is a fundamental strip with its vertical edges identified. On the covering space the charged <world line> is still the solution of part <1/b>; on the quotient it is
$$
\bar x(t)=x(t)\pmod P.
$$
A jump between opposite strip edges is a coordinate seam, not a discontinuity of the physical <world line>. At its $n$th complete lap, $x=nP$, so
$$
\boxed{\gamma_n=1+\frac{anP}{c^2},\qquad
t_n=\frac ca\sqrt{\left(1+\frac{anP}{c^2}\right)^2-1},\qquad
v_n=c\sqrt{1-\gamma_n^{-2}}.}
$$
For $a=qE_x/m>0$, the <speed> increases on each return while remaining below $c$. Its <kinetic energy> increases by
$$
\boxed{K_{n+1}-K_n=qE_xP.}
$$
There is no single-valued electrostatic <potential energy> $-qE_x\bar x$ on a circle: the nonzero <electromotive force> is $\oint\mathbf E\cdot d\mathbf l=E_xP$. A globally defined <magnetic vector potential> one-form $A_x=-E_xt$ instead makes the driving explicit. In an embedded circuit, <Faraday's law> realizes this <electromotive force> through changing magnetic flux. More generally an external apparatus maintaining the field supplies the increasing particle <energy>. In an isolated dynamical field-plus-particle system, backreaction would change the field. Thus this is not a cyclic machine extracting work from a conservative potential, and it does not violate <conservation of energy>.
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