  Thermodynamic uncertainty relations (TURs) express a fundamental tradeoff
between the precision (inverse scaled variance) of any thermodynamic current by
functionals of the average entropy production. Relying on purely variational
arguments, we significantly extend these inequalities by incorporating and
analyzing the impact of higher statistical cumulants of entropy production
within a general framework of time-symmetrically controlled computation. This
allows us to derive an exact expression for the current that achieves the
minimum scaled variance, for which the TUR bound tightens to an equality that
we name Thermodynamic Uncertainty Theorem (TUT). Importantly, both the minimum
scaled variance current and the TUT are functionals of the stochastic entropy
production, thus retaining the impact of its higher moments. In particular, our
results show that, beyond the average, the entropy production distribution's
higher moments have a significant effect on any current's precision. This is
made explicit via a thorough numerical analysis of swap and reset computations
that quantitatively compares the TUT against previous generalized TURs. Our
results demonstrate how to interpolate between previously-established bounds
and how to identify the most relevant TUR bounds in different nonequilibrium
regimes.
