A First Bound on the Moffat Energy and Thorium--229 Clock as a Probe of the Nonlocal Time-Energy Structure

The new Thorium–229 nuclear clocks give us a new laboratory system in which to test if modification of the Time-Energy uncertainty principle is correct. In this paper we derive the nonlocal Time-Energy uncertainty principle and then apply the published ${}^{229}$Th nuclear clock data as a first probe of the nonlocality scale $E_M$. We note that since these experiments were not designed to test nonlocal quantum field theory the constraints are interpreted as motivating bounds and not definitive exclusions. By using the direct clock-energy channel, we find conservative lower bounds on $E_M$ at the tens of MeV scale, with optimistic present-data estimates reaching the hundred MeV scale. Including the known nuclear sensitivity enhancement of the ${}^{229}$Th transition gives us stronger model dependent bounds in the GeV range, while nuclear-scale reference-energy scenarios can reach the TeV range. The conclusion we draw from this is that nuclear clocks already provide an experimental route from nonlocal time–energy uncertainty to measurable laboratory bounds on non-Planckian nonlocality.