Which of the following does NOT constitute a condition where a train may run early?

The situation where a train may run early requires certain conditions to be met, and option C states that this can happen "at all stations at any time." This is not accurate because running early is generally subject to specific rules and conditions.

Permitting early departures has very defined parameters, which are designed to maintain safety, order, and adherence to schedules within the railroad system. For instance, running early may only be sanctioned at certain locations like Jamaica when indicated by a blue light or at terminals without scheduled arrivals, where there is no conflicting train traffic or other operational concerns.

In contrast, the other choices outline specific scenarios or locations where early running is allowed under certain circumstances. Those include formal permissions at certain stations, such as those indicated by blue lights or stations without scheduled arrivals, where operational efficiency may warrant an early departure. Therefore, the selection confirming that running early is permitted "at all stations at any time" incorrectly implies an absolute flexibility that does not exist within the operational protocols of LIRR, making it the correct answer for the question.