Then real problem is that the trigger on the bomb is ill specified. It's not as complete as you might think. The entire point of these simultaneity exercises is to demonstrate that simultaneity is not preserved between frames of reference. It's an odd artifact of relativity, and that's why it needs thought experiments like these.

Thus, a physical bomb cannot blow up "if the presidents do not sign simultaneously," because simultaneity is specific to the reference frame one is talking about. The most natural answer is "in the reference frame within which it is at rest." That would be the only meaningful natural assumption.

You could have the bomb blow up if the presidents do not sign simultaneously in the platform frame. This could be done by signalling to the bomb from an observe in the platform frame, or by the bomb doing relativistic calculations to adjust its readings to put them into the platform frame.

However, you cannot do both. The whole point of these experiments is that you can't do both because simultaneity is not preserved between reference frames. This is the same paradox as a bus with a bomb that will blow up if the bus dips below 55mph, but the bus is already at 0 mph in its own frame of reference. In the bus case, it is because the magnitude of absolute velocity is not preserved between Newtonian reference frames, and in the presidents on the train case it is because simultaneity is not preserved between relativistic reference frames.

Both the train bomb which blows up if the presidents do not sign simultaneously and the bus bomb that blows up if the bus dips below 55mph are incompletely defined scenarios because the triggers do not specify the reference frame. In the case of the bus bomb, we will intuitively translate this as 55mph "in the ground frame," and that intuition fills the gap.

In the train scenario, the typical intuition is simultaneous "in the bomb's reference frame (aka the train reference frame)." If everyone agrees to this intuition, we have no paradox.

A coin can land heads up. A coin can land tails up. A coin cannot land heads up and tails up at the same time.

The real problem is that the trigger on the bomb is ill specified. It's not as complete as you might think. The entire point of these simultaneity exercises is to demonstrate that simultaneity is not preserved between frames of reference. It's an odd artifact of relativity, and that's why it needs thought experiments like these.

Thus, a physical bomb cannot blow up "if the presidents do not sign simultaneously," because simultaneity is specific to the reference frame one is talking about. The most natural answer is "in the reference frame within which it is at rest." That would be the only meaningful natural assumption.

You could have the bomb blow up if the presidents do not sign simultaneously in the platform frame. This could be done by signalling to the bomb from an observe in the platform frame, or by the bomb doing relativistic calculations to adjust its readings to put them into the platform frame.

However, you cannot do both. The whole point of these experiments is that you can't do both because simultaneity is not preserved between reference frames. This is the same paradox as a bus with a bomb that will blow up if the bus dips below 55mph, but the bus is already at 0 mph in its own frame of reference. In the bus case, it is because the magnitude of absolute velocity is not preserved between Newtonian reference frames, and in the presidents on the train case it is because simultaneity is not preserved between relativistic reference frames.

Both the train bomb which blows up if the presidents do not sign simultaneously and the bus bomb that blows up if the bus dips below 55mph are incompletely defined scenarios because the triggers do not specify the reference frame. In the case of the bus bomb, we will intuitively translate this as 55mph "in the ground frame," and that intuition fills the gap.

In the train scenario, the typical intuition is simultaneous "in the bomb's reference frame (aka the train reference frame)." If everyone agrees to this intuition, we have no paradox.

A coin can land heads up. A coin can land tails up. A coin cannot land heads up and tails up at the same time.