Grimble said:
Every event is fixed in every frame at the space coordinates where the event occurred, just as it is fixed at the time coordinate when the event occurred.
The word "fixed" implies "does not move". That means you are imputing a state of motion (not moving) to an event.
Grimble said:
Will you please, please, please stop being so enigmatic and explain what you mean?
Here is the math. The specification of the scenario is that there are events with the following coordinates in the rest frame of M (coordinates are given as ##(x, t)## pairs, and I have chosen the units of space and time so all the following coordinate values in this frame come out to be nice unit integers):
Event A: a light flash strikes the embankment and the rear end of the train at ##(-1, 0)##
Event B: a light flash strikes the embankment and the front end of the train at ##(1, 0)##.
Event M: Light from both light flashes reaches the observer on the embankment at ##(0, 1)##.
Event O: The observer at the midpoint of the embankment and the observer at the midpoint of the train are co-located (just passing each other) at ##(0, 0)##.
Now, assume that the train moves at speed ##v < 1## in the positive ##x## direction relative to the embankment (we are using units in which ##c = 1##). Then we can easily compute the events at which the light flashes from event A and event B meet the observer at the midpoint of the train. They will be:
Event MB': Light from the flash at the front of the train reaches the observer at the midpoint of the train at ##( \frac{v}{1+v}, \frac{1}{1+v} )##.
Event MA': Light from the flash at the rear of the train reaches the observer at the midpoint of the train at ##( \frac{v}{1-v}, \frac{1}{1-v} )##.
It is obvious that these two events are
not the same--i.e., they are different points in spacetime. In other words, you have been using the symbol M' under the assumption that it referred to some single event--but it doesn't, because
there is no single event where the light flashes reach the observer at the midpoint of the train. In other words:
you have been reasoning from a false premise. That is why you have been getting false conclusions.
Using the correct premise, we can see that since MB' obviously occurs before MA', the flash from the front of the train will reach the observer at the midpoint before the flash at the rear of the train, just as Einstein said.
But just to be sure, let's transform everything into the train's rest frame. The Lorentz transformation will be ##x' = \gamma \left( x - v t \right)##, ##t' = \gamma \left( t - v x \right)##. So the coordinates in the primed frame (the train frame) of all the events above come out to be as follows:
Event A: ##(- \gamma, \gamma v)##
Event B: ##(\gamma, - \gamma v)##
Event M: ##(- \gamma v, \gamma)##
Event O: ##(0, 0)##
Event MB': ##(0, \sqrt{\frac{1-v}{1+v}})##
Event MA': ##(0, \sqrt{\frac{1+v}{1-v}})##
Note that events MB' and MA' both happen at ##x' = 0##, i.e., at the midpoint of the train, but
at different times.
All this is bog standard SR, and this is what we mean when we say you are incorrect.