Six trains on three axes: a Lorentz-covariant analysis of a multi-frame light-signal thought experiment

I analyse a thought experiment in which six trains move at 0.99c along three mutually perpendicular embankments. Lightning strikes, fixed mirrors and train-mounted mirrors generate light signals between the trains and the embankments. I define every event in the embankment frame and transform each event into the train frames with Lorentz transformations. I then compare special relativity (SR) with a Galilean emission rule in which reflected light inherits the velocity of the mirror. Four results follow. First, the nine bodies form seven inertial frames, not nine, and give 21 unordered pairs with only three distinct relative speeds. Second, separation speeds in the embankment frame reach 1.98c for collinear trains and 1.40c for perpendicular trains, while every relative velocity stays below c (0.99995c and 0.99980c). Third, every light path between embankment mirrors takes the same time in both models, so these paths cannot test SR. Only light clocks carried by the trains discriminate: for a clock of height one light-second, SR predicts a round trip of 14.18 s of embankment time and the emission rule predicts 2.00 s. Existing tests of source-velocity independence and of time dilation already decide this case in favour of SR. Fourth, a perpendicular pair of trains exposes a Wigner rotation of 73.9°, which resolves the apparent conflict between the "length" of one train and the "width" of another. The geometry gives a complete, checkable event bookkeeping for teaching and for testing claims about relativistic kinematics.