[Jaaanosik (OP)]

If y≠y' then does it mean the Lorentz transformation is 'broken'?

Why would it be?

The Lorentz transformation says y=y'.


Figure 7: The train light round-trip observed from the platform inertial reference frame.

It is obvious how this is the true for the train light round-trip for any chosen time in t=0s and t=4s platform time interval.


Figure 10: The second red photon in the train grid of inertial observers at time t'=2s'.

We see y≠y' for the platform light round-trip between t>2s and t<=4s.
The proper time between two inertial grids cannot be synchronized.
The Lorentz transformation inherently determines who is looking at a specific event.

[Origin]

We see y≠y'

I must be missing something because it looks like y=1 and y'=1 on your illustrations.

[Jaaanosik (OP)]

We see y≠y'

I must be missing something because it looks like y=1 and y'=1 on your illustrations.

That's true for the event of reflection.
I wrote: We see y≠y' for the platform light round-trip between t>2s and t<=4s.

[Jaaanosik (OP)]

Here is a space-time diagram to ponder.



Figure 11: Train car space-time diagram for the train time t'=2s'.

Just to demonstrate the problem. The event of reception at the platform observer does not exist for the train grid of inertial observers yet. It is in the future.
There are some additional arrows, just ignore them at the moment.

[Halc]

Changing the perspective as shown in Figure 9 and Figure 10, suddenly y!=y'.
Is that correct? Do you agree?

In all your diagrams, for any event anywhere, the y coordinate of that event in S is idential to the y' coordinate in S'. So your assertion of y != y' (assuming that y and y' are the coordinate of some event) is unfounded, and your descriptions do not show any such thing.

When Kryptid agreed that y != y', that was when y and y' were the names of the axes in their respective frames, and indeed, the axes are not the same from one frame to the next.
But then you changed y and y' to mean a coordinate of some event, in which case they are the same for any event between frames which have a zero velocity difference component in the y direction.

Figure 7: The train light round-trip observed from the platform inertial reference frame.

Fig 7 does not show a round trip in S. It shows a light pulse starting at one end of the platform and ending at the other. It takes 4 seconds (in S) to do this.

Figure 10: The second red photon in the train grid of inertial observers at time t'=2s'.

OK, for some reason you've taken a special interest in this unlabeled pulse event where the red arrow ends. The y coordinate in S of that event is identical to the y' coordinate in S' of that same event. I cannot fathom why you take such an interest in that particular event.

We see y≠y' for the platform light round-trip between t>2s and t<=4s.

Between t>2s and t <= 4s, no round trip is taken in S. There is no y or y' for any worldline in either frame, so no y to equal y' or not. A line is not an event, and thus has no single set of coordinates.

The proper time between two inertial grids cannot be synchronized.

Grids don't have a proper time. If you're saying that clocks stationary in different respective inertial frames cannot be synchronized over time, then SR agrees with that.

The Lorentz transformation inherently determines who is looking at a specific event.

The LT makes no mention of observers. It works just fine in the complete absence of observers.

Here is a space-time diagram to ponder.

Figure 11: Train car space-time diagram for the train time t'=2s'.

Nice picture!  The lower ~horizontal grey bit is the train at t'=2 seconds in S', yes. The other grey bar seems to be the platform position events corresponding to t=4s in S

Just to demonstrate the problem. The event of reception at the platform observer does not exist for the train grid of inertial observers yet. It is in the future.

SR does not posit a past/present/future ontology for events. Doing so would contradict the first premise of SR. So any mention of these is meaningless.
The event of reception at the platform observer(s) do very much exist in your picture. You label at least one of them and everything.

And y = y' for every single event in the diagram.

If your issue is that the reception events at the two ends of the platform that are simultaneous in the S frame are not simultaneous in the S' frame, that is indeed the case, as per relativity of simultaneity.  You seem not to have identified any issue with SR at all. Your issue seem to be related to an attempt to drag an additional premise presentism into a theory that has a premise (both premises actually) that would be in contradiction with presentism. But no argument based on presentism has been part of any of the discussion before the immediate prior post.

[Jaaanosik (OP)]

SR does not posit a past/present/future ontology for events. Doing so would contradict the first premise of SR. So any mention of these is meaningless.
The event of reception at the platform observer(s) do very much exist in your picture. You label at least one of them and everything.

What do you mean by past/present/future ontology for events?
What is the first premise of SR?

[Jaaanosik (OP)]

SR does not posit a past/present/future ontology for events. Doing so would contradict the first premise of SR. So any mention of these is meaningless.
The event of reception at the platform observer(s) do very much exist in your picture. You label at least one of them and everything.

The past/present/future is clearly defined by the space-time interval.