But, since the velocity add equations are used to derive the the acceleration equations based on everything I read from you, you did agree with the SR acceleration equations.

It that true?

I have no problem with the clock postulate or the relativistic rocket equations. They aren't applicable to this thought experiment. You could, no doubt, design some variant of your thought experiment where a non-instant acceleration plays a role. If you did that, then I would note two things: First, the relativistic rocket equation (or, acceleration equation) that you’ve been considering would still be insufficient to fully describe O from the perspective of O’. Second, the result would need to be consistent with the result we already have. Both points are made here:

http://books.google.... frame"&f=falseat the end of page 168 and page 169.

There is a tool, however, that would allow you to describe things from the accelerated observer’s perspective completely (from the start of the thought experiment to the end), exactly, and easily. It is general relativity. General Relativity is a generalization of special relativity for the case of accelerated motion.

Whether you use general relativity or Rindler coordinates, the conclusion will be the same. In the line parallel to acceleration, clocks are dilated slower in the negative direction as a function of distance and faster in the positive direction as a function of distance during the acceleration. In an acceleration version of your thought experiment, while O accelerates there is no significant distance between O and O’. When O’ accelerates, O is further ahead in the direction of acceleration (higher in gravitational potential in the general relativistic sense and greater in X in Rindler coordinates). It’s clock will therefore run fast relative to O’ during the acceleration which is consistent with the result we’ve already found. The clock moves from t=2 to t=3.125 from the perspective of O’. It ran fast.

To accelerate is to move from one system of coordinates to another. There are two frames of reference for O'. In the first, t' = 2.5 is simultaneous with t = 2. In the second, t' = 2.5 is simultaneous with t = 3.125. Because your thought experiment relies on "when" t' = 2.5 (a matter of simultaneity) it is important not to neglect this detail.

May I see your equations to prove this?

Given on wiki here:

From the first equation of the Lorentz transformation in terms of coordinate differences

[math]\Delta t = \gamma \left(\Delta t' - \frac{v \,\Delta x}{c^{2}} \right)[/math]

it is clear that two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are colocal in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′.

Special RelativityIn step 3:

3. After time t', O' will acquire v in precisely the same way as O in precisely the same direction.

From the perspective of O', t jumps from 2 to 3.125 (because the plane of simultaneity shifts)

set Δt' = 0, v = -.6 (negative because the change in velocity is toward the other frame) and Δx = 1.5 (the distance in the first frame) and you should get the correct shift of 1.125. In fact, let's do that now,

[math]\Delta t = \gamma \left(\Delta t' - \frac{v \,\Delta x}{c^{2}} \right)[/math]

[math]\Delta t = 1.25 \left(0 - (-0.9) \right)[/math]

[math]\Delta t = 1.125[/math]

There you have it. By the Lorentz transformations, when O' changes frames t changes by 1.125. It also should be clear from the properties of Minkowski spacetime. The diagram I made is perhaps helpful in that regard.

Here, also, is a very good wikibook on the topic:

Special Relativity/Simultaneity, time dilation and length contractionThe first section on phase shifting describes the part you are doing wrong well. In particular, that switching frames of reference moves the present instant along the time axis in a manner that depends on distance. The Andromeda paradox is a good example of this. Over very large distances (to the Andromeda galaxy) the present instant shifts quite a lot even with a small velocity change.

I think your question has been answered rather well and any inconsistencies are now shown to result from your refusal to apply relativity properly. Perhaps someone else has a different perspective that you would find helpful.

~modest