[Bored chemist]

Two photons (which have been emitted from the Sun and Andromeda on the same moment

"at the same moment" from whose point of view?

[xersanozgen (OP)]

The "closing speed" between the particles as measured in the rest frame of the accelerator, can exceed c, however, the relative speed between the particles as measured by either particle cannot equal or exceed c.  For example, if both particle A and B have speeds of 0.999c relative to the accelerator, either particle will measure the  relative speed between the particles as being 0.999999499c

Yes, you  expressed clearly the nuance.

When the closing speed value exceeds  c, an observer (that is located on a particle) cannot determine other particle's simultaneous location because of limited/finite value of light's velocity; and he cannot measure its real relative speed (or decreasing speed of the distance between two objects).

If an observer is an actor of the experiment or analysis; his visual abilities or restrictions causes wrong perception. Therefore we must consider God's eye/sight or the status without observer.

[xersanozgen (OP)]

This formula is also valid if you throw two rockets off a single rocket.

Consider that the Earth is one rocket, and the LHC throws two rockets (bunches of protons) in different directions.

Apply this formula, and you will see that the bunches of protons never exceed c compared to the Earth, and they never exceed c relative to each other.

Yes you can be right; so, if an astronout pushes his shuttle, he moves away from shuttle. However the space shuttle also will move away from astronout by an amount speed toward opposite direction. Both of them get speeds according to LCS/space; arithmetic total of their speeds is the meaning of relative speed according to each one of them and this value cannot exceed c. Please attention: The shuttle and astronout have their own masses; masses and pushing force are the reason of motions.

However, if we realise a photo-flash from shuttle toward opposite direction, the photon packet moves away from a point (that is marked on LCS or space) by the vaue of velocity c . The shuttle had passed over this emitting point at the emitting moment. The distance between the shuttle and photon packet increase by bigger speed than c. The shuttle keeps its motion independently toward anywhere or present orbit. The photon packet goes due to electro-magnetic cycle in space; the shuttle or their source does not cause the motion of photons; so, the "action-reaction" concept is not mentioned.

[xersanozgen (OP)]

The velocity addition formula describes how time dilation affects perception in the proper time of a travelling object. It is all to do with relative velocities.

We have a possibility to use a co-reference frame (The Earth/local places) for mechanic problems. Co-reference frame is useful for analyzing.

We can use similar possibility for light kinematics due to space or virtual/imaginery Light Coordinate System. The velocity of light is already the value c according to space; we can use the universal value of the other actors' speed.

Besides we must not use the observer as an actor in the experiments and analyses.

[xersanozgen (OP)]

Two photons (which have been emitted from the Sun and Andromeda on the same moment

"at the same moment" from whose point of view?

God's sight or theorical imagination.

[xersanozgen (OP)]

Is (it) 1.27 million  light years or merely 1.27 million years?

Both.

In a vacuum, light travels at "the speed of light".
- So in space, light travels 1 light-year in 1 year.
- And 1.27 million  light years in 1.27 million years

If you look at photons emitted now by the Sun and Andromeda, they will pass at the half-way point, which is 1.27 million  light years away, in 1.27 million years.

Some members are insistent about single c  for the limit  of relative speeds.

If we consider a relativity  between an observer and a photon packet of Andromeda, this insistence became valid; but we consider two photon packets that are approaching to each other.

Relativity concept is an alternative method to solve some problems; however this method is unsuccessful for determinig the meeting point (special and classic relativity); meeting point of this method is the inert partner directly.

Realistic/practical/main analyzing method is done by co-reference frame; it does not confuse. If we assign one of  test objects as reference frame (inert); we already suppose that its location is fixed; but this is not reality. We use this method for artificial faciliteness.

Please don't fall to a mental trap; you may trust your own intellectual performance.

Of course, Einstein is our an idol/hero (even for me because of  the "E = mc² " relation and Bose-Einstein density). However, we can improve our paradigm for cosmological analyses and light kinematics and we can calculate better  the age of Universe (my result: 19.28 Gyrs). The mentality of SR prevents cosmological analyses.