The answer for the total mass of the Universe?

[Ibix]

However, what could potentially refute the conclusions drawn from my second proof?

In Newtonian physics, acceleration is frame invariant, so all frames measure the same acceleration $a$.

 

[PeterDonis]

what could potentially refute the conclusions drawn from my second proof?

Um, the fact that it's wrong?

observers moving at constant velocities relative to one another will observe the same physical phenomena

This is not correct. What is correct is that the laws of physics will be the same in all inertial frames. But that does not mean that all physical phenomena will be the same. For example, if you and I are moving relative to each other, we will measure the same light beam to have different energy and frequency--different physical phenomena. But the relationship between the light beam's energy and momentum will be the same for both of us--same laws of physics.

So your so-called "proof" is based on a false premise and is invalid.

 

[WeirdUniverse]

But If they have different physics phenomena in different inertial frames, doesn't mean that we would know if we are moving or not?

 

[PeterDonis]

If they have different physics phenomena in different inertial frames, doesn't mean that we would know if we are moving or not?

You would know you were moving or not moving relative to something else--such as the light source in my example. "Moving" is relative. You would not know you were moving or not moving in any absolute sense.

 

[WeirdUniverse]

You mean the universe is actually absolute, but it appears relative to us?

 

[PeterDonis]

You mean the universe is actually absolute, but it appears relative to us?

"The universe" is way too vague. The geometry of spacetime and the stress-energy tensor distribution that gives rise to it via the Einstein Field Equation are absolute, yes. But those things have nothing to do with any particular reference frame.

 

[Dale]

Yes, it does because both objects were added with the same velocity so the difference in their velocities don't change.

Exactly. That is how the principle of relativity works. Different reference frames are related by the Galilean transformation which is simply adding the same velocity to everything.

But I'm not talking about their collisions,

Nor am I. I am talking about an arbitrary force acting between them according to Newton's laws.

If we push a box, either the box gained the acceleration of a or the rest masses gained the acceleration of -a as we have experienced

This is false as I showed above. $\vec a \ne -\vec A$ in general.

the total energy must be the same for that two inertial frames

This is not correct. Energy is frame variant. In every frame, energy is conserved. But different frames will disagree on the total energy.

At this point you need to either engage with the corrections that you have received and start learning the physics that you are missing or there will be no point in further discussion. Do not re-assert the mistakes that have already been pointed out to you. If you do not understand why they are mistakes, then ask for clarification on the specific points that you do not understand. But they are in fact mistakes that indicate some misunderstandings of Newtonian physics.