mathematic

This is an interpretation. I doubt if it is orthodox, rather it is an interesting idea that some take seriously as a possibility.

Highly unlikely. In order for life (as we know it) to exist, the background has to be cool enough, which would imply a temperature cooler than infra-red.

The CMB started out as blackbody radiation of 3000 deg K. This is somewhat cooler than the surface of the sun. I don't know exactly how it would look, but I suspect it would be red.

On earth, liquids and solids are also mostly empty space. When matter is really squashed, you have white dwarfs or neutron stars - even black holes, all of which have much higher densities than ordinary matter.

You are correct in that there are 3x3x2=18 possibilities. Your immediate thought is correct. You seem to be looking for something more complicated, but there isn't any.

Your question is too vague. What is the statement that you are asking to be proved?

Your statement is unclear. Are you mixing organism in test tubes or do have 3 test tubes for each organism separately? Also test tube with 0 is a separate case, since it doesn't matter which organism it is.

If you go fast enough so that radio waves get blue shifted to gamma rays, your body will perceive them as gamma rays.

What is the precise definition of "entail" in this context?

Question for you, why do think having components answers the question in any way? Elementary particles have properties.

The formula is correct for |x| < 1 (or -1 < x < 1). The series does not converge for other values of x. If you are familiar with complex numbers, the condition |x| < 1 applies here also.

For each abs term, divide the x axis into 2 parts depending on whether the argument is + or -. For example, abs(x+1) divide at x = -1. Then for each part you have either x+1 (x > -1) or -(x+1) (x < -1), without abs. Then analyze the expression for each part. In your example, you may need several pieces, but for each piece it should be easy enough.

Yes. Also (-infinity,2.5), but not both at the same time.

The point of the question is that there is more than one x for each value of f(x) (with one exception). Therefore to define the inverse properly, you have to breakup the x domain into pieces, so that there is only one x for each value of f(x).

The physics inside a black hole has never been worked out, so your question really can't be answered. Photons might spiral into the center, but that is just a guess.

If you travel at the speed of light (which you can't) anything is possible.

I think you mean "associative" not "distributive". The answer is yes. As for proofs, note that the addition of vectors can be viewed as separate additions for each component, where the commutative and associative laws hold (ordinary arithmetic).

What are the elementary that makes them any different?

What is the distinction you are making between "elementary" ad "fundamental"?

Negative and positive are arbitrary. The important point is that protons and electrons have opposite charge.

As a practical matter, there is no way to avoid photons.

These two items are completely unrelated. However they are both well documented phenomena.

What is your point (question)?

By definition an ionized gas is a plasma.

Photons are the particles of light, radio, x-rays, etc. Your question is confusing.