Vladimir
You asked me to comment on your essay.
"And mathematics and physics have one foundation - essential primary structure of Nature"
This is the key. And physical existence is the physically existent state of whatever comprises it at any given time. It is a sequence of such discrete and definitive states, altering at an extremely high speed, each degree of difference being vanishingly small.
In trying, through all the quotes, to find your definition of physical existence/reality, I find:
"Physical reality is formed of all the world's material objects, both substantial and non-substantial (e.g., electromagnetic, gravitational and other fields), as well as all motion processes and internal changes that happen with these objects"
Now, there are several questions here, which can be boiled down to:
1 What is an object? When is a dustbin, car, you, etc that object you are referring to? It alters, ie it is different, yet we still label it as being the same thing. Albeit we might say it has changed. But that is contradictory, because a difference is a difference. Physically, it is either one thing, or it is something else. In other words, there are no objects (except in the sense of whatever is the elementary substance(es)). What exists at any given time is the physically existent state of whatever comprises the reality, which is then superseded by the next state.
2 Then one asks, what is 'non-substantial', 'a process', 'internal changes', in terms of reality. Because for something to be reality, it must have a physical presence. And once one starts defining that, as it occurs, in terms of physical state, then this problem disappears.
I am not quite sure what your point is re consciousness, etc. However, none of this has anything to do with physical existence. It is independent of the mechanisms whereby it is detected, and existed before it was detected. Obviously, there must be something, so that it can then be detected (forget the detail that what is detected is a physically existent representation of what occurred). In other words, all considerations about measurement/whatever affecting reality are nonsense.
Paul