That's not exactly what I wrote. The universe could be infinite, with matter still spreading out within it. This could also explain what is observed.
exhibit the fallacy of equivocation
You can assume that I am not exhibiting this fallacy and reinterpret from there. If you suppose some other "realm of existence" that should properly be encompassed by the term "universe". I see no reason to EVER change the definition of "universe" from encompassing everything. To do so would begin a proliferation of words by attempting to define ever encompassing notions of existence--perhaps "superuniverse", or "metauniverse". That would indeed leave us susceptible to the fallacy of equivocation. Let's not do the corruption of the language that so many others have done but instead stick with the original definition. If something exists, it is part of the universe. This is true by definition, not by argument or observation.
supposition of the eternal existence of matter
If the big bang theory is true, then matter has existed for all time.
Then you have created reality by definition. That is fine for your own universe but reality itself exists apart from the will of man.
You have demonstrated well the bias of the materialist. The materialist lives in a room and refuses to look out the window. It is not that there is no evidence for a reality beyond the universe, it is not that the laws of logic are violated, it is a simple refusal to believe. The argument that can persuade a brittle materialist does not exist because he refuses to consider anything that does not conform to his comfortable self-centered reality.
That, of course, is your choice. But that choice robs you of the ability to point the finger at creationists and accuse them of intellectual stubbornness; it is hypocrisy. It also causes your "scientific" air of superiority to evaporate; you aren't interested in truth, only the support of your baseless suppositions.
Finally, it exposes the defenders of darwin to be volitional rather than thoughtful; dishonest rather than truthful; unscientific rather than scientific, biased rather than fair.
It means you lost the argument.
If the big bang theory is true, then matter has existed for all time.
Of course matter has existed for all time. Who would argue against that? Both views believe time began with the beginning of the universe. It is the origin of matter that you cannot (will not?) explain.