> Everything that I can measure objectively, including the act of measuring and the concepts of "measure" and "objectivity" happen within the theater of my consciousness

Correction: they happen in what you believe to be a theatre of consciousness. Belief without evidence is religion.

You can demonstrate that such a theatre might exist by demonstrating some phenomenon which cannot be accounted for via mechanistic processes. All attempts have failed so far, as the intuition pumps are typically fallacious.