Roger Penrose's Theory of Consciousness Through the Lens of the Theory of Axiomatic Necessity (TNA): Objective Reduction versus Failure of Local Closure

This paper analyzes Roger Penrose’s Orch-OR theory of consciousness through the formal framework of the Theory of Axiomatic Necessity (TNA). By examining Objective Reduction (OR) as a physical threshold where quantum superposition collapses into definite states, we demonstrate that Penrose's non-computable foundational layer maps directly onto the admissibility domain (N₁), rather than emerging from the operational domain (N₀) of biological silicio-like structures. Through the theorem of Failure of Local Closure, we argue that the brain's micro-tubular network does not generate consciousness internally, but rather acts as a highly specialized structural antenna tuned to non-derivable conditions of admissibility. Ultimately, this formulation reconciles Gödelian incompleteness in human thought by proving that conscious experience requires an irreducible external structural support (N₁) that escapes the operational limits of pure algorithmic computation.