The Physicality of Logic: On the Refutation of Incompleteness, Undecidability, and Computational Complexity Gaps as Features of Physical Reality

Three foundational results of twentieth-century mathematical logic — Gödel's incompleteness theorems, Turing's undecidability of the halting problem, and the P vs NP complexity gap — have been widely taken to impose permanent limitations on what a physical theory can achieve. This paper argues that all three applications rest on the same category error: the attribution to physical reality of properties that belong only to formal systems operating over infinite domains. Under the framework of Minimal Physically Derivable Theories (MPDT), established by the Uniqueness Theorem, physical reality is constituted by a finite discrete substrate whose total information capacity is bounded. A formal system whose…