The Physicality of Logic: The Structural Inapplicability of Incompleteness, Undecidability, and Computational Complexity Gaps to Finite Physical Domains

Here's the metadata, reflecting the full accumulated revision from the live v4/DOI 19491912 record through v9. Title The Physicality of Logic: The Structural Inapplicability of Incompleteness, Undecidability, and Computational Complexity Gaps to Finite Physical Domains Authors Daniel L. Burnstein (Ormstown, Québec, Canada) — ORCID: 0000-0002-7966-4250 Resource type Preprint (match whatever the existing DOI 19491912 record currently uses) Description (Zenodo description field) This version substantially revises and retitles the previously published record. The paper argues that 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 — rest on a category error when applied to physical reality: the attribution to a finite, bounded substrate of properties that belong only to formal systems operating over infinite domains. Under Minimal Physically Derivable Theories (MPDT), a formal system whose intended domain is the finite preonic substrate does not satisfy the preconditions of any of the three results — Gödel's theorems require arithmetical sufficiency over an infinite model; Turing's proof requires an infinite tape; P vs NP requires asymptotic growth over unbounded input size. This is a Level 1 result, conditional on QGD's axioms and structural inapplicability rather than a refutation of these results in the infinite systems they were proved about. The paper develops physical constructivism — four principles establishing that mathematical objects exist only when physically constructed — and, new in this version, two quantitative results extending that programme: a Physical Cost Invariance theorem showing the number of physical binding events required to construct any positive integer v is exactly v−1 regardless of construction path, and a Minimum Momentum Cost theorem showing the total physical momentum required is bounded below by (v−1)·c̃, achieved uniquely by sequential construction. Both results are new derivations connecting integer arithmetic directly to physical process, independent of the paper's logic-focused main argument. Companion to: P9 (Fermat's Last Theorem), P27 (Physically Derivable Set Theory), P36 (Continuous Mathematics as Finite Prescriptions), P40 (The Constructibility Criterion).