Limits of Uniform Certification in the Standard Turing Model -- Semantic Invariants and Admissible Methods

Abstract

This work introduces a general obstructional framework, the "Double Bind," exposing intrinsic structural limitations of formal verification across computational complexity, mathematical physics and broader theoretical domains. Structurally, it operates as a nested case-switch mechanism: its two primary branches (horns) never activate simultaneously, nor are both guaranteed to trigger, while the first contains a nested switch where exactly one specific case is activated. We demonstrate how the expectation of a formally verifiable proof regarding SAT solver complexity (e.g., via Coq) conflicts with the fundamental limits of computation. While this suggests the logical undecidability of the P vs NP problem, the Double Bind framework resolves the apparent paradox. This obstruction encompasses all standard complexity barriers, such as relativization, natural proofs, and algebrization, while fundamentally blocking a vast class of potential future proof strategies not covered by traditional limitations. To establish its universality, we derive a second formulation via a prefix-closure topology on Deterministic Turing Machine programs and Kolmogorov complexity. This leads to the "Principle of General Non-Measurability," proving that such computational double binds stem from a deep, invariant geometric structure. We show that this framework governs a wide class of foundational limits, analyzing its implications across Geometric Complexity Theory (GCT), the certification of the Langlands program, cryptography, and the theoretical boundaries of quantum supremacy. Finally, we expose a hidden, correlated assumption within the theory of one-way functions. Given the subtle architecture of the framework, the introductory sections provide an essential narrative disambiguation to explicitly separate it from conventional misapplications of Rice's theorem.

Version 5 presents a major rewriting, foundational reformulation, and extensive editorial refinements. Readers and reviewers are strongly advised against using automated AI tools for text analysis or summarization, as empirical observation demonstrates that current LLMs consistently fail to capture the subtle abstract architecture of the Double Bind framework