An Algebraic Proof of P ≠ NP

README Title: An Algebraic Proof of P ≠ NPAuthor: Zhang Xun (ORCID: 0009-0004-5396-8952)Version: v1.0 (Preprint), 2026-10-18 Overview This preprint presents a claimed proof that P ≠ NP within the framework of the FBGT axiom system and Super-Algebraic Cluster Theory (SAG). The argument is developed using group theory, algebraic topology, and functional operator topology rather than the classical Turing machine model. Core Construction The proof proceeds through the following chain: Cognitive Hypercategory (CogTop) and observer endofunctor FobsFobs are defined over a four-dimensional compact cognitive manifold MM with boundary. SAG Reasoning-Step Group G=π1(F,γ0)G=π1(F,γ0): algorithms are algebraized as group words, with time complexity identified as word length T(n)≡L(w)T(n)≡L(w). Dimensional Ascent Barrier: Hahn–Banach extension fails at the Tarski boundary (Fisher degeneracy set), where the Lie bracket [∇,∇∗]≠0[∇,∇∗]=0 triggers spectral transition from discrete to continuous spectrum. Non-removable Homology Classes: Baire category arguments show polynomial-length words cannot contract non-trivial homology classes supported on non-first-category sets, yielding HS(Σ)>0HS(Σ)>0. Threefold Blockade: algebraic (word-length constraint ∣S∣p(n)<2Ω(n)∣S∣p(n)<2Ω(n)), geometric (bandwidth overflow), and measure-theoretic (ΔT>ΔCΔT>ΔC) arguments jointly establish P ≠ NP. Appendices Appendices A–F provide the AX-30 Fisher-anchored executable definition, a minimal BIBD worked example, dimensional-analysis negative results, three epistemic honesty statements, a review of the Fisher matrix's status in FBGT, and task priorities. Notably, Appendix D explicitly acknowledges that the AX-30 semantics remain conjectural and require authoritative confirmation. Status and Caveats This is a preprint built on the author's own axiomatic system (FBGT/SAG). The proof's validity depends entirely on acceptance of axioms A0–A12 and the CogTop/SAG framework. The appendices themselves record unresolved gaps (e.g., the C=Φ(T)C=Φ(T) mapping is not derived; Fisher-matrix provenance in core FBGT texts is unverified). Independent verification of the axiom system and the Turing-machine embedding functor is required before the result can be assessed by the broader complexity-theory community. Framework Versions Version v410.5281/zenodo.22533545 Sep 6, 2026 Version v310.5281/zenodo.21618022 Jul 27, 2026 Version v210.5281/zenodo.21614404 Jul 27, 2026 Version v110.5281/zenodo.21595680 Jul 26, 202 Keywords: P ≠ NP, cognitive topology, SAG, reasoning-step group, topological rigidity, algorithm rigidity. Licenses CCBY4‘0 知识属于全人类。论文工作只表明人类数学家探索的方向和到过的地方。’