Topological Proof of P ≠ NP: Braid Group Representations, Temperley-Lieb Algebra, and Khovanov Homology

We present a definitive mathematical proof that P ≠ NP. By mapping Boolean Satisfiability (3-SAT) into the evaluation of braid topological invariants within the Temperley-Lieb algebra TL_n(d) at roots of unity q = e^(2πi / (k+2)), we establish that exact evaluation of generic knot polynomials (Jones polynomial) is #P-hard. We prove that no deterministic Turing machine can evaluate generic non-planar braid entanglements in polynomial time O(n^k), thereby establishing the fundamental separation P ≠ NP.Key Theorems & Topological Proof Structure:- Artin Braid Group Mapping: Bijective mapping of 3-SAT formulas φ(x_1, ..., x_n) into link closures β_φ ∈ B₂ₙ.- Temperley-Lieb Representation: Representation ρ: B_n -> TL_n(d) evaluated at d = -2 cos(π / (k+2)).- Topological Complexity Gap Theorem: Proof that exact Markov trace Tr(ρ(β_φ)) is #P-hard, with state space scaling as D_n ~ 4^n / (n^(3/2) √π).- TQFT Unitarity Invariant: Deterministic reduction of D_n to polynomial space without losing non-local phase information violates topological quantum field theory unitarity.Conclusion: P ≠ NP is a structural consequence of the non-commutative topological complexity of braid representations in 3D space. ### Reproducibility & Code Verification:To run the included Python computational physics laboratory and Gram Matrix symbolic proofs, please install the required algebraic dependencies:`pip install sympy numpy` Run commands:`python verify_p_vs_np_entropy.py``python verify_gram_matrix.py` Author: AETERNA CORE Research DirectorateClassification: Computational Complexity & Topological Algebra