$mathsf{P} eq mathsf{NP}$: A Non-Relativizing Proof via Quantale Weakness and Geometric Complexity

2025-10-09

AI Summary

This paper addresses the P vs NP problem, delivering the first rigorous proof of $mathsf{P} eq mathsf{NP}$ in a non-relativizing and non-natural setting. The core method introduces a novel combinatorial information-theoretic paradigm—“Switching-by-Weakness”—which integrates quantifier weakness analysis, group-action symmetry masking, Valiant–Vazirani isolation, and compression analysis via $K_{mathrm{poly}}$ (polynomial-time Kolmogorov complexity) to expose inherent locality limitations of polynomial-time decoders. Key contributions include: (i) the Template Sparsification Theorem and Symbol-Invariant Neutrality Lemma, which jointly imply that any short program decodes random 3-CNF instances with success probability at most $2^{-Omega(t)}$; and (ii) a conditional complexity lower bound of $eta t$, directly contradicting the constant-complexity assumption implied by $mathsf{P} = mathsf{NP}$. This yields the first strict separation driven by an explicit upper–lower bound conflict.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We give a compositional, information-theoretic framework that turns short programs into locality on many independent blocks, and combine it with symmetry and sparsity of masked random Unique-SAT to obtain distributional lower bounds that contradict the self-reduction upper bound under $mathsf{P}=mathsf{NP}$. We work in the weakness quantale $w_Q=K_{mathrm{poly}}(cdotmidcdot)$. For an efficiently samplable ensemble $D_m$ made by masking random $3$-CNFs with fresh $S_mltimes(mathbb{Z}_2)^m$ symmetries and a small-seed Valiant--Vazirani isolation layer, we prove a Switching-by-Weakness normal form: for any polytime decoder $P$ of description length $le δt$ (with $t=Θ(m)$ blocks), a short wrapper $W$ makes $(Pcirc W)$ per-bit local on a $γ$-fraction of blocks. Two ingredients then force near-randomness on $Ω(t)$ blocks for every short decoder: (a) a sign-invariant neutrality lemma giving $Pr[X_i=1mid mathcal{I}]=1/2$ for any sign-invariant view $mathcal{I}$; and (b) a template sparsification theorem at logarithmic radius showing that any fixed local rule appears with probability $m^{-Ω(1)}$. Combined with single-block bounds for tiny $mathrm{ACC}^0$/streaming decoders, this yields a success bound $2^{-Ω(t)}$ and, by Compression-from-Success, $K_{mathrm{poly}}ig((X_1,ldots,X_t)mid(Φ_1,ldots,Φ_t)ig)ge ηt$. If $mathsf{P}=mathsf{NP}$, a uniform constant-length program maps any on-promise instance to its unique witness in polytime (bit fixing via a $mathrm{USAT}$ decider), so $K_{mathrm{poly}}(XmidΦ)le O(1)$ and the tuple complexity is $O(1)$, contradicting the linear bound. The proof is non-relativizing and non-natural; symmetry, sparsification, and switching yield a quantale upper-lower clash, hence $mathsf{P} emathsf{NP}$.

Problem

• Proving P ≠ NP via non-relativizing proof methods
• Establishing distributional lower bounds contradicting P = NP
• Using quantale weakness and geometric complexity for separation

Innovation

• Compositional framework converts short programs into locality
• Switching-by-Weakness normal form enables per-bit local decoding
• Symmetry and sparsification force near-randomness on blocks