Turing complete Navier-Stokes steady states via cosymplectic geometry

2025-07-10

AI Summary

This work addresses the fundamental question of whether steady solutions to the Navier–Stokes equations on three-dimensional Riemannian manifolds can achieve universal computation (i.e., Turing completeness). Methodologically, it introduces a cosymplectic geometric framework, establishing—for the first time—a link between nonzero harmonic 1-forms and cosymplectic structures to construct Turing-complete steady Navier–Stokes solutions. The key contribution is proving that viscosity does not obstruct universal computability when the manifold admits a nonzero harmonic 1-form—equivalently, when its first de Rham cohomology group is nontrivial. This result generalizes the classical Beltrami–Reeb correspondence and extends contact-geometric techniques to viscous fluid systems. By integrating harmonic form theory, cosymplectic manifold geometry, and constructive methods for steady solutions, the work provides the first rigorous geometric criterion bridging fluid dynamics and computability theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning with Manifolds
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Social Networks and Social Media: Computational social science
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

Problem

• Construct Turing complete Navier-Stokes steady states
• Show computational universality with nonvanishing harmonic 1-forms
• Extend Beltrami-Reeb correspondence via cosymplectic geometry

Innovation

• Construct Turing complete Navier-Stokes solutions
• Use nonvanishing harmonic 1-forms for universality
• Extend Beltrami-Reeb flow via cosymplectic geometry