Existence of a Periodic Attractor for the 3D Navier-Stokes Equations on ${\mathbb{T}}^3$

This paper presents a framework for solving the Navier-Stokes equations on the 3D torus $T^3$ . In this work, a review is made and a sequel follows with regard to the author’s work in 2021 in [1] where the Geometric Calculus method was used to obtain Equation (6) in that reference. Developing an algebraic and differential procedure to rewrite the Navier-Stokes equations in a form that renders solutions in a particular i-th direction of fluid flow, the expressions solved for give finite time singularities in the form of the LambertW function on the surface of the ball embedded in ${\mathbb{T}}^3={\left[0,1\right]}^3$ . In this paper, it is argued that these singularities for a single $u_i$ direction can be shifted outwards by using fixed point methods via nth compositions of LambertW functions. In this paper, the solution to Equation (6) is provided, which proves that the triple product $u_x{u}_y{u}_z$ has both singular and no finite time blowup. We develop a geometric-analytic framework for the three-dimensional incompressible Navier-Stokes equations on the torus ${\mathbb{T}}^3$ that yields a periodic global attractor and clarifies the role of singular structures arising in nonlinear velocity interactions. Building on a componentwise reformulation of the Navier-Stokes equations, the nonlinear transport term is resolved explicitly in physical space, revealing a rigid normal-form structure governed by compositions of the Lambert $W$ function. Singular solutions initially appear on embedded spherical manifolds through branch-point behavior of $W$ , but we show that