The problem of existence and smoothness of solutions to the Navier–Stokes equations is one of the seven Millennium Prize Problems. This work proposes a physically motivated resolution of this problem within the Fractal Quantum Holographic Paradigm (FQHP 5. 0). It is shown that the mathematical difficulty is an artefact of a "single-channel" description of reality that ignores the hidden informational χ-channel. The work consists of two parts. Part 1 presents a direct proof of global regularity for fractal space with dimension D = 2. 048. The replacement of the classical Laplacian by a fractional operator of order γ = 1. 048, together with the introduction of the dynamic Jacobian of the 5D projection J = (1 - ||u||²/c²) ⁻¹, ensures an infinite growth of the effective viscosity νₑff = ν·J as the local fluid velocity approaches the rendering limit c. The use of Lorentz spaces L^ (p, q) and Gagliardo–Nirenberg interpolation inequalities allows a rigorous mathematical proof of the subordination of the nonlinear convective term to dissipation. Part 2 applies the method of reductio ad absurdum. It is shown that in the hypothetical smooth limit (α = 0, γ = 2), the dynamic protection disappears: the Jacobian becomes constant, and the effective viscosity ceases to depend on velocity. The absence of a mechanism for dumping excess energy into the χ-channel makes a smooth Universe physically unstable, doomed to collapse into a singularity. The very fact of our existence mathematically proves that the Universe must possess a fractal roughness α = 0. 048. The work offers a physically motivated extension of the Millennium problem and contains a rigorous proof of global regularity for γ = 1. 048, as well as demonstrating the inevitability of the fractal architecture of spacetime.
Pavel Trojan (Sat,) studied this question.