The Complete Mathematical Framework of Fractal Universality: Triadic Decomposition and the Global Regularity of Navier-Stokes with its complete demonstration as required by PDE rigor and Clays' bar
Year: 2025
Type: article
Abstract: This work presents the complete mathematical formulation of the Fractal Universality Axiom (FUA) and its application to the resolution of the Navier-Stokes global regularity problem. I establish a unified framework connecting microscopic activation processes to macroscopic fluid dynamics through triadic fractal decomposition. The theory demonstrates how three fundamental threads—carrier, envelope, and coupling—generate universal fractal dimensions D (carrier) t ≈ 0.63 and D (envelope) t ≈ 0.81 that govern scale-invariant phenomena across physical domains. I provide a rigorous proof of global existence and smoothness for the incompressible Navier-Stokes equations in R 3 × [0, ∞) for all smooth initial data, resolving the Clay Mathematics Institute Millennium Problem. The proof leverages the triadic decomposition to control the vortex stretching term through a universal aggregation operator U that distributes energy across scales, preventing finite-time singularity formation. The framework is mathematically grounded in Littlewood-Paley theory, with explicit dissipation functionals and energy estimates in classical Sobolev spaces. Beyond fluid dynamics, I demonstrate applications in neural network optimization through centroidal fractal envelope training, achieving computational efficiency while maintaining the universal dimensional setpoint D ≈ 0.81. This work establishes fractal universality as both a fundamental physical principle and a powerful mathematical tool, with rigorous proofs of dimensional elevation, compositional closure, and cross-domain invariance.
Author: Morcillo, Patrick
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7092361758
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