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| Hovedforfatter: | |
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| Format: | Recurso digital |
| Sprog: | engelsk |
| Udgivet: |
Zenodo
2025
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| Fag: | |
| Online adgang: | https://doi.org/10.5281/zenodo.18025843 |
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Indholdsfortegnelse:
- <p>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.</p>