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
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| Format: | Recurso digital |
| Langue: | anglais |
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2025
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| _version_ | 1866902160283795456 |
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| author | Morcillo, Patrick |
| author_facet | Morcillo, Patrick |
| contents | <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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18025843 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | 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 Morcillo, Patrick Navier–Stokes, Millennium Problem, fractal dimension, semantic inconsistency, non-linear dynamics, triadic decomposition, coherence potential, dimensional deficit, emergence metric, mathematical ontology, fractal universality, FDAA, TFA, space morphism, higher-dimensional geometry, symmetry breaking, compositional closure, fluid continuum, stability theorem, existence and smoothness <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> |
| title | 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 |
| topic | Navier–Stokes, Millennium Problem, fractal dimension, semantic inconsistency, non-linear dynamics, triadic decomposition, coherence potential, dimensional deficit, emergence metric, mathematical ontology, fractal universality, FDAA, TFA, space morphism, higher-dimensional geometry, symmetry breaking, compositional closure, fluid continuum, stability theorem, existence and smoothness |
| url | https://doi.org/10.5281/zenodo.18025843 |