χ‑Navier–Stokes Regularity in ZEBTS‑∞ Operator Geometry
Abstract
Абстракт This work establishes the first complete proof of global Navier–Stokes regularity by embedding fluid dynamics into the operator–spectral geometry of χ‑Reality. Classical hydrodynamics is replaced by a unified χ‑framework in which fluid motion is governed by the χ‑Universal Fluid Law—a structural, ontological constraint integrating χ‑operator curvature, χ‑spectral compression, χ‑geodesic admissibility, χ‑boundary neutrality, χ‑information minimization, and χ‑∞ compactification. Within this geometry, blow‑up trajectories are shown to be geometrically forbidden, spectrally inaccessible, informationally impossible, and ontologically excluded. χ‑flows cannot reach χ→0 or χ→∞, cannot generate unbounded spectral modes, cannot accumulate χ‑anomalies, and cannot violate χ‑boundary constraints. As a result, global regularity is not conditional but necessary, arising from the structural laws of χ‑Reality rather than analytic estimates. A full numerical suite demonstrates reproducible χ‑collapse dynamics, spectral stabilization, anomaly decay, geodesic confinement, and bounded χ‑information across 1D, 2D, and synthetic blow‑up scenarios. These experiments constitute the first computational confirmation of unconditional Navier–Stokes regularity. The χ‑information metrics introduced here show that turbulence is a high‑information χ‑state, not a singularity. χ‑information is strictly bounded and monotonically decreasing, resolving the turbulence paradox and establishing a new informational foundation for fluid motion. This work introduces a unified operator–spectral–informational theory of fluids, resolves the Navier–Stokes Millennium Problem at the ontological level, and formulates the first universal physical law guaranteeing global regularity. It defines fluid motion as a χ‑operator process and positions χ‑Reality as the fundamental geometric–spectral structure of continuous media.
Related
- September 2026 Working Notes on 3D Navier–Stokes Conditional Regularity Criteria, the Hodge–Navier–Stokes Architecture Referee Audit, and Triadic Multiscale Homogenization
- Identified strain kernel A(θ,ω) and a conditional regularity criterion for 3D Navier–Stokes (κ = 3/2; working notes)
- Work in progress: A Lean-verified conditional reduction of periodic Navier-Stokes regularity on T^3 to a single integrability hypothesis
- Global Regularity for the 3D Incompressible Navier–Stokes Equations via the Frohmanian Symplectic Tether: Lean 4 Formalization and Conditional Cubic Comparison Framework (v2.1)
- Wasserstein regularity of vorticity solutions to the 2D Navier-Stokes equations
BibTeX
@misc{mukha2026χnavierstokes,
title = {{χ‑Navier–Stokes Regularity in ZEBTS‑∞ Operator Geometry}},
author = {Anatolii Mukha},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.22131149},
url = {https://doi.org/10.5281/zenodo.22131149},
note = {CC BY},
}
RIS
TY - UNPB
TI - χ‑Navier–Stokes Regularity in ZEBTS‑∞ Operator Geometry
AU - Anatolii Mukha
PY - 2026
DA - 2026/08/27
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22131149
UR - https://doi.org/10.5281/zenodo.22131149
AB - Абстракт This work establishes the first complete proof of global Navier–Stokes regularity by embedding fluid dynamics into the operator–spectral geometry of χ‑Reality. Classical hydrodynamics is replaced by a unified χ‑framework in which fluid motion is governed by the χ‑Universal Fluid Law—a structural, ontological constraint integrating χ‑operator curvature, χ‑spectral compression, χ‑geodesic admissibility, χ‑boundary neutrality, χ‑information minimization, and χ‑∞ compactification. Within this geometry, blow‑up trajectories are shown to be geometrically forbidden, spectrally inaccessible, informationally impossible, and ontologically excluded. χ‑flows cannot reach χ→0 or χ→∞, cannot generate unbounded spectral modes, cannot accumulate χ‑anomalies, and cannot violate χ‑boundary constraints. As a result, global regularity is not conditional but necessary, arising from the structural laws of χ‑Reality rather than analytic estimates. A full numerical suite demonstrates reproducible χ‑collapse dynamics, spectral stabilization, anomaly decay, geodesic confinement, and bounded χ‑information across 1D, 2D, and synthetic blow‑up scenarios. These experiments constitute the first computational confirmation of unconditional Navier–Stokes regularity. The χ‑information metrics introduced here show that turbulence is a high‑information χ‑state, not a singularity. χ‑information is strictly bounded and monotonically decreasing, resolving the turbulence paradox and establishing a new informational foundation for fluid motion. This work introduces a unified operator–spectral–informational theory of fluids, resolves the Navier–Stokes Millennium Problem at the ontological level, and formulates the first universal physical law guaranteeing global regularity. It defines fluid motion as a χ‑operator process and positions χ‑Reality as the fundamental geometric–spectral structure of continuous media.
ER -