| | |

Projective Rank-one Closure for Terminal Navier–Stokes Saturation

Byadmin

Internal ID: CGI-RSR-000006
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics, Physics
Research Topics: Partial differential equations / fluid dynamics, Clay Millennium Problems, Navier-Stokes

Abstract


We study terminal rank-one saturation mechanisms in a dyadic analysis of the three-dimensional incompressible Navier–Stokes equations. Starting from a high–high OBCI closure module for comparable high-frequency interactions, we analyze the remaining determining-scale paraproduct strain branch using localized output Gram matrices. Positive spectral mass away from the top eigenvalue gives a coherence-rank defect and hence a square-function gap. The terminal branch ledger reduces the possible nondepleted rank-one configurations through Beltrami depletion, finite-beat damping, orthogonal channel splitting, the middle-eigenvalue strain criterion, and velocity- and vorticity-direction criteria. The final moving-frame one-component branch has the form U = ϕv, |v| = 1. Rank-one output coherence makes this a projective problem: the active output selects a projective direction [v]. If [v] is flat, the branch is fixed-frame and trivial by incompressibility. If [v] is nonflat and visible in the output space, it produces a secondary projective mode and hence positive coherence-rank defect. The only remaining case is an axial/scalar-angle projective degeneracy, which is routed through the velocity-direction, vorticity-direction, planar/2D3C, or splitting alternatives. Thus no terminal nondepleted rank-one output-coherent saturation branch persists under the stated modules and criteria.

Available Document

DOI: 10.5281/zenodo.19970451

Citation:
Petersen, B. L. (2026). Projective Rank-one Closure for Terminal Navier–Stokes Saturation. Zenodo. https://doi.org/10.5281/zenodo.19970451

Source Code and Supporting Materials

N/A

Summary and Notes

The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive directional alignment among The CG viewpoint interprets dangerous nonlinear growth as a problem of excessive coherence among interacting modes.

Rather than asking only how large the velocity field becomes, this approach asks how organized the nonlinear output becomes. In particular, it studies whether high-frequency interactions can collapse into a single dominant output channel capable of sustaining critical saturation.

The current Navier–Stokes manuscripts translate this idea into localized output Gram matrices. These matrices measure whether nonlinear contributions combine through one rank-one coherent channel or whether secondary independent channels force a spectral gap.

Key organizing ideas:

  • near rank-one cases are tested against the intrinsic geometry of the Navier–Stokes symbol
  • dangerous growth requires persistent rank-one output coherence
  • localized Gram matrices make output coherence measurable
  • spectral mass away from the top eigenvalue creates a coherence-rank defect
  • coherence-rank defect produces a square-function / bilinear gap

Related Work

CGI-RSR-000005: Rank-One Coherence Obstructions in High–High Navier–Stokes Interactions.

Similar Posts