A unified quaternion–complex framework for the incompressible Navier–Stokes equations: Geometric structure, an energy-competition reformulation of regularity, and flow diagnostics

We present a unified quaternion–complex framework for the incompressible Navier–Stokes equations that exposes the geometric structure underlying viscous fluid motion. Introducing complex coordinates z=x+iy and the complex velocity F=u+iv, we show that the convective nonlinearity decomposes exactly as (u·∇)F=F ∂zF+F* ∂z¯F, separating an analytic part from a conjugate, non-analytic part. In three dimensions, we construct the corresponding decomposition through quaternions in a form that is polynomial in the velocity field and its gradient and therefore free of the inverse operator: the channel fields are the anticommutator and commutator halves of the quaternion product of the velocity with its gradient, ΦmA=12{Q,∂mQ}⋆=−12 ∂m|Q|2 and ΦmC=12[Q,∂mQ]⋆=Q×∂mQ, measuring respectively the variation of the speed and the turning of the velocity direction. The central structural results are the exact pointwise identity |ΦA|2+|ΦC|2=|Q|2|∇Q|2 and the exact pointwise orthogonality Re(ΦmA ⋆ ΦmC)=0, which split the weighted gradient energy into two channels sharing a single budget, together with the exact convective decomposition (u·∇)Q=12∇|Q|2+ωQ×Q into Bernoulli and Lamb components. Both channels originate in the convective nonlinearity, and viscous dissipation enters only through the Laplacian; the channels are kinematic, in the same sense as the classical strain–rotation split. Using this decomposition, we recast the global regularity question as a sharply stated energy-competition inequality between the two channels, equivalently an a priori enstrophy bound, prove that regularity follows when this inequality holds, and state its unconditional establishment as an open problem; we do not claim a resolution of the Clay Millennium problem. Direct numerical simulations at 643, 1283, and 2563 verify every exact identity to machine precision, provide a first dynamical measurement of the channel competition, showing the global partition saturating in the band 0.62–0.70, and quantify the relation of the resulting geometric flow diagnostic to the Q-criterion and λ2. The framework further yields a scale-by-scale cascade in which the classical transfer-conservation law acquires an exact channel decomposition, an exact asymptotic similarity law for the channel structure of the laminar boundary layer, Rex T→f/2f′, and a similarity-preserving channel-based closure with computed wall shear f″(0)≈0.332 06+0.0483 αQ.