Navier–Stokes as relational difference redistribution: A mechanism model for incompressible flow under closure constraints

We formulate an incompressible redistribution framework in which flow is described
as the rate of change of an accumulated transverse difference field. The active field
$\mathbf P$ and accumulated field $\mathbf D$ are linked by
$\partial_t\mathbf D=\mathbf P$, leading to an extension of the incompressible
Navier--Stokes equations with three distinct mechanisms: viscous diffusion,
conservative memory, and linear relaxation. The classical incompressible
Navier--Stokes system is recovered exactly when the memory and relaxation parameters
vanish.

The model admits a natural energy balance in which the memory contribution
$c^2|\nabla\mathbf D|^2/2$ is conservative, while viscosity and linear relaxation
provide the dissipative channels. Its linearized spectrum predicts a finite band of
underdamped modes when $c^2>\nu\lambda$, with analytically determined edges
\[
k_\pm=\frac{c\pm\sqrt{c^2-\nu\lambda}}{\nu}.
\]
This yields a direct experimental or numerical test: the presence and location of
a ringing band determine whether the proposed closure is quantitatively applicable.
The result provides a compact, falsifiable model of incompressible flow with
accumulated memory.