The Band-Limited Navier--Stokes Equation: Derivation from Finite Information Bounds

The classical Navier--Stokes existence and smoothness problem addresses whether initial smooth velocity fields on $\mathbb{R}^3$ can evolve under an unprojected partial differential equation (PDE) to form finite-time blowups or infinite-speed rotational singularities. In this paper, we demonstrate that fluid singularities are not open dynamic puzzles awaiting viscous suppression, but unphysical artifacts arising from the \emph{continuum blind spot}---the continuous PDE assumption of infinite spatial divisibility ($\Delta x \to 0$, $k \to \infty$). Anchored directly to the companion framework of information-bounded mutual renewal (Stieger, 2026), physical fluid fields operating under finite information capacity (Bekenstein entropy bounds and Shannon--Nyquist cardinal sampling) are intrinsically band-limited Shannon interpolants. We establish the complete kinematic framework of the Band-Limited Navier--Stokes (BL-NS) system: (1) isotropic rotational recoverability ($SO(3)$) uniquely selects the maximal spherical Fourier passband $\mathcal{B}_{K_0}(0)$ over a cubic Brillouin zone; (2) the projection operator $\mathcal{P}_\Omega$ is the exact, mandatory continuous image of discrete point-wise lattice interactions; (3) the pressure field $p_\Omega$ is explicitly solved via an autonomous band-limited Poisson equation; (4) the non-linear commutator $\|\mathbf{C}_\Omega(\mathbf{u}_\Omega)\|_{L^2}$ obeys an explicit $K_0^{5/2}$ scaling bound with exact annular constant $4\sqrt{14\pi}/3$; (5) prior to any potential singular epoch $T^*$, BL-NS trajectories converge to classical Navier--Stokes trajectories at rate $\mathcal{O}(K_0^{-1})$; and (6) 19-slot stencil dispersion explicitly derives how the local quarter-turn coin $V^2 = -\mathbb{I}_4$ generates the nonlocal spatial Hilbert envelope $\mathbf{w}_\Omega = \mathcal{H}[\mathbf{u}_\Omega] + \mathcal{O}(a^2 k^2)$. Physical fluids are thereby proved to be kinematically immune to finite-time blowup.