A Unified Formulation of Fluid Mechanics and a Helmholtz–Leray Projection Framework for the Classical Navier–Stokes Sector

This record provides the full technical version of the manuscript submitted to the Journal of Mathematical Fluid Mechanics under manuscript number JMFM-D-26-00193. This paper proposes a unified formulation of fluid mechanics based on a position field and the transported velocity, density, and volume-element fields. The central object is a Taylor-type stress generated intrinsically by the finite-difference transport operator acting on the momentum–acceleration density, rather than by prescribing an external phenomenological constitutive law. The resulting system consists of exact transport laws for the mass-volume measure and the volume element, together with the corresponding Taylor-stress momentum balance. A Green–resolvent representation of the general solution is constructed at the operator level. The quadratic Taylor sector yields the effective elliptic principal part, while the higher Taylor tail is retained as a controlled Neumann–Duhamel correction. Under the standard Millennium conditions, interpreted in the classical projected-sector sense defined in the manuscript, the Helmholtz–Leray projection of the unified dynamics coincides with the classical incompressible Navier–Stokes system.