Artian Fluid Mechanics: Navier-Stokes from Exact Finite-Address Conservation

Abstract

From finite transport packets to the Navier-Stokes equation This paper asks what a fluid equation becomes when the continuum is not the source object. The primitive state is a finite graph of completed address events. Material is counted in Artian-mass units, \[ B_w=\frac{M_w}{m_A}, \] and nonnegative directed packets define the signed link current \[ J_{ww'}=T_{w\to w'}-T_{w'\to w}, \qquad J_{ww'}=-J_{w'w}. \] That antisymmetry gives exact finite-cell material conservation. Directed momentum packets then split uniquely into bulk advection, a counterflow residual, and A4 interaction impulse. The resulting conservative hydrodynamic projection is \[ \partial_T\rho+\nabla\!\cdot(\rho\mathbf v)=0, \] \[ \partial_T(\rho\mathbf v) +\nabla\!\cdot(\rho\mathbf v\otimes\mathbf v) =\nabla\!\cdot\sigma+\rho\mathbf g. \] For a spinless simple fluid, discrete rotational Noether balance makes the stress symmetric. Objectivity, parity, isotropy, symmetry, and first-gradient locality then leave one legal linear tensor form: \[ \boxed{ \sigma =-p\mathbf I+2\mu\mathbf D+\zeta(\nabla\!\cdot\mathbf v)\mathbf I }. \] The master equation is therefore \[ \boxed{ \rho(\partial_T+\mathbf v\!\cdot\nabla)\mathbf v =-\nabla p+\mu\nabla^2\mathbf v +\left(\zeta+\frac{\mu}{3}\right) \nabla(\nabla\!\cdot\mathbf v)+\rho\mathbf g }. \] The transport signs are fixed by the completed-record entropy balance, \[ \dot s_{\rm prod} =\frac{2\mu}{T}\mathbf D{:}\mathbf D +\frac{\zeta}{T}(\nabla\!\cdot\mathbf v)^2 +\frac{\kappa}{T^2}|\nabla T|^2 +\frac{k_B}{V_{\rm SQ}}\mathcal C \ge0, \] so \(\mu,\zeta,\kappa\ge0\) in the declared near-equilibrium class. Inversion symmetry of the address kernel eliminates every odd spatial derivative; the first finite-address correction is fourth order and suppressed by \(\ell_A^2\). The paper also constructs an explicit positive channel-envelope kinetic dynamics whose finite state space is forward invariant. It yields global evolution at every finite substrate tick and excludes point-supported physical blow-up. This is a theorem about the QTT finite state space; the abstract smooth-continuum Clay problem remains a different mathematical object. Version: 2.0 Concept DOI: 10.5281/zenodo.20122608 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 Scientific status: GREEN: EXACT FINITE MATERIAL-PACKET CONSERVATION GREEN: EXACT FINITE MOMENTUM-PACKET BALANCE GREEN: CAUCHY CONTINUUM PROJECTION GREEN IN DECLARED CLASS: NEWTONIAN STRESS UNIQUENESS GREEN: ENTROPY-SIGN TRANSPORT CONE GREEN: FINITE-LEDGER GLOBAL EVOLUTION STANDARD-PHYSICS CONSISTENCY: NAVIER-STOKES FORM NOT CLAIMED: GLOBAL SMOOTHNESS OF THE ABSTRACT CONTINUUM PDE Public anchors: Entropy and Second-Law reference theorem Inertia and Newton's Second Law Stationary-action theorem Renewal Dust discipline Derivation Atlas QTT Lexicon