IncompressibleNSOperator
plain-language theorem explainer
The incompressible Navier-Stokes operator structure encodes a complete discrete flow on a finite lattice, with velocity divergence-free, transport given by conservative flux differences, and pair amplitudes plus stretch factors derived from normalized vorticity and local strain. Researchers modeling discrete fluids or proving J-cost decrease would cite it to access the full surface including stretching bounds and viscous absorption. The definition supplies a constructor that assembles these fields from the minimal core operator data.
Claim. An incompressible Navier-Stokes operator on a lattice with $N$ sites consists of a lattice topology, grid spacing $h>0$, viscosity $nu>0$, a discrete state, a divergence-free velocity field, a transport flux scalar field, a site permutation, and derived positive pair amplitudes and stretch factors at each site such that the stretching contribution at site $i$ is at most the pairwise stretching change of those values and the indexed pair budget is absorbed by the negative total viscous term.
background
The module works on a finite lattice whose topology supplies neighbor maps along three axes. Vector fields assign a real value to each site and axis; scalar fields assign a real to each site. The core operator carries only the physical data: topology, $h$, $nu$, the discrete state from the 2D Galerkin truncation, a divergence-free velocity, and transport flux. Conservative transport at a site is the difference between the flux and the flux at its permuted image. Pair amplitude at a site is the absolute normalized vorticity; the stretch factor is the local strain ratio $1+dt|nabla u|$.
proof idea
The declaration is a structure whose fields are stated explicitly. The accompanying constructor populates the pair fields by calling the core-derived amplitude and stretch functions, their positivity theorems, the core stretching bound lemma, and the viscous absorption property already present in the core operator.
why it matters
This supplies the full operator surface required by downstream constructions such as the operator pair budget, its non-negativity, its absorption by viscosity, and the J-cost monotonicity theorem from the core alone. It closes the compatibility layer between the minimal physical data and the pair-event interface used for recognition-cost arguments. In the discrete Navier-Stokes setting it guarantees that stretching contributions remain controlled by viscosity, supporting the broader monotonicity results that feed into the Recognition framework.
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