IndisputableMonolith.NavierStokes.JcostMonotonicity
This module establishes monotonicity of the J-cost for discrete incompressible Navier-Stokes flows on a three-direction lattice. It shows that stretching terms are absorbed into the viscous budget once sitewise RCL bounds are summed. Researchers applying Recognition Science to lattice hydrodynamics would cite these results. The argument proceeds by combining local absorption lemmas with global summation over the CoreNSOperator data.
claim$\frac{d}{dt}J_{\rm cost}\leq 0$ holds for flows governed by the discrete NS operator once the lattice stretching contribution is absorbed into the viscous budget via summed RCL pair bounds.
background
The module imports DiscreteNSOperator, which defines a finite three-direction lattice topology together with discrete differential operators and the CoreNSOperator that carries only the physical lattice flow data. From the flow's velocity gradient and Laplacian it derives the pair-budget and viscous-absorption fields. Recognition Science supplies the J-cost functional $J(x)=(x+x^{-1})/2-1$ and the Recognition Composition Law that controls its multiplicative behavior on the lattice.
proof idea
The module organises its argument through four sibling declarations. stretching_absorbed_by_viscosity establishes local absorption of stretching by viscosity. dJcost_dt_nonpos_of_operator lifts the local bound to a global non-positive time derivative. MonotonicityCert packages the result as a certificate, and monotonicityCert supplies the concrete instance. All steps rely on summation of sitewise RCL bounds supplied by the upstream DiscreteNSOperator module.
why it matters in Recognition Science
The module supplies the monotonicity step required for any Recognition Science treatment of discrete Navier-Stokes dynamics. It directly realises the absorption claim stated in its own doc-comment and thereby connects the J-uniqueness property (T5) to lattice fluid models. No downstream declarations are recorded yet, indicating this is an early foundational block for the NavierStokes domain.
scope and limits
- Does not treat compressible or viscous-free flows.
- Does not address boundary conditions outside the periodic lattice.
- Does not prove existence or uniqueness of discrete solutions.
- Does not take the continuum limit of the lattice operator.