IndisputableMonolith.NavierStokes.VortexStretching
VortexStretching module supplies J-cost bounds for stretching factors of the form 1+eps together with viscous domination and pair-absorption lemmas on the discrete lattice. It is used by the DiscreteMaximumPrinciple to close the self-improving sub-Kolmogorov argument. The proofs are short algebraic comparisons that invoke the Recognition Composition Law balance and the nonnegativity of J.
claim$J(1 + 0) = 0$ and $J(1 + 0) = 0$ for eps >= 0, with the bound $J(1 + eps) <= eps^2 / 2$ together with viscous-rate and pair-budget absorption estimates derived from the CoreNSOperator and RCL pairs.
background
The module sits inside the discrete incompressible Navier-Stokes development on the RS lattice. DiscreteNSOperator supplies the finite three-direction lattice topology, the CoreNSOperator carrying velocity-gradient and Laplacian data, and the derived pair-budget and viscous-absorption fields. StretchingPairs isolates the paired stretching/compression events and the exact RCL identity J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) that governs their cost.
proof idea
The module assembles a chain of algebraic lemmas. Jcost_one_plus_eps is obtained by comparing eps^2 / (2(1+eps)) against eps^2/2 using 1+eps >= 1. viscous_dominates_of_product_bound and subKolmogorov_pair_absorption apply the same comparison to the viscous term and the pair budget. master_absorption and exponential_vorticity_decay combine the local bounds into global decay statements.
why it matters in Recognition Science
The lemmas close the absorption step required by DiscreteMaximumPrinciple, which then shows that the sub-Kolmogorov condition is self-improving under the discrete evolution and therefore need not be imposed externally. This removes the last external hypothesis from the lattice regularity claim in the Recognition Science framework.
scope and limits
- Does not treat the continuous Navier-Stokes equations.
- Does not establish regularity outside the discrete three-direction lattice.
- Does not remove the finite-lattice cutoff.
- Does not compute explicit numerical values for the absorption constants.