IndisputableMonolith.NavierStokes.FRCBridge
The FRCBridge module defines finite-volume vorticity profiles on RS lattices, packaging siteCount, omegaMax, omegaRms, and finiteVolumeControl to enforce norm equivalence via sqrt(siteCount) scaling of RMS values. Researchers studying discrete approximations to Navier-Stokes regularity would cite these structures to control pointwise amplitudes on finite windows. The module imports the phi-ladder cutoff from PhiLadderCutoff and assembles supporting lemmas on normalized ratios and J-cost bounds to bridge lattice data to the energy cascade.
claimA finite-volume vorticity profile on an RS lattice consists of a natural number $N$ (siteCount), scales $omega_{max}$ and $omega_{rms}$, and a control datum ensuring that the pointwise supremum of the vorticity satisfies $||omega||_infty leq sqrt(N) cdot omega_{rms}$.
background
This module sits inside the Navier-Stokes regularity program that uses the phi-ladder as an ultraviolet cutoff on the RS discrete lattice. The central object is the finite-volume vorticity profile, whose fields record the number of active sites in a finite window together with the maximum and root-mean-square vorticity scales. The finiteVolumeControl field supplies the norm-equivalence relation that replaces the usual Sobolev embedding on the continuum: pointwise amplitude is controlled by sqrt(siteCount) times the RMS scale. The imported PhiLadderCutoff module supplies the algebraic core that terminates the energy cascade once these lattice quantities are bounded.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
FRCBridge supplies the lattice-specific vorticity profile that feeds the main results on Navier-Stokes regularity from the phi-ladder cutoff. Its definitions enable the subsequent lemmas (normalizedRatio_pos, finiteVolume_Jcost_bound, frc_holds_on_RS_lattice) that close the discrete energy estimates. The structure directly supports the combinatorial argument that the phi-ladder terminates the cascade on the RS lattice, as outlined in the parent PhiLadderCutoff module.
scope and limits
- Does not establish existence of global smooth solutions to the continuous Navier-Stokes equations.
- Does not derive explicit numerical values for the vorticity scales omegaMax or omegaRms.
- Does not address boundary conditions or forcing terms outside the finite lattice window.
- Does not prove convergence of the lattice model to the continuum limit.
depends on (1)
declarations in this module (17)
-
structure
FiniteVolumeProfile -
def
normalizedRatio -
theorem
normalizedRatio_pos -
theorem
normalizedRatio_ge_one -
theorem
normalizedRatio_le_sqrt_siteCount -
theorem
Jcost_le_self_of_one_le -
theorem
finiteVolume_Jcost_bound -
def
RSLatticeFRC -
def
LatticeFRC -
theorem
frc_holds_on_RS_lattice -
theorem
frc_holds_on_RS_lattice_exists -
structure
ConditionalCompletionRoute -
theorem
close_conditional_loop -
structure
LatticeRegularityCertificate -
theorem
lattice_regular_via_direction_constancy -
structure
FRCBridgeCert -
def
frcBridgeCert