IndisputableMonolith.NavierStokes.FRCBridge
The FRCBridge module defines finite-volume vorticity profiles on RS lattices, packaging siteCount, omegaMax, omegaRms, and the finiteVolumeControl bound that equates pointwise and RMS norms via sqrt(siteCount). Researchers extending the phi-ladder cutoff to discrete Navier-Stokes would cite these structures to control energy cascades on finite windows. The module is a collection of definitions and elementary lemmas on normalized ratios and J-cost bounds.
claimA finite-volume vorticity profile consists of siteCount : \mathbb{N}, \omega_{\max}, \omega_{\mathrm{rms}} : \mathbb{R}, and finiteVolumeControl witnessing \| \omega \|_{\infty} \le \sqrt{\mathrm{siteCount}} \, \omega_{\mathrm{rms}} on an RS lattice.
background
This module belongs to the Navier-Stokes domain and imports PhiLadderCutoff, whose doc states it formalizes the algebraic and combinatorial core of the argument that the φ-ladder provides an ultraviolet cutoff terminating the Navier-Stokes energy cascade on the RS discrete lattice.
FiniteVolumeProfile is the central object: siteCount counts active lattice sites in the finite window, omegaMax and omegaRms package the relevant scales, and finiteVolumeControl supplies the norm-equivalence input that pointwise amplitude is controlled by sqrt(siteCount) times the RMS scale.
Sibling definitions supply normalizedRatio (with positivity, lower and upper bounds) together with Jcost_le_self_of_one_le and finiteVolume_Jcost_bound that prepare the lattice FRC statements.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies the discrete-lattice bridge required by the phi-ladder cutoff argument for Navier-Stokes regularity. It feeds the main results listed in PhiLadderCutoff on ultraviolet termination of the energy cascade. It touches the T7 eight-tick octave and D=3 spatial dimensions by discretizing the RS lattice while leaving the continuous limit and full time-dependent regularity open.
scope and limits
- Does not prove global regularity of the Navier-Stokes equations.
- Does not treat the continuous-space limit of the lattice.
- Does not include time evolution or forcing terms.
- Does not derive the phi-ladder cutoff itself.
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