IndisputableMonolith.NavierStokes.RM2U.BetInstantiation
The BetInstantiation module packages data for an ancient Navier-Stokes Galerkin element at truncation level N, from which the RM2U profile is extracted via radial coefficient A(r). Researchers applying the Recognition Science RM2U gate to Navier-Stokes would cite it when bridging Galerkin approximations to tail and tightness conditions. The module achieves this by importing the core radial spec, energy identities, non-parasitism hypothesis, RM2 closure steps, and tail flux instantiation to assemble the required Bet1 and Bet2 integrability.
claimAn ancient NS Galerkin element at truncation level $N$ is a data package containing the radial coefficient $A(r)$ for $rgeq 1$ that satisfies the Bet1 and Bet2 integrability conditions, enabling extraction of the RM2U profile from the tail/tightness gate.
background
The module sits inside the RM2U spec layer for Navier-Stokes, which fixes time $t$ and spherical direction parameter $b$ to reduce the problem to a scalar radial coefficient $A(r)$ for $rgeq 1$. The energy identity supplies the algebraic reduction of the tail-flux boundary term to two integrability obligations. Non-parasitism isolates the single hard hypothesis, while RM2 closure converts coercive $ell=2$ control into finiteness of the log-critical shell moment and boundedness of the affine obstruction.
proof idea
This is a definition and instantiation module with no internal proofs. It assembles the argument by importing the five upstream modules (Core, EnergyIdentity, NonParasitism, RM2Closure, TailFluxInstantiation) to connect Galerkin extraction to the Bet1 integrability conditions and complete the RM2U pipeline for ancient elements.
why it matters in Recognition Science
The module supplies the concrete Galerkin-level data that feeds rm2u_pipeline_complete and the overall RM2U closure in the Recognition Science treatment of Navier-Stokes. It fills the step in navier-dec-12-rewrite.tex where truncation data verifies tail flux vanishing and non-parasitism, linking directly to the coercive $ell=2$ to RM2 transition.
scope and limits
- Does not prove the non-parasitism hypothesis itself.
- Does not establish existence of ancient solutions beyond Galerkin truncation.
- Does not address the full three-dimensional Navier-Stokes regularity question.
- Limits analysis to the radial coefficient extracted at fixed truncation level N.
depends on (5)
declarations in this module (10)
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structure
GalerkinAncientElement -
theorem
bet1_for_galerkin -
theorem
bet1_original_for_galerkin -
theorem
nonParasitism_for_galerkin -
theorem
tailFlux_vanishes_for_galerkin -
theorem
rm2_closed_for_galerkin -
theorem
bet2_for_galerkin -
theorem
rm2u_pipeline_complete -
structure
BetInstantiationCert -
theorem
betInstantiationCert