IndisputableMonolith.NavierStokes.RM2U.EnergyIdentity
The EnergyIdentity module supplies the base energy identity for the RM2U ℓ=2 radial operator on fixed-time slices. It isolates the radial coefficient A(r) for r ≥ 1 without time derivatives or forcing. Navier-Stokes analysts working on RM2U closure for ancient solutions cite this when deriving L2 coercivity bounds. The module organizes supporting definitions and lemmas drawn from Core and SkewHarmGate to feed Bet1 and tail-flux arguments.
claimThe RM2U ℓ=2 radial operator on a time slice is the second-order radial differential operator acting on the scalar coefficient A(r) for r ≥ 1, with the associated energy identity obtained by integration by parts that yields the quadratic form ∫ (|∂_r A|^2 + 6 r^{-2} |A|^2) r^2 dr plus boundary terms at infinity.
background
RM2U.Core specifies the tail/tightness gate as a 1D radial coefficient problem: fix time t and spherical direction b, then work with the scalar radial function A(r) for r ≥ 1. SkewHarmGate supplies the integration-by-parts lemmas that produce skew/self-adjoint cancellations and control the harmonic/affine tail modes whose only obstruction is a boundary term at infinity. The present module packages the energy identity for the ℓ=2 case in this setting, with no time evolution or forcing included.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds BetInstantiation (connecting weighted L2 moments to RM2U closure), NonParasitism (isolating the hard gate), ProjectedPDE (hypothesis interface for the projected evolution), and TailFluxInstantiation (coercive bounds via tail vanishing). It supplies the energy-identity step required by the RM2U pipeline in navier-dec-12-rewrite.tex before the Bet1 integrability route and the single non-parasitism hypothesis can be applied.
scope and limits
- Does not include time derivatives or external forcing.
- Does not address the full 3D Navier-Stokes projection.
- Does not prove the non-parasitism statement.
- Does not instantiate concrete Bet1 or tail-flux vanishing.