IndisputableMonolith.NavierStokes.RM2U.ProjectedPDE
The module states an abstract hypothesis that the RM2U operator admits a splitting rm2uOp = F - A_t allowing its pairing against A r^2 to be bounded on finite radial intervals. Researchers deriving energy identities for projected Navier-Stokes models cite it as a time-slice scaffolding step. The module contains no proofs and imports only the EnergyIdentity algebraic spine.
claimHypothesis: there exist operators $F$ and $A_t$ such that rm2uOp $= F - A_t$, so that the pairing of rm2uOp against $A r^2$ is bounded on each interval $[a,R]$.
background
The module belongs to the RM2U branch of the Navier-Stokes formalization and imports the EnergyIdentity file. That upstream module supplies the algebraic spine: it defines the tail-flux boundary term and invokes the SkewHarmGate lemma to convert the statement that the tail flux vanishes at infinity into two concrete integrability obligations on the boundary term and its derivative. The present module adds only the abstract splitting hypothesis needed to bound the projected operator pairing on time slices.
proof idea
This is a hypothesis module with no proofs. Its structure consists of a single abstract declaration that the operator admits the decomposition rm2uOp = F - A_t, together with the statement that the resulting pairing against A r^2 remains controllable on each compact radial interval.
why it matters in Recognition Science
The module supplies the hypothesis interface required by the Option A energy-identity chain in the RM2U development. It feeds the projected-PDE analysis that ultimately aims at the full Navier-Stokes regularity question by isolating the time-slice pairing bound before integrability conditions are imposed.
scope and limits
- Does not define explicit expressions for the operators F or A_t.
- Does not prove existence of the splitting or the resulting bound.
- Does not treat the full time-dependent evolution, only fixed-time slices.
- Does not connect the bound to the three-dimensional Navier-Stokes equations.