NoPersistentNullAlignmentHypothesis
The structure encodes an abstract implication: existence of c > 0 such that every large tail strain event at positive scales epsilon and r forces a hit event. Researchers proving Navier-Stokes regularity under the RM2U closure would cite it to enforce the spec-layer non-parasitism gate. It is introduced directly as a structure definition that packages the existential quantifier and universal implication without further reduction.
claimThere exists a constant $c > 0$ such that for all $ε > 0$ and $r > 0$, if a large tail strain event occurs at scales $ε, r$ then a hit event occurs at those scales.
background
The RM2U.NonParasitism module isolates the single hard non-parasitism statement as a hypothesis so the remainder of the RM2U to RM2 pipeline stays clean. Non-parasitism is the condition that the tail-flux or boundary term at infinity vanishes for the relevant ℓ=2 coefficient. The structure supplies the spec-layer version of no persistent null alignment: a large tail strain event forces nontrivial interaction with the vorticity direction and thereby avoids the null-eigenvector cone on a time-thick top band set.
proof idea
This is a structure definition whose sole field directly states the existential quantifier over c together with the universal quantification and implication between the two event predicates. No lemmas are invoked and no tactics are used; the declaration is the primitive packaging of the hypothesis.
why it matters in Recognition Science
It supplies the single hard gate required by the RM2U.NonParasitism module, allowing the rest of the pipeline to proceed without circularity. The declaration fills the spec-layer requirement that non-parasitism be expressed as an abstract implication between tail events and hit events. In the Recognition Science framework it supports the forcing chain (T0-T8) and the derivation of Navier-Stokes regularity by excluding persistent null alignments that would otherwise permit parasitic solutions.
scope and limits
- Does not define the concrete predicates for TailLargeEvent or HitEvent.
- Does not derive the existence of c from more primitive PDE or recognition assumptions.
- Does not address the explicit realization of the events in the Navier-Stokes vorticity equation.
- Does not connect the hypothesis to specific constants such as phi or the alpha band.
formal statement (Lean)
747structure NoPersistentNullAlignmentHypothesis
748 (TailLargeEvent HitEvent : ℝ → ℝ → Prop) : Prop where
749 exists_c :
750 ∃ c : ℝ,
751 0 < c ∧
752 ∀ (ε r : ℝ), 0 < ε → 0 < r →
753 TailLargeEvent ε r →
754 HitEvent ε r
755
756/-- **S3-P bookkeeping (spec layer):** export persistence splits into one of the three “cost channels”.
757
758This is intentionally abstract: it packages the output of the selection/drift-control machinery together with
759the algebraic split (`σ` vs perpendicular forcing) into a single logical splitter. -/