action_form_verified
The definition marks the gravitational wave action as verified when expanded around an FRW background for a positive scale factor and a transverse traceless tensor perturbation, with real parameters alpha and lag coupling. Workers deriving RS-consistent wave equations or energy densities would cite this as the base verification step. It proceeds by direct definition to the constant true proposition.
claimLet $a(t)$ be a scale factor satisfying $a(t)>0$ for all $t$, and let $h_{TT}(t)$ be a tensor field obeying the transverse condition (sum over one spatial index vanishes) and traceless condition (sum over diagonal vanishes). For real parameters $α$ and $C_{lag}$, the action form is verified.
background
The ScaleFactor structure supplies a positive function $a:ℝ→ℝ$ drawn from the FRW metric. TensorPerturbation encodes a field $h_{TT}:ℝ→(Fin 3→Fin 3→ℝ)$ together with the transverse sum condition and traceless diagonal sum condition. The lag coupling is fixed by the eight-tick resonance definition as $C_{lag}=φ^{-5}$. The scale function from large-scale structure supplies discrete powers $φ^k$ for mode analysis.
proof idea
The declaration is a definition that directly sets the proposition to True. No lemmas are invoked; it functions as an explicit verification marker for the tensor action form.
why it matters in Recognition Science
This definition anchors the quadratic tensor contribution in the GW action expansion module. It incorporates the lag coupling from eight-tick resonance and the phi-ladder scaling from the forcing chain. No downstream uses are recorded yet, leaving open its insertion into explicit wave propagation or power-spectrum calculations.
scope and limits
- Does not derive the explicit quadratic coefficients from the underlying Lagrangian.
- Does not address scalar or vector perturbation sectors.
- Does not incorporate nonlinear or higher-order interaction terms.
- Does not specify numerical values for alpha or the lag coupling beyond the RS definition.
formal statement (Lean)
32def action_form_verified (scale : ScaleFactor) (h : TensorPerturbation) (α C_lag : ℝ) : Prop :=
proof body
Definition body.
33 True
34
35end GW
36end Relativity
37end IndisputableMonolith