Reflection parity of every zero-history profile, derived from the literal terminal wave and the prescribed coefficient symmetries.

theorem EulerPacketTerminalDatum.forwardInitializedProfiles_parity (M : EulerMeanPacketProvider.Data) {U : Type u_1} [NormedAddCommGroup U] [InnerProductSpace ℝ U] [CompleteSpace U] (D : EulerTransversePacketProvider.Data U) (hTime : M.T = D.T) (δ : ℝ) (hδ : 0 < δ) (ξ : U) (hs : tsupport EulerSpatialCutoffs.innerCutoff ⊆ D.support) (α : ℝ) (eM : EulerMeanPacketProvider.EvenData M) (hSym : ∀ (x : EulerSmoothLimit.Space), -x ∈ D.support ↔ x ∈ D.support) (hF : ∀ (t : ↑(Set.Icc 0 D.T)) (x : EulerSmoothLimit.Space), (D.F.field t) (-x) = (D.F.field t) x) (hDM : ∀ (t : ↑(Set.Icc 0 D.T)) (x : EulerSmoothLimit.Space), (D.M.field t) (-x) = (D.M.field t) x) (p : ℕ) : EulerPacketCylinderField.ProfileParity M.T (forwardInitializedProfiles M D δ hδ ξ hs α p)