Asymptotic Vanishing of the Renormalised Curvature Invariant in Coleman–De Luccia Geometries

We prove that the renormalised Kretschmann curvature invariant vanishes asymptotically at the true-vacuum pole of Coleman–De Luccia (CDL) bounce geometries with a nondegenerate quadratic true-vacuum minimum and positive vacuum energy. For the explicit quartic double-well potential V (ϕ) =14 (ϕ2−1) 2+δϕV () =14 (²-1) ²+ (ϕ) =41 (ϕ2−1) 2+δϕ with parameters δ=κ=0. 01==0. 01δ=κ=0. 01, whose CDL bounce on the compact O (4) O (4) O (4) -symmetric geometry ρ∈0, L0, Lρ∈0, L (L≈125. 87L125. 87L≈125. 87) is rigorously certified by a Krawczyk interval-arithmetic existence proof, we show that the renormalised invariant Kren=Kphys−24Ht4Kₑ₄₍=K₏₇ₘₒ-24Hₜ⁴Kren=Kphys−24Ht4 tends to zero as the field approaches the true vacuum. AsymptoticVanishing The proof has three logically distinct components. First, the Weyl tensor vanishes identically for any smooth O (4) O (4) O (4) -symmetric metric, implying that the Kretschmann scalar reduces entirely to the Ricci sector. Second, inserting the Einstein equation and the Hamiltonian constraint yields an on-shell identity expressing the Ricci invariant purely in terms of the scalar kinetic energy and the potential. Third, regularity at the south pole forces the scalar derivative to vanish and the field to approach the true-vacuum value, which drives both the kinetic term and the potential excess to zero. As a result, the renormalised curvature invariant decays to zero in the true-vacuum limit. The argument is analytic once the existence of the bounce solution is established; numerical calculations serve only to certify the existence of the solution and to illustrate the quadratic plateau regime near the true vacuum. The proof requires neither conformal rescalings nor coordinate transformations of the tail region. The structure extends to any potential with a nondegenerate quadratic true-vacuum minimum and positive vacuum energy. Degenerate vacua with vanishing second derivative lead instead to a qualitatively different power-law tail regime and are treated in a companion work.