Uniqueness of c1 = c2 in the interface effective action

Prove or disprove that setting c1 = c2 in the quadratic extrinsic-curvature sector of the effective interface action that matches the Lorentzian Schwarzschild exterior (region I) to the pure quadratic-gravity powerball interior (region II) across the constant-radius hypersurface r = r_* is the only viable choice that yields a controlled, finite limit as δ → 0. Concretely, determine whether any choice with c1 ≠ c2 in the action S_interface^(I–II) = (G/8π) ∫ d^3y sqrt(-h) [ζ R^(3) + c1 K^2 − c2 K_ab^2 + O(ℓ_P^{-1})] can avoid divergences and remain consistent with the generalized junction conditions in the δ → 0 limit.

Background

To connect the Schwarzschild exterior (described by general relativity) to the complex powerball interior (described by pure quadratic gravity), the authors introduce an effective interface action localized on the constant-radius hypersurface r = r_*. This interface includes intrinsic curvature and quadratic extrinsic-curvature terms with couplings ζ, c1, and c2. Near the would-be horizon, K grows large as δ → 0, so generic K^2 and K_ab^2 terms can introduce divergences unless their coefficients are tuned.

Using the scalar Gauss relation on the hypersurface, choosing c1 = c2 removes the dangerous combination and effectively renormalizes ζ. The authors adopt c1 = c2 = 0 thereafter but explicitly note that they cannot prove c1 = c2 is the only viable option. Establishing uniqueness (or constructing counterexamples) would clarify the space of consistent interface theories and the robustness of the matching procedure.