This paper revisits a class of Euclidean solutions in Type IIB string theory that involve an axion and a dilaton field. In a fixed axion-charge sector the author separates two kinds of solutions. The special, extremal case with E = 0 is the BPS (Bogomol’nyi–Prasad–Sommerfield) instanton. Turning on E > 0 deforms that endpoint into non‑BPS wormholes with a smooth throat. The main technical result is that the quadratic fluctuation operator around the E = 0 instanton, after the right constraints are imposed, factorizes as H = Q†Q and therefore has a positive‑semidefinite spectrum on the reduced physical domain.

To get this result the paper sets up the charge‑sector radial equations in two commonly studied reductions: four‑dimensional flat space and an AdS5 (anti‑de Sitter in five dimensions) truncation. The axion equation gives a conserved charge and the geometry obeys a Hamiltonian constraint. Cutting a complete two‑ended solution at the symmetric minimal sphere (the “neck”) changes the variational problem and requires care. By imposing the Hamiltonian constraint, fixing gauge freedom, enforcing the charge‑sector boundary condition, and removing collective zero modes, the author reduces the quadratic action to a physical Hessian. On that reduced domain the Hessian factorizes as Q†Q, which implies its singular‑value spectrum is non‑negative.

Why this matters: factorization into Q†Q is a strong structural statement. It means that, for the E = 0 BPS instanton and on the properly reduced space of fluctuations, there are no negative directions in the quadratic action. This places the spectrum of Type IIB wormhole‑type saddles near that endpoint on firmer mathematical ground. In plain terms, the instanton endpoint behaves like a stable base point for small deformations when the correct physical constraints are applied.

Important caveats follow from the analysis itself. The positive‑semidefinite result is an endpoint theorem for the E = 0 BPS instanton. It is not a direct proof of full stability for the E > 0 non‑BPS wormhole: the same radial equations govern both cases, but the fluctuation problems and boundary domains differ. The AdS5 truncation follows the same endpoint logic but brings a different boundary‑value problem and requires a dedicated holographic boundary analysis for statements about non‑BPS wormholes there. The paper emphasizes that stability statements made in one variational ensemble (for example fixing a neck value versus fixing a canonical momentum at the neck) do not transfer automatically to the other.