Inverse Problem for Multiple Kinks in the Deformation Method: Impossibility for Two‑Minima Seeds and Complete Rigorous Solution for Multi‑Minima Seeds

The deformation method of Bazeia, Losano and Malbouisson generates new scalar field potentials from a seed potential and a smooth, strictly monotonic deformation function. The inverse problem—reconstructing the deformation function from desired kink properties—was recently solved for a single kink. The present paper closes the remaining open problem of multiple kinks. We prove a strengthened impossibility theorem: for any seed potential with exactly two minima, the requirement that the deformation be a diffeomorphism forces the image of the vacua to be strictly increasing; consequently, at most one BPS kink can exist. This definitively eliminates the prospect of multiple alternating kinks from two‑minimum seeds, even allowing degenerate vacua. For seeds with three or more minima, we show that a strictly monotonic chain of vacua can host arbitrarily many non‑degenerate kinks with independently prescribed masses and leaf lengths. The proof is constructive and relies on an endpoint controllability lemma and a rigorous extension of the single‑leaf existence theorem. The results provide a complete and rigorous solution to the inverse problem for multiple BPS kinks in the deformation method.