Inverse Deformation and Reflectionlessness: A Calibration Test Recovering the Rescaled \(φ^4\) Kink

We present a numerical pipeline that couples the Bazeia–Losano–Malbouisson deformation method with the integrability condition of a reflectionless kink stability operator. The pipeline is validated on a benchmark case: the \(φ^4\) seed, a leaf of length \(Δ=1\), and the target kink mass \(M=1/3\). Using a convex combination of a very broad Gaussian and a wide regularised function, the bisection solver finds the mixing coefficient \(α = 2097125/2097152 ≈ 0.999999\) that exactly satisfies the mass constraint. The resulting \(q(φ)\) is essentially constant \(1/2\), leading to the affine deformation \(f(ψ)=2ψ-1\) and the rescaled \(φ^4\) potential \(V(ψ)=2ψ²(1-ψ)²\). The numerical reflection coefficient at several momenta is \(|R| 10⁻³\) (zero within integration accuracy), and the discrete spectrum is \(\{0.0104,\;3.007,\;3.996\}\), matching the Pöschl–Teller values \(\{0,3,4\}\) to within \(0.4%\). This calibration confirms that the pipeline correctly identifies the trivial integrable point and is ready for extensions that search for non‑trivial reflectionless deformations. The accompanying Mathematica notebook pp03.nb implements the full algorithm with built‑in feasibility checks.