arXiv:2611.13556v1 Announce Type: new Abstract: Incremental input-to-state stability ($\delta$-ISS) guarantees that the trajectories of a system converge towards one another, up to a gain on the difference of their inputs. Some existing learning-based designs that combine incremental stability with safety achieve only incremental input-to-state practical stability, whose non-vanishing offset can make the guarantee conservative. We consider control-affine systems with a known, full-row-rank input matrix and an unknown drift learned using Gaussian process regression. The prescribed safe set is a superlevel set of a smooth function satisfying a radial condition that ensures a diffeomorphism onto $\mathbb R^n$ and it need not contain any equilibrium of the uncontrolled system. Using this transformation, we design a smooth feedback law and derive sufficient conditions that, under the stated learning assumptions, guarantee with high probability both forward invariance of a compact subset of the safe set and $\delta$-ISS on that subset. The incremental estimate uses the distance induced by the transformation and contains no practical offset. Simulations on a Van der Pol oscillator and a Lotka-Volterra system illustrate the proposed approach.