Infinite Boundary-Jet Ambiguity for Proper Embeddings into a Smooth Strongly Convex Domain

For every n >= 2, we construct one bounded domain D in C^(n+1) with smooth strongly convex boundary and a circle of distinct proper holomorphic embeddings of B^n into D. All embeddings extend smoothly to the closed ball and have the same infinite jet at one fixed boundary point. The domain can be chosen arbitrarily close to the unit ball in any prescribed finite smoothness norm on its defining function. A holomorphic function flat at the boundary point supplies the graphs, and a compactly supported perturbation of the quadratic defining function places all graphs in the same target. For comparison, an elementary orthogonality argument shows that a proper rational ball map of degree at most d is determined by its d-jet at the interior origin, even among all holomorphic maps to the target ball. The boundary example concerns a prescribed boundary point; it does not resolve the general optimal dimension-dependent interior jet order or close the full two-part AIM question. This is an unrefereed preprint; no absolute priority claim is made.