Maximal Gaps between PL and Linear Embedding Dimensions

For a finite simplicial complex K, let e_PL(K) and e_L(K) denote its minimum piecewise-linear and simplex-wise linear embedding dimensions. We prove that, for every d>=3, a finite d-complex satisfies e_PL(K)=d and e_L(K)=2d+1; this is the largest possible gap in that dimension. In dimension two we obtain the pair (3,5). These examples answer the minimum-dimension question recorded by Brehm and Sarkaria in the 2006 Oberwolfach report. The proof adapts Newman's Radon-pair, order-type and Janson method to make a random nonembedding obstruction survive every small set of facet deletions. Removing overlaps then produces a linear hypergraph, to which the PL realization lemma of Lee and Nevo applies. The construction is probabilistic and existential; no manifold hypothesis or explicit small facet list is asserted. This is an unrefereed, AI-assisted preprint.