No coarse Hilbert embedding for unrestricted persistence diagrams

We prove that the space of all finite persistence diagrams, with no common cardinality or coordinate bound, does not coarsely embed into Hilbert space for any Wasserstein exponent \(1\le p\le2\). The metric uses the usual \(\ell^\infty\) ground distance and matching to the diagonal. For each finite family of rational signed measures on a square grid, a sufficiently large common background lets unit transports be matched through intermediate grid points. After scaling and translation, the resulting diagram distances are exactly the \(1/p\) powers of the signed transportation distances. A finite-subset criterion then transfers a hypothetical coarse embedding to the signed planar grids, contradicting the theorem of Naor and Schechtman.