Finite-time blowup for a Schrödinger envelope coupled to ionization

We construct a finite-time singularity for a three-dimensional focusing Schrödinger envelope coupled to an ionization density. For one strictly positive ionization coefficient, with cubic absorption and no avalanche term, a nonzero compactly supported smooth optical datum and zero initial density produce a smooth solution whose gradient norm tends to infinity at a finite time. The density remains nonnegative. Thus global smooth propagation for all admissible coefficients and Schwartz data is false. The construction continues a scalar self-similar profile to positive ionization, then solves the optical and density perturbations together. A fixed scalar linear theory and compactly supported finite-dimensional corrections allow the singular solution to start from compact smooth data.