PopCore is a fabrication technique that laser-cuts 3D models from paper-foam-paper sandwich materials. Its key elements are two laser-cut lever mechanisms that allow users to break off surrounding residue material, thereby “excavating” joints efficiently and with very high precision, which PopCore produces by laser cutting from the top and bottom. This produces flush joints, folded edges that are perfectly straight, and no burn marks—giving models a homogeneous, clean look. This allows applying personal fabrication to new fields, including industrial design, architecture, and packaging design, that require a visual finish beyond what traditional personal fabrication delivers. We present the algorithms and a software tool that generates PopCore automatically. Our user study participants rated PopCore models significantly more visually appealing (7.9/9) than models created using techniques from the related work (4.7/9 and 2.3/9) and suitable for presentation models (11/12 participants), products (10/12 participants) and high-end packaging (10/12 participants).