Generic Mean Curvature Flows with Cylindrical Singularities

The paper proves isolatedness, stability, and denseness of nondegenerate cylindrical singularities via a two-scale normal form and a pseudolocality extension, together with a centering map that eliminates positive modes; these methods are coherent and well-documented. By contrast, the model’s solution hinges on selecting the Gaussian center y by zeros of ∇y F_{y,1}(M_t) and then asserts, without justification, that the unique zero y_*(t) equals 0. That step is essential for its isolatedness argument and is not established; it also bypasses the nontrivial centering/orthogonality machinery the paper implements. The model also treats surjectivity onto center modes and the LS-neighborhood persistence at a heuristic level, whereas the paper supplies a precise centering map and cone dynamics. Hence the paper’s argument is sound, but the model’s proof has a critical gap.