arXiv:2611.14349v1 Announce Type: new Abstract: For nonlocal interface evolution problems such as the Mullins-Sekerka flow, certain parametric schemes discretize only the interface and approximate the Dirichlet-to-Neumann map by boundary elements. Their error analysis requires bounds on how this discrete operator changes as the polygon moves. In that analysis, the polygons are only Lipschitz and close to a smooth curve only in a Lipschitz norm, the perturbations are controlled only in the energy norm, and the kernels are singular on the diagonal. We study the symmetric Galerkin boundary element matrix on the polygons in a Lipschitz neighborhood of a smooth reference curve. Its first variation, tested against discrete Neumann solutions, is bounded by the energy norm of the perturbation alone. The second-order remainder of the associated mass form is bounded by the product of the energy norm and the Lipschitz norm of the perturbation, so that only one of the two factors needs a uniform bound. On the interpolating polygon, the matrix is consistent with the continuous operator without the logarithmic loss that a pointwise comparison of the kernels would suggest, and its shape derivative is consistent with that of the continuous operator. All bounds are uniform in the mesh size and depend only on the constants of the reference curve.