Invertibility of the linearized matrix S without a small-temperature assumption

Prove quantitative invertibility (e.g., strict diagonal dominance or a uniform lower bound on the smallest singular value) of the matrix S defined in equation (6.6), for some admissible cutoff function χ satisfying Assumption 6.1, without requiring the thermal parameter θ to be sufficiently small; doing so would remove the θ ≤ θ0 restriction from the main theorem on effective soliton velocities.

Background

The core analysis introduces a proxy linear system with coefficient matrix S that arises from (a regularized, formally differentiated version of) the asymptotic scattering relation. The main theorem currently assumes θ ≤ θ0 to ensure S is strictly diagonally dominant and thus invertible.

The authors believe S should be quantitatively invertible for an appropriate choice of the cutoff χ, but they lack a general proof; establishing this would extend their result to all θ>0 (see also Remark 6.7).