A sharp Weyl remainder for constant-index transmission eigenvalues

We prove a Weyl formula with remainder $O(r^{d-1})$ for the interior transmission problem with a constant refractive index $n>1$ in an arbitrary bounded smooth domain in $\mathbb R^d$, $d\ge2$. The count includes every nonzero complex energy of modulus at most $r^2$, with its algebraic multiplicity for the original maximal-domain transmission operator. The leading coefficient is $\omega_d(2\pi)^{-d}|\Omega|(1+n^{d/2})$. The argument combines the known Dirichlet-to-Neumann approximation with a sector estimate for a normalized boundary family. At inverse frequency scale $h$, its trace-class part has norm $O(h^{1-d})$ at scaled energy height $h$, and the sector estimate controls an actual logarithm of its determinant. Pole cancellation in a strip then gives the same order for the contour argument, including the endpoint contribution at every counting radius. No convexity assumption is used.