The heat trace for domains with curved corners

The heat trace of a planar polygon contains corner terms depending only on the opening angles, while the heat trace of a smooth planar domain contains curvature terms along the boundary. We show that, for curvilinear polygons, these two phenomena first interact at order $t^{1/2}$. We compute this first corner-curvature heat invariant and prove a sharp sign law for its Dirichlet angular factor: its sign is determined solely by whether the corner is convex or reflex. More precisely, we derive the local heat trace expansion through order $t^{1/2}$, for both Dirichlet and Neumann boundary conditions. The new coefficient decomposes into the usual smooth-boundary contribution and a sum of local curved-corner terms, each depending only on the interior angle $\alpha$ and the one-sided limiting curvatures $\kappa_{\pm}$ of the adjacent arcs. In the Dirichlet case, the curved-corner contribution has the form $C_{1/2}(\alpha,\kappa_+,\kappa_-) = c_{1/2}(\alpha)\frac{\kappa_+ + \kappa_-}{4\sin(\alpha/2)}$, with $c_{1/2}(\alpha)$ given by an explicit sector heat kernel integral. We determine its sign for every $0<\alpha<2\pi$. The sign law has a spectral consequence: it gives a new obstruction to a curvilinear polygon being Dirichlet-isospectral to a straight-sided polygon. In particular, every convex curvilinear polygon Dirichlet-isospectral to a straight-sided polygon must itself be straight-sided, removing the assumption of straight corners from the theorem of Enciso and G\'omez-Serrano.