Heat Kernel and Closed Geodesic Asymptotics for Nilpotent Coverings

We establish all order long-time asymptotic expansions for heat kernels on nilpotent coverings and for prime closed geodesics in fixed central classes of nilpotent quotients of compact hyperbolic surfaces. The exact lattice-side input is the finite-dimensional rational Floquet-Bloch theory of the companion paper: rational Kirillov restrictions give exact finite-dimensional fibers, and a generalized Pytlik functional gives exact Fourier-inversion and normalized-trace identities. At a rational parameter $p/q$ the decomposition is exact, and the fluctuation of the fiber integrand is controlled only by $q$. Hence the large-denominator comparison with the smooth Kirillov or Schr\"odinger normal form is uniform on the rational support of the Pytlik functional; irrational parameters do not enter the rigorous trace argument. For general nilpotent models, coefficient-weighted spectral sums are justified to every fixed order by positive Rockland estimates, the Plancherel-Mellin formula, and a trace-level order-balance argument. In contrast with approaches which usually give leading terms or integrated Edgeworth-type asymptotics, the method gives genuinely local, pointwise higher-order heat-kernel expansions. The same representation-theoretic quantity governs the leading term in the closed-geodesic asymptotics, producing a nilpotent Chebotarev-type phenomenon. The Heisenberg model is computed to the first correction term, and the Engel model is represented through the resolvent and heat-kernel calculus of the quartic oscillato