arXiv:2610.25921v1 Announce Type: new Abstract: The polyatomic Boltzmann equation extends the classical kinetic description to gases whose molecules possess internal energy in addition to their translational motion. The distribution function then has one more variable than in the monatomic case, and the collision operator becomes an eight-fold integral whose evaluation dominates the cost of solving the equation. In this work, we propose a fast Laguerre--Fourier spectral method for the polyatomic Boltzmann equation with continuous internal energy, which evaluates the collision operator in $O(K^2MN^4\log N + K^3MN^4)$ online operations after a solution-independent precomputation, where $N$ is the number of Fourier modes in each velocity direction, $K$ is the number of Laguerre modes and $M$ is the number of quadrature points on the sphere. Specifically, the distribution function is expanded in generalized Laguerre functions in the internal energy and in Fourier modes in velocity. A sequence of changes of variables then separates the internal-energy integrals from the velocity convolutions, so that the former no longer involve the distribution function and are computed once and stored. Numerical experiments are presented for both the spatially homogeneous and the spatially inhomogeneous Boltzmann equation. They confirm the spectral accuracy of the proposed method and its agreement with the direct simulation Monte Carlo method.