Certified spectral approximation of transfer operators and the Gauss map

We prove that the full spectral picture of a transfer operator (every isolated eigenvalue, eigenvector, and Riesz projector outside the essential spectral radius) can be approximated to arbitrary precision by finite-rank discretizations, with no spectral pollution. The method is a~posteriori: once a computable approximation bound is available (from compactness or a Doeblin--Fortet--Lasota--Yorke inequality, which may require hyperbolicity constants and adapted Banach spaces), the spectral gap and multiplicity are certified from computed data via a resolvent perturbation bound, with no further dynamical input. This applies both to compact operators on a single Banach space and to quasi-compact operators satisfying a Doeblin--Fortet--Lasota--Yorke inequality, extending Li's resolution of the Ulam conjecture from the invariant density to the entire discrete spectrum with certified error bounds at every finite truncation. As a benchmark, we certify the first $50$ nonzero eigenvalues of the Gauss--Kuzmin--Wirsing operator to at least $90$ rigorous decimal digits, together with their eigenvectors, Riesz projectors, and spectral gap, yielding a certified spectral expansion for Gauss--Kuzmin distributions with explicit error bounds and providing a rigorous answer to the Gauss--Babenko--Knuth problem on the spectral data of the Gauss map.