<p>The Koopman–Hill projection method offers an efficient, numerically validated approach for stability analysis of linear time-periodic systems and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, it has previously only been motivated via the Koopman framework, involving the optimistic truncation of an ill-posed bi-infinite initial value problem, therefore lacking any rigorous theoretical guarantees. We close this gap for a class of dynamical systems with exponentially decaying Fourier coefficients by presenting a closed-form bound for the difference between the fundamental solution matrix and its Koopman–Hill approximation. The bound converges to zero as the truncation order goes to infinity. It is derived using constructive series expansions for fixed truncation order, making the results fully independent from the ill-posed bi-infinite Koopman lift. The bound is not sharp, but nevertheless provides a solid theoretical foundation for the Koopman–Hill projection method. In addition, it enables conservative but reliable inference of Floquet multipliers and associated stability properties. The same methodology applied to a subharmonic Koopman–Hill formulation yields a bound with improved convergence rate. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and illuminate how it can be used in practice to assess the accuracy of computed Floquet multipliers quantitatively.</p>