The Resonance Bias Framework: Resonance, Structure, and Arithmetic in Quadrature Error

We study the trapezoidal rule for periodic functions on uniform grids and show that the quadrature error exhibits a rich deterministic structure, beyond traditional asymptotic or statistical interpretations. Focusing on the prototype function f(x) = sin^2(2 pi k x), we derive an analytical expression for the error governed by a resonance function chi_P(y), closely related to the Dirichlet kernel, roots of unity, and discrete Fourier analysis on the group Z/PZ. This function acts as a spectral filter, connecting the integration error to arithmetic properties such as k/P and geometric phase cancellation, visualized as vector averaging on the unit circle. We introduce the Resonance Bias Framework (RBF), a generalization to arbitrary smooth periodic functions, leading to the error representation B_P[f] = sum_{k != 0} c_k chi_P(k/P). Although this is mathematically equivalent to the classical aliasing sum, it reveals a deeper mechanism: the quadrature error arises from structured resonance rather than random aliasing noise. The RBF thus provides an interpretable framework for understanding integration errors at finite resolution, grounded in number theory and geometry.