Certified Error-Cancellation Scales for Gauss–Legendre Quadrature under Endpoint Double Scaling

Abstract Fixed-integrand quadrature error bounds do not describe what happens when a parameter-dependent endpoint singularity approaches the integration interval as the Gauss–Legendre order increases. We study the ordinary, unmodified rule for nearly singular kernels on [0,2], under the coupled scaling δ ≍ 𝑁−4 , where 𝑁 = n + 1/2. Under explicit joint C 1 regularity and positivity conditions, the signed quadrature error has a locally uniform limit determined by endpoint data and a convergent Bessel-zero series. Analytic tail bounds and outward-rounded interval enclosures establish precisely one simple limiting zero in each of two disjoint parameter intervals. With additional C 1 regularity of the coefficients frozen at the singular parameter, we derive a next-order 𝑁 −2 correction comprising a canonical contribution and two endpoint-slope terms. A lacunary Legendre-polynomial construction shows that joint C 1 regularity alone does not guarantee this correction. The next-order formula determines the displacement of any selected sequence of exact finite-order error-cancellation roots near either limiting zero, without assuming uniqueness at finite quadrature order. For a conservative bipolar electrostatic oscillator, the endpoint-slope terms vanish, leading to an explicit 𝑁 −6 correction to its cancellation amplitude. Independent numerical checks of several kernel families and the oscillator complement the analytic results. The conclusions characterize signed-error transitions and isolated cancellation scales for the classical quadrature rule rather than proposing an alternative integration scheme.