Fourier-Constrained Quadratic Energy: Global Optimizers, Free Boundaries, and Certified Windows

This versioned working preprint develops a Fourier-constrained quadratic energy minimization problem over even, nonnegative mass-one L2 densities on [-1/2,1/2]. An explicit cosine divided-difference family gives a global three-regime optimizer structure for 0<c<1, uniqueness of a symmetric free support interval, two regularity transitions, and a small-support asymptotic. The manuscript also records rigorous internal rational-interval certificates for Fourier/energy test-function windows motivated by consecutive moderate gaps between zeros of the Riemann zeta function. The number-theoretic interpretation beyond the width range explicitly stated by Angon--Ross--Tang--Zhao is distinguished as a further source-verification task. This is an evolving non-refereed research preprint, not a final closure or proof of the Riemann hypothesis. Further versions may extend or correct the theory.