Sharp mixed-potential Fourier pairs and planar universal optimality

We study planar energy comparison and sharp Fourier auxiliaries for completely monotone potentials of squared distance. The attributed OpenAI Math energy comparison and Gaussian construction cover locally finite density-one configurations, including nonperiodic competitors without a separation bound and cases of infinite energy. For positive density and finite triangular reference energy, we construct one real radial Fourier pair satisfying all sixteen auxiliary conclusions simultaneously. A common positive representing measure may have infinite total mass. Finite reference energy also yields physical integrability and the reverse Fourier identity for the same pair. For periodic configurations, an exact excess identity and the selected Gaussian family's first-cell margins bound ordered short pairs, including nonzero periodic self-images. Actual Voronoi facets, deletion, and reinsertion control the fraction of defective closed six-neighbor neighborhoods on the original period cell. An explicit geometric coefficient has an inverse-square tolerance bound. Its Gaussian and nonzero positive-mixture consequences give cube-root excess estimates and separate zero-excess endpoints. The mixture statement requires a finite potential and finite competitor and reference energies, but not finite total representing mass. An explicit ninth-root consequence makes both local tolerance and exceptional fraction tend to zero. Global alignment and equality classification remain separate tasks. Formal proof companion: the Lean software package is identified by reserved DOI 10.5281/zenodo.23826711. It is being prepared for release; complete package qualification and independent replay remain pending. The formal companion does not yet establish an independently replayed verification of the complete manuscript. 2020 Mathematics Subject Classification: Primary 52C15; Secondary 42B10, 52C05. License scope: Barry Z. Cynamon’s original text and figures are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Material adapted from the identified OpenAI Math sources retains its Apache License 2.0 terms and attribution. These licenses apply to their respective material; the CC BY grant does not relicense upstream material. See the manuscript’s attribution and assistance statement and Appendix B.1.