Experimental Survey of Discrete Minimizers of the $p-\text{frame}$ Energy

We provide a detailed analysis of results from a large-scale computational exploration of real and complex (weighted) point configurations that minimize p-frame energies, uncovering phase transition behavior exhibited by the minimizers. We utilize numerical linear programming methodologies to offer complementary lower bounds that support our experimentally obtained upper bounds on minimal energy values. Furthermore, we present the development of an exceptionally symmetric weighted design consisting of 85 points, which outperforms the current best known lower bounds for a minimal-sized weighted design in the realm of five-dimensional complex projective space. In conclusion, based on our thorough observations and in-depth analysis, we conjecture that the support of this novel weighted design is universally optimal.