Exact Configurations and Collision Scales for Exponential Energy
Overview
Points on an interval minimize the sum of exp(-d) over pairwise distances d, with repetitions allowed. The paper determines every minimizing position and the exact number of repeated points at each endpoint. On the unit interval this proves Emmerich's empirical one-third law in Section 4 of arXiv 2603.28179v2. It also answers the nearby-q question at the end of Appendix A.3 for fixed point count and interval length. For exp(-d^q) below q=1, moving cluster points approach the endpoints on different exponential scales; at special lengths the outermost scale instead has a logarithmic correction. Above q=1 the endpoint multiplicities persist, including at these lengths. Finite-size energy corrections and a common leading factor for the exponential scales are proved in the same article. The global transitions in Conjectures A.1 and A.2 remain outside this local result.
Original abstract (English)
We determine the minimum of the exponential pair energy among all ordered n-point configurations on a compact interval, allowing repeated points. The unique minimizer consists of equal endpoint clusters and an equally spaced interior block. A scalar equation specifies every position, and explicit interval-length thresholds determine the cluster multiplicity. On the unit interval this proves the empirical one-third law reported by Emmerich. We also determine the first finite-size corrections to the minimum energy. For the kernels exp(-d^q) as q increases to one, each moving point in an endpoint cluster approaches the boundary on a distinct exponential scale. At a length threshold the outermost point instead has an explicit (1-q) log(1/(1-q)) scale. From above, the endpoint multiplicities persist for every fixed interval length, including the thresholds. These results describe both the exact critical configuration and its two different local perturbation regimes.
Original public questions
Michael T. M. Emmerich: Critical phase transitions in minimum-energy configurations for the exponential kernel family e^{-|x-y|^q} on the unit interval — Section 4 empirical one-third law and the final nearby-q perturbative question in Appendix A.3, arXiv v2. This is not a solution of Conjectures A.1/A.2 or the full large-n phase diagram.
Mathematical review
The manuscript has completed internal review. Complete formalization is not yet established for this public version.
Review standard