We report the beam energy and collision centrality dependence of fifth and sixth order cumulants ($C_5$, $C_6$) and factorial cumulants (${\kappa }_5$, ${\kappa }_6$) of net-proton and proton number distributions, from center-of-mass energy ($\sqrt{s_{NN}}$) 3 GeV to 200 GeV $\mathrm{Au}+\mathrm{Au}$ collisions at RHIC. Cumulant ratios of net-proton (taken as proxy for net-baryon) distributions generally follow the hierarchy expected from QCD thermodynamics, except for the case of collisions at 3 GeV. The measured values of $C_6/C_2$ for 0%–40% centrality collisions show progressively negative trend with decreasing energy, while it is positive for the lowest energy studied. These observed negative signs are consistent with QCD calculations (for baryon chemical potential, ${\mu }_B\le 110\mathrm{MeV}$) which contains the crossover transition range. In addition, for energies above 7.7 GeV, the measured proton ${\kappa }_n$, within uncertainties, does not support the two-component ($\mathrm{Poisson}+\mathrm{binomial}$) shape of proton number distributions that would be expected from a first-order phase transition. Taken in combination, the hyperorder proton number fluctuations suggest that the structure of QCD matter at high baryon density, ${\mu }_B\sim 750\mathrm{MeV}$ at $\sqrt{s_{NN}}=3\mathrm{GeV}$ is starkly different from those at vanishing ${\mu }_B\sim 24\mathrm{MeV}$ at $\sqrt{s_{NN}}=200\mathrm{GeV}$ and higher collision energies.

• Received 22 July 2022
• Revised 21 November 2022
• Accepted 11 January 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.082301