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 distributions, from $\sqrt{s_{NN}} = 3 - 200$ GeV Au+Au collisions at RHIC. The net-proton cumulant ratios generally follow the hierarchy expected from QCD thermodynamics, except for the case of collisions at $\sqrt{s_{NN}}$ = 3 GeV. $C_{6}/C_{2}$ for 0-40% centrality collisions is increasingly negative with decreasing $\sqrt{s_{NN}}$, while it is positive for the lowest $\sqrt{s_{NN}}$ studied. These observed negative signs are consistent with QCD calculations (at baryon chemical potential, $\mu_{B} \leq$ 110 MeV) that include a crossover quark-hadron transition. In addition, for $\sqrt{s_{NN}} \geq$ 11.5 GeV, the measured proton $\kappa_{n}$, within uncertainties, does not support the two-component shape of proton distributions that would be expected from a first-order phase transition. Taken in combination, the hyper-order proton number fluctuations suggest that the structure of QCD matter at high baryon density, $\mu_{B}\sim 750$ MeV ($\sqrt{s_{NN}}$ = 3 GeV) is starkly different from those at vanishing $\mu_{B}\sim 20$MeV ($\sqrt{s_{NN}}$ = 200 GeV and higher).
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