  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).
