A search is reported for near-threshold structures in the J/ψJ/ψ invariant mass spectrum produced in proton-proton collisions at sqrt[s]=13 TeV from data collected by the CMS experiment, corresponding to an integrated luminosity of 135 fb^{-1}. Three structures are found, and a model with quantum interference among these structures provides a good description of the data. A new structure is observed with a local significance above 5 standard deviations at a mass of 6638_{-38}^{+43}(stat)_{-31}^{+16}(syst) MeV. Another structure with even higher significance is found at a mass of 6847_{-28}^{+44}(stat)_{-20}^{+48}(syst) MeV, which is consistent with the X(6900) resonance reported by the LHCb experiment and confirmed by the ATLAS experiment. Evidence for another new structure, with a local significance of 4.7 standard deviations, is found at a mass of 7134_{-25}^{+48}(stat)_{-15}^{+41}(syst) MeV. Results are also reported for a model without interference, which does not fit the data as well and shows mass shifts up to 150 MeV relative to the model with interference.
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