Events where the two leading jets are separated by a pseudorapidity interval devoid of particle activity, known as jet-gap-jet events, are studied in proton-proton collisions at $\sqrt{s} =$ 13 TeV. The signature is expected from hard color-singlet exchange. Each of the highest transverse momentum ($p_\mathrm{T}$) jets must have $p_\mathrm{T}^\text{jet}$ $\gt$ 40 GeV and pseudorapidity 1.4 $\lt$ $|\eta^\text{jet}|$ $\lt$ 4.7, with $\eta^\text{jet1} \eta^\text{jet2}$ $\lt$ 0, where $\text{jet1}$ and $\text{jet2}$ are the leading and subleading jets in $p_\mathrm{T}$, respectively. The analysis is based on data collected by the CMS and TOTEM experiments during a low luminosity, high-$\beta^*$ run at the CERN LHC in 2015, with an integrated luminosity of 0.66 pb$^{-1}$. Events with a low number of charged particles with $p_\mathrm{T}$ $\gt$ 0.2 GeV in the interval $|\eta|$ $\lt$ 1 between the jets are observed in excess of calculations that assume only color-exchange. The fraction of events produced via color-singlet exchange, $f_\text{CSE}$, is measured as a function of $p_\mathrm{T}^\text{jet2}$, the pseudorapidity difference between the two leading jets, and the azimuthal angular separation between the two leading jets. The fraction $f_\text{CSE}$ has values of 0.4-1.0%. The results are compared with previous measurements and with predictions from perturbative quantum chromodynamics. In addition, the first study of jet-gap-jet events detected in association with an intact proton using a subsample of events with an integrated luminosity of 0.40 pb$^{-1}$ is presented. The intact protons are detected with the Roman pot detectors of the TOTEM experiment. The $f_\text{CSE}$ in this sample is 2.91 $\pm$ 0.70 (stat) $^{+1.08}_{-1.01}$ (syst) times larger than that for inclusive dijet production in dijets with similar kinematics.
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