Search for pair-produced vector-like top quarks decaying to Lorentz-boosted top quarks and scalars in proton-proton collisions at $\sqrt{s}=$ 13 TeV 2026 The first search for pair-produced vector-like top quarks, $\mathrm{T}$, decaying to a new scalar boson $\phi$ and a standard model (SM) top quark is presented. The search targets events containing a photon pair originating from the decay of one of the scalar bosons, accompanied by all-hadronic decays of the SM top quark pair. The analysis is optimized for boosted top quark signatures in which the decay products of the top quarks are reconstructed as large-radius jets. The search is based on proton-proton collision data at a center-of-mass energy of $\sqrt{s}=13~\mathrm{TeV}$, collected by the CMS experiment at the CERN LHC during 2016, 2017, and 2018, corresponding to an integrated luminosity of $138~\mathrm{fb}^{-1}$. The search is performed in the two-dimensional $(m_{\mathrm{T}}, m_{\phi})$ plane. No significant deviation from the standard model prediction is observed. Assuming a branching fraction of 100$\%$ for $\phi \to \gamma\gamma$, upper limits at 95$\%$ confidence level are set on the $\mathrm{T}\bar{\mathrm{T}}$ production cross section. Vector-like top quark masses below $1.39~\mathrm{TeV}$ are excluded. NOTE