The spin-one Motzkin chain is gapped for any area weight t<1

We consider the spin-one Motzkin chain with area weight $t>0$. We resolve three questions from the literature about this model. We prove (i) existence of a uniform spectral gap for all $t<1$ as conjectured by Zhang--Ahmadein--Klich \cite{zhang2017novel} (ii) an explicit formula for the long-distance limit of the string order parameter, which implies it is non-vanishing at small $t$, confirming a conjecture by Barbiero et al. \cite{barbiero2017haldane}, and (iii) that gaplessness for $t>1$ is robust and extends to hard boundary conditions, answering a question of Zhang--Klich \cite{zhang2017entropy}. These conclusions rest on an effective approximate description of the ground states of finite open Motzkin chains in terms of height-controlled imbalanced Motzkin walks.