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We show that if a subset $K$ in the Heisenberg group (endowed with theCarnot-Carath\'{e}odory metric) is contained in a rectifiable curve, then itsatisfies a modified analogue of Peter Jones's geometric lemma. This is aquantitative version of the statement that a finite length curve has a tangentat almost every point. This condition complements that of \cite{FFP} except apower 2 is changed to a power 4. Two key tools that we use in the proof are ageometric martingale argument like that of \cite{Schul-TSP} as well as a newcurvature inequality in the Heisenberg group.

The traveling salesman problem in the Heisenberg group: upper bounding curvature