In an influential paper titled "The Benefits of Relaxing Punctuality", Alur, Feder, and~Henzinger introduced Metric Interval Temporal Logic~(MITL) as a fragment of the real-time logic Metric Temporal Logic~(MTL) in which exact or punctual timing constraints are banned. Their main result showed that model checking and satisfiability for~MITL are both EXPSPACE-Complete.\par Until recently, it was widely believed that admitting even the simplest punctual specifications in any linear-time temporal logic would automatically lead to undecidability. Although this was recently disproved, until now no punctual fragment of~MTL was known to have even primitive recursive complexity (with certain decidable fragments having provably non-primitive recursive complexity).\par In this paper we identify a `co-flat' subset of~MTL that is capable of expressing a large class of punctual specifications and for which model checking (although not satisfiability) has no complexity cost over~MITL. Our logic is moreover qualitatively different from~MITL in that it can express properties that are not timed-regular. Correspondingly, our decision procedures do not involve translating formulas into finite-state automata, but rather into certain kinds of reversal-bounded Turing machines. Using this translation we show that the model checking problem for our logic is EXPSPACE-Complete, and is even PSPACE-Complete if timing constraints are encoded in unary.