  We show that two semi-infinite positive temperature polymers coalesce on the
scale predicted by KPZ (Kardar-Parisi-Zhang) universality. The two polymer
paths have the same asymptotic direction and evolve in the same environment,
independently until coalescence. If they start at distance $k$ apart, their
coalescence occurs on the scale $k^{3/2}$. It follows that the total variation
distance of two semi-infinite polymer measures decays on this same scale. Our
results are upper and lower bounds on probabilities and expectations that
match, up to constant factors and occasional logarithmic corrections. Our
proofs are done in the context of the solvable inverse-gamma polymer model, but
without appeal to integrable probability. With minor modifications, our proofs
give also bounds on transversal fluctuations of the polymer path. Since the
free energy of a directed polymer is a discretization of a stochastically
forced viscous Hamilton-Jacobi equation, our results suggest that the
hyperbolicity phenomenon of such equations obeys the KPZ exponent.
