We use a fourth-order fluid expansion to make general predictions for asymptotically nonlinear material functions in oscillatory shear, a characterization protocol sometimes known as medium-amplitude oscillatory shear. The calculation applies to any viscoelastic fluid in the terminal regime defined by the limit of Deborah number much less than one. Two viscous nonlinearities appear at third order, with shear stress scaling as omega(3), and are interrelated by a constant multiplicative factor. Two elastic nonlinearities appear at fourth order, with shear stress scaling as omega(4), and are also interrelated by a constant multiplicative factor. These nonlinear measures are decoupled from the linear material functions G' and G '' because they depend on different expansion coefficients. Experimental measurements are presented for all four asymptotic shear material functions using a linear and well-entangled homopolymer of polyisoprene. The experimental observations are consistent with both the predicted frequency scaling and the predicted interrelations in the terminal regime. Signs and magnitudes cannot be universally predicted, leaving these as free parameters that depend on the specifics of the material microstructure or constitutive model. These general results explain previous observations involving different materials and constitutive models, and provide an important reference for future experiments, analytical results, and numerical computations of these rheological fingerprints. (C) 2014 The Society of Rheology.