For more than half a century the theoretical landscape for single chain dynamics for dense polymeric solutions and melts below the entanglement threshold has been dominated by the Rouse model for independent phantom chains, supported by ideas of hydrodynamic screening. There exists, however, a large body of literature from experiments, Monte Carlo, and molecular dynamics simulations on the deviations from the Rouse behavior for unentangled homopolymer melts, showcased mostly in the subdiffusive behavior of center-of-mass of tagged chains at intermediate times, with the subdiffusion exponent reported in the range 0.75-0.85. The influence of the surrounding chains of length Ns on the motion of a single tagged chain of length N is a key test, by which, through high-precision numerical simulation of unentangled melts, we show that the Rouse model fails. Our central results are that at intermediate times the tagged chains center-of-mass moves subdiffusively, proportional to t(alpha) with subdiffusion exponent alpha = 0.87 +/- 0.03 as opposed to alpha(Rouse) = 1, and that its crossover time to Fickian behavior is directly controlled by the relaxation time of the surrounding chains when the latter are shorter. The terminal relaxation time for the tagged chain and the long time diffusion coefficient are then sensitive to N-s. Both measured exponent flow, that is plots of d alpha(t)/d ln (t) vs alpha(t) where alpha(t) is the effective exponent between and t, and successful blob scaling arguments support the anomalous value of alpha as a true exponent. We find the same exponent in the scaling of Rouse mode amplitude correlation functions and directly related exponent for the monomeric diffusion. We show that the consequences of these results on the dynamics of a tagged monomer and the chains segmental orientation autocorrelation function agree very well with rheological measurements and NMR relaxometry experiments. We reflect back on a history of related experimental anomalies and discuss how a new theory might be developed.