  Understanding fundamental limits of the various technologies suggested for
future 5G and beyond cellular systems is crucial for developing efficient
state-of-the-art designs. A leading technology of major interest is
non-orthogonal multiple-access (NOMA). In this paper, we derive an explicit
rigorous closed-form analytical expression for the optimum spectral efficiency
in the large-system limit of regular sparse NOMA, where only a fixed and finite
number of orthogonal resources are allocated to any designated user, and vice
versa. The basic Verd\'u-Shamai formula for (dense) randomly-spread
code-division multiple-access (RS-CDMA) turns out to coincide with the limit of
the derived expression, when the number of orthogonal resources per user grows
large. Furthermore, regular sparse NOMA is rigorously shown to be spectrally
more efficient than RS-CDMA across the entire system load range. It may
therefore serve as an efficient means for reducing the throughput gap to
orthogonal transmission in the underloaded regime, and to the ultimate
Cover-Wyner bound in overloaded systems. The results analytically reinforce
preliminary conclusions in [1], which mostly relied on heuristics and numerical
observations. The spectral efficiency is also derived in closed form for the
suboptimal linear minimum-mean-square-error (LMMSE) receiver, which again
extends the corresponding Verd\'u-Shamai LMMSE formula to regular sparse NOMA.
