  Works of Hooley and Heath-Brown imply a near-optimal bound on the number $N$
of integral solutions to $x_1^3+\dots+x_6^3 = 0$ in expanding regions,
conditional on automorphy and GRH for certain Hasse--Weil $L$-functions; for
regions of diameter $X\ge 1$, the bound takes the form $N\le C(\varepsilon)
X^{3+\varepsilon}$ ($\varepsilon>0$). We attribute the $\varepsilon$ to several
subtly interacting proof factors; we then remove the $\varepsilon$ assuming
some standard number-theoretic hypotheses, mainly featuring the Ratios and
Square-free Sieve Conjectures. In fact, our softest hypotheses imply
conjectures of Hooley and Manin on $N$, and show that almost all integers
$a\not\equiv \pm 4 \bmod{9}$ are sums of three cubes. Our fullest hypotheses
are capable of proving power-saving asymptotics for $N$, and producing almost
all primes $p\not\equiv \pm 4 \bmod{9}$.
