  We present an analytical derivation of the volume fractions for random close
packing (RCP) in both $d=3$ and $d=2$, based on the same methodology. Using
suitably modified nearest neigbhour statistics for hard spheres, we obtain
$\phi_{\mathrm{RCP}}=0.65896$ in $d=3$ and $\phi_{\mathrm{RCP}}=0.88648$ in
$d=2$. These values are well within the interval of values reported in the
literature using different methods (experiments and numerical simulations) and
protocols. This order-agnostic derivation suggests some considerations related
to the nature of RCP: (i) RCP corresponds to the onset of mechanical rigidity
where the finite shear modulus emerges, (ii) the onset of mechanical rigidity
marks the maximally random jammmed state and dictates $\phi_{\mathrm{RCP}}$ via
the coordination number $z$, (iii) disordered packings with
$\phi>\phi_{\mathrm{RCP}}$ are possible at the expense of creating some order,
and $z=12$ at the FCC limit acts as a boundary condition.
