  We study the exceptionally short (32-41 ms) precursors of two
intermediate-duration thermonuclear X-ray bursts observed with RXTE from the
neutron stars in 4U 0614+09 and 2S 0918-549. They exhibit photon fluxes that
surpass those at the Eddington limit later in the burst by factors of 2.6 to
3.1. We are able to explain both the short duration and the super-Eddington
flux by mildly relativistic outflow velocities of 0.1$c$ to 0.3$c$ subsequent
to the thermonuclear shell flashes on the neutron stars. These are the highest
velocities ever measured from any thermonuclear flash. The precursor rise times
are also exceptionally short: about 1 ms. This is inconsistent with predictions
for nuclear flames spreading laterally as deflagrations and suggests
detonations instead. This is the first time that a detonation is suggested for
such a shallow ignition column depth ($y_{\rm ign}$ = 10$^{10}$ g cm$^{-2}$).
The detonation would possibly require a faster nuclear reaction chain, such as
bypassing the alpha-capture on $^{12}$C with the much faster
$^{12}$C(p,$\gamma$)$^{13}$N($\alpha$,p)$^{16}$O process previously proposed.
We confirm the possibility of a detonation, albeit only in the radial
direction, through the simulation of the nuclear burning with a large nuclear
network and at the appropriate ignition depth, although it remains to be seen
whether the Zel'dovich criterion is met. A detonation would also provide the
fast flame spreading over the surface of the neutron star to allow for the
short rise times. (...) As an alternative to the detonation scenario, we
speculate on the possibility that the whole neutron star surface burns almost
instantly in the auto-ignition regime. This is motivated by the presence of 150
ms precursors with 30 ms rise times in some superexpansion bursts from 4U
1820-30 at low ignition column depths of ~10$^8$ g cm$^{-2}$.
