  Context: Star formation efficiency (SFE) theories are currently based on
statistical distributions of turbulent cloud structures and a simple model of
star formation from cores. They remain poorly tested, especially at the highest
densities. Aims: We investigate the effects of gas density on the SFE through
measurements of the core formation efficiency (CFE). With a total mass of
$\sim2\times10^4$ M$_\odot$, the W43-MM1 ridge is one of the most convincing
candidate precursor of starburst clusters and thus one of the best place to
investigate star formation. Methods: We used high-angular resolution maps
obtained at 3 mm and 1 mm within W43-MM1 with the IRAM Plateau de Bure
Interferometer to reveal a cluster of 11 massive dense cores (MDCs), and, one
of the most massive protostellar cores known. An Herschel column density image
provided the mass distribution of the cloud gas. We then measured the
'instantaneous' CFE and estimated the SFE and the star formation rate (SFR)
within subregions of the W43-MM1 ridge. Results: The high SFE found in the
ridge ($\sim$6% enclosed in $\sim$8 pc$^3$) confirms its ability to form a
starburst cluster. There is however a clear lack of dense cores in the northern
part of the ridge, which may be currently assembling. The CFE and the SFE are
observed to increase with volume gas density while the SFR steeply decreases
with the virial parameter, $\alpha_{vir}$. Statistical models of the SFR may
well describe the outskirts of the W43-MM1 ridge but struggle to reproduce its
inner part, which corresponds to measurements at low $\alpha_{vir}$. It may be
that ridges do not follow the log-normal density distribution, Larson
relations, and stationary conditions forced in the statistical SFR models.
