  We seek to prove the means, motive and opportunity of 2 < z < 5 dropout
galaxies for large-scale structure. Together with low-z tracers, these samples
would map practically every linear mode and facilitate a tomographic
decomposition of the CMB lensing kernel over an unprecedented volume, thereby
yielding a proxy for (the time evolution of) matter density fluctuations that
provides compelling tests of horizon-scale General Relativity, neutrino masses
and Inflation-- viz., curvature, running of the spectral index and a
scale-dependent halo bias induced by (local) primordial non-Gaussianity.
  Focusing on color-color selection, we estimate the completeness,
contamination, and spectroscopic survey speed of tailored Lyman-break galaxy
(LBG) samples. We forecast the potential of CMB lensing cross-correlation,
clustering redshifts and Redshift-Space Distortions (RSD) analyses. In
particular, we estimate: the depth dependence of interlopers based on CFHTLS
data and propagate this to biases in cosmology; new inferences of (non-linear)
halo bias at these redshifts and depths using legacy data; detailed forecasts
of LBG spectra as would be observed by DESI, PFS, and their successors. We
further assess the relative competitiveness of potential spectroscopic
facilities based on an intuitive figure-of-merit and define a modernisation of
traditional selections to the photometric system of LSST where necessary.
  We confirm these science cases to be compelling for achievable facilities in
the next decade by defining a LBG sample of increasing Lyman-alpha equivalent
width with redshift, which delivers both percent-level RSD constraints on the
growth rate at high-z and measurements of CMB lensing cross-correlation at z=3
and 4 with a significance measured in the hundreds. Finally, we discuss the
limitations and avenues for improvement beyond this initial exploration
(abridged).
