  Layered cobaltates embody novel realizations of correlated quantum matter on
a spin-1/2 triangular lattice. We report a high-resolution systematic
photoemission study of the insulating cobaltates (Na1/2CoO2 and K1/2CoO2).
Observation of single-particle gap opening and band-folding provides direct
evidence of anisotropic particle-hole instability on the Fermi surface due to
its unique topology. Kinematic overlap of the measured Fermi surface is
observed with the $\sqrt{3}$x$\sqrt{3}$ cobalt charge-order Brillouin zone near
x=1/3 but not at x=1/2 where insulating transition is actually observed. Unlike
conventional density-waves, charge-stripes or band insulators, the on-set of
the gap depends on the quasiparticle's quantum coherence which is found to
occur well below the disorder-order symmetry breaking temperature of the
crystal (the first known example of its kind).
