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We present preliminary results for $B_K$ from the MILC ultrafine lattices,based on a partial ensemble of 305 configurations. We use HYP-smeared improvedstaggered valence quarks. The analysis is done using fitting forms based onboth SU(2) and SU(3) staggered chiral perturbation thery. For the SU(2)analysis, we find that the result using the NLO fit function is consistent withthat from a partial NNLO fit. For the SU(3) analysis, where we have to usepartially constrained fits due to the number of fit parameters, we find thatour two preferred fits ("N-BB1" and "N-BB2") are also consistent, both witheach other and with the results of the SU(2) fits. These results are used incompanion proceedings to improve the control over the continuum extrapolation.

First results for $B_K$ on the ultrafine ($a=0.045$ fm) ensemble