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Swift intensive accretion disk reverberation mapping of four AGN yielded light curves sampled $\sim$200-350 times in 0.3-10 keV X-ray and six UV/optical bands. Uniform reduction and cross-correlation analysis of these datasets yields three main results: 1) The X-ray/UV correlations are much weaker than those within the UV/optical, posing severe problems for the lamp-post reprocessing model in which variations in a central X-ray corona drive and power those in the surrounding accretion disk. 2) The UV/optical interband lags are generally consistent with $ \tau \propto \lambda^{4/3} $ as predicted by the centrally illuminated thin accretion disk model. While the average interband lags are somewhat larger than predicted, these results alone are not inconsistent with the thin disk model given the large systematic uncertainties involved. 3) The one exception is the U band lags, which are on average a factor of $\sim$2.2 larger than predicted from the surrounding band data and fits. This excess appears due to diffuse continuum emission from the broad-line region (BLR). The precise mixing of disk and BLR components cannot be determined from these data alone. The lags in different AGN appear to scale with mass or luminosity. We also find that there are systematic differences between the uncertainties derived by javelin vs. more standard lag measurement techniques, with javelin reporting smaller uncertainties by a factor of 2.5 on average. In order to be conservative only standard techniques were used in the analyses reported herein., Comment: 25 pages, 9 tables, 6 figures, published in ApJ. The paper has been revised to conform to the published version. Please note that the data (Table 2) are available at the Digital Repository at the University of Maryland, https://drum.lib.umd.edu/handle/1903/21536