The DEC claims they have made no specific representations that an air permit is forthcoming within a week. in fact, my feeling is that they would never represent to anyone any specific timeframe about a permit that is underway.
The DEC is comprised of more than one person. With whom did you speak and what is their position/role? The other posters probably spoke with other persons from DEC. That's the most likely explanation for why you were told something different than they were.
At least 3 people say 'your feeling' is wrong. Why? Because they CALLED, and reported back. Even I've called, and the people at the DEC are very friendly and outspoken.....