Sure, we can debate semantics here if you want. I'm fine swapping in "faithfully execute" into my prior comment though, that's basically what I meant without going word for word constitutional.
My point remains though. "Faithful execution" is in the eye of the beholder and is up for debate. One person may see the Department of Education as faithfully executing the congressional mandate while another could see it as poorly run, inefficient, or point to our education level relative to other countries. Both would have good arguments to make.
Further, I don't read Congress's power to approve the budget as part of the mandate for a department. Congress isn't saying "spend XX billion and build an education department," they're saying "build an education department and don't spend more than XX."
That can surely be debated in a legal context, but I think you would be hard pressed to find many average people that would read a budget as a "spend every penny" mandate. Many corporations operate this way, and while in my experience people will spend their full budget to avoid a decrease next year they are also well aware of the absurdity of that.
The impoundment law itself was/is controversial and this will surely be challenged in court on those grounds. The question still remains, though, whether any miscarriage of the law us found in departments being shut down. Its too early, mainly because they at least appear to be acting rashly, but that doesn't mean these departments have been faithfully executing to date.