The tl;dr version is the title plus, the last paragraph. FWIW, some background really helps.
Background: I have always thought at least and at the very least are the grammatical and idiomatic phrases I could use. Today, I ran into this comment:
Welcome to Chem.SE! This appears to be a homework question. Please see the policy on posting homework questions here; at the least, please include more details on your thought process so far in trying to answer this question. - Brian yesterday
This raised my alarm for being ungrammatical; however, as free dictionary says, at the least does exist, so does at very least:
Question: Now this is very baffling. At least doesn't have a definite article, while at the very least does. They are so much more common than their "evil twins" that I reckon some would only consider them grammatical.
So, my question is why? What does the definite article the indicate in at the very least that it doesn't and shouldn't in at least? This doesn't seem to be a matter of pure stylic choice, as there seems to be a consensus about which version of the idioms to use.
Edit: I'm not looking to find contextual differences. I'm concerned about a grammatical explanation about what's going on.