Is there a mathematical term meaning "the original assumption"?

Question

Is there a term which refers strictly to the foundational assumption(s) that a proof is based off of? I don't mean this in the sense of an axiom, but rather, for example, stating that $m \in \mathbb{R}$ and $n \in \mathbb{Z}$ at the start of the proof. The term could then be used to refer to the collective of these two assertions unambiguously at any point in the proof.

Answer 1

There isn't really. So you have to make your own. One common way is

Assume $$m\in\Bbb R,\quad n\in\Bbb Z\tag{*}$$

and you can then refer to it as $(\text *)$ at any point.

Answer 2

The usual term for this is "hypothesis" (or often "hypotheses", when there is more than one assumption). The hypotheses of a theorem are the assumptions made in the statement of the theorem which must hold in order for the conclusion to follow. More generally, a "hypothesis" of an argument is an assumption you make at the start of the argument which will prove that some consequences follow from the assumptions.

That said, the term "hypothesis" is not as strict and unambiguous in its meaning as you seem to be asking for. Since it can broadly refer to any assumption made in an argument, the exact meaning varies with context and it is not a totally precise term on its own. Still, it is probably the term that mathematicians most frequently use with the meaning you ask for.