There is only two things

... that qualify as proof: conclusive evidence (Empirical Proofs) and deductive argument (Anecdotal Proofs).

The mode to do so is by means of what is known as "the chain".

In order to produce C; the thing to be (dis)proven, you need to do so

A : in an unbroken (thus logically valid) chain of establised facts/events

B : by eliminating all other possibilities (rendering what remains sound)

When not A and B, no C.

An assertion without evidence can therefore be dismissed the same way since those who assert must prove.

. . .

- can the theist construct a deductive argument showing the biblical God exists?

- can the atheist construct a deductive argument showing the biblical God non-existent?

Well, there you go then.