Confusion about articles before unique objects in math

When a mathematician says that There exists a B they know nothing else about B except that B must exist (in the appropriate mathematical sense of exist, which I won't go into here). Existence and non-existence can be proved without definiteness.

The appropriate article for something that has just entered the discourse without any information except its existence is the indefinite article. It's used for initial introductions.

• A man dressed in black entered the room.
• I saw a strange-looking dog this morning.
• Every field must have an additive identity.

Once an entity has been introduced to the discourse, it can be referenced with the definite article.

• The man in black looked around and left.
• I'd never seen the dog before.
• The additive identity must have certain properties.

Logically, the existential quantifier (∃, pronounced there exists or for some) is indefinite, while the universal quantifier (∀, pronounced for each, for every, or for all) is definite, though not necessarily presupposed to exist.

Using a is not a mistake but the beginning of the full description.

It is common to speak of the object with a and then go into the description which identifies it as unique. Then it is described as the or this mapping to include the details of the description.

If you were to use the to describe it from very the beginning it would be assumed that the description as unique has been delayed or made elsewhere.