Apologies if this question is a bit simplistic/vague for MO:

I'm looking for an all-purpose definition in the literature of when a sufficiently generic filter "canonically codes" a generic real. Examples of what I mean come from everyone's favorite notions of forcing to add certain reals "on purpose":

In Cohen forcing, the generic real is obtained by taking the union of the generic real.

In random forcing, the generic real is obtained by taking the intersection of the generic filter.

In Mathias or Laver forcing, the generic real is obtained by taking the union of stems in the generic filter.

In Sacks forcing, the generic real is obtained by taking the intersection of (branch spaces of trees in) the generic filter... etc, etc.

What I am **not** looking for is anything having to do with the "other" reals that are added "by accident" through such forcing.

I can, of course, come up with a bespoke definition that covers the above cases, but I was hoping for something general and known.

**Edit:** To be more precise, I am looking for something that is an *injective function* from (sufficiently generic) filters to reals. A preliminary definition might be something like: "there is a definable (from the poset $\mathbb{P}$) injection from generic filters for $\mathbb{P}$ to reals". In the case of Cohen forcing, the function would be the map which just takes the union of the generic filter.

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