Sorta related to my other thread and sorta not. I want to simplify representing a badugi by ignoring the suits and just saying it has all 4 suits and it qualifies. But now if I want to find my equity with a 3 card hand against that badugi it matters exactly how the badugi suits are distributed:

A23 vs 4567 badugi - it won't matter at all I just need to remove one out from the deck it doesn't matter whether it's the 4,5,6, or 7.
A23 vs A267 badugi - now it matters a lot how the badugi suits are distributed. If I need a heart it will be the 6 or 7 50% of the time so I need to discount a half out.

Is there a way to express this discounting factor generically?

Do you know your opponent's hand?

No, that's my point. Depending on exactly what he has I would know the exact discounting amount, so I want a way to generalize it for whatever hands he can possibly have

So I wrote some code and found an interesting result. I am nearly certain these outputs are correct and I didn't mess up the code:

A23 against any K badugi: 9.20 outs (from 10)
A23 against any Q badugi: 8.22 outs (from 9)
A23 against any J badugi: 7.24 outs (from 8)
A23 against any T badugi: 6.27 outs (from 7)
A23 against any 9 badugi: 5.30 outs (from 6)
A23 against any 8 badugi: 4.35 outs (from 5)
A23 against any 7 badugi: 3.40 outs (from 4)
A23 against any 6 badugi: 2.47 outs (from 3)
A23 against any 5 badugi: 1.54 outs (from 2)
A23 against any 4 badugi: 0.41 outs (from 1)
So my methodology was to start with As 2h 3c and then run it against every possible badugi with highest card == rank X (so not X or better!!). I also correctly filtered out combinations of the X-high badugi that contained an As, 2h, or 3c so as not to duplicate hands that can't really exist. I believe my resulting distribution is correct. I then took the equity and converted it into number of outs as shown above.
So if we look at the amount of outs vs the "theoretical max possible outs" which is (X-3) as the rank X increases the outs need to be discounted more. This makes sense to me because there are more combinations that block the suit we need the higher we go. Ex: A23Q badugi will block the Q of diamonds we need some % of the time but 456Q badugi will block a rank we need 100% of the time.
Here's my problem: the rate of change of this discounting decreases as the rank increases. From a logical POV if there were infinite ranks of poker hands we would expect the total outs to converge on (X-4) when the overlapping ranks minimize to effectively zero. But as my output shows the rate of discounting is decreasing as the rank rises. Can anyone explain this?? My intuition would have been the opposite of what my data shows, that the rate of change of discounted outs would increase as the wideness of the overall range increases.

If the rate of discounting increases, it can't converge. It has to approach the limit at a decreasing rate.

Hmm good point! I still cannot get my mind around the "why" in this case though. I would think as the range widens we would discount more and more frequently (put another way, we would collide with the suits of the A23 hand less often)

This should be "from 0.5" I think, which may also be a clue to answering your question.