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I found alternative forms of present and imperfect conjunctive forms of 'esse' on the german-latin dictionary website https://latin.cactus2000.de/showverb.php?verb=esse&form=esse :

Form table

I realized that the first and second person plural does exist for neither of them.
The case was proven to be true by visiting another website and looking up the form table.

Furthermore a quick Packhum search shows that 'fuat' e.g. is used multiple times by many authors while 'fuamus', which would be the first person plural, can not be found anywhere.

But why is that?

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Why no fuamus?

There is an enormous difference in the frequencies of different forms of any word. It is instructive to compare the more common versions first: The form sit gives 6985 hits, while simus gives 232.

If we assume that the ratio of frequencies of the personal forms is the same for both variants (which may or may not be valid but is a reasonable guess), then the 25 hits for fuat should lead us to expect 25×232/6985 ≈ 0.8 hits for fuamus. Having none is very consistent with this number.

But this only means that it makes sense for the word fuamus to not appear in the extant literature, not that it is an invalid Latin word. The distinction is important.

Morphological analogy in general

The Latin morphology is very regular. Seeing fuat for the third person singular leads us to reasonably assume fuamus for the first person plural. But given the size of the Latin vocabulary and the amount of forms that a verb has, it is unreasonable to expect all the imaginable forms to exist in the Roman literature.

Sometimes an absent form is conspicuous: a verb appearing thousands of times but always in the third person strongly suggests that the first and second person forms should be frowned upon.

But sometimes absence is a matter of chance, as I would say is likely in your specific case. Sometimes people read too much into attestations and declare forms impossible if no written Roman record supports it. A native speaker of Latin is familiar with how conjugation works and will have no trouble interpreting fuamus if they are familiar with simus, sit, and fuat even if they have never heard fuamus before.

If you want to know whether a given form like fuamus really exists (whatever that might even mean), you can always ask a question on this site. There is no general answer that covers all possible words as there are many things at play: variant forms (e.g. -e vs. -i in third declension ablative), expected numbers by comparison (as above), mentions by Roman grammarians, semantic reasons, syntactic behaviour of the word in question (e.g. transitivity), homographs, and probably more.

Tables with grain of salt

Do not take conjugation tables too seriously, unless they come directly from a reputable grammar. An online source like the one you link to is not the most reliable kind of sources. While it looks like a sound reference for a quick check, it is not a good tool for understanding whether fuamus or foremus exists.

Ratio comparisons with grains of salt, too

I want to make clear that the calculation of ratios above is just a rough and quick estimate but not very nuanced.

If we repeat the exercise from above for foremus starting from forem (44 hits), essem (416), and essemus (97), we should expect about 10 instances of foremus. That makes the absence a little more alarming, but my model is crude. To get a better picture, you first need to understand the difference between esset and foret and their usage differences in different authors and analyze the instances in that context. Then the number of instances of essemus that you would have expected to be foremus might go from ten to zero.

Even if you want to do a context-free comparison, you can do something more statistically sensible than just counting a simplistic expected number.

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  • Thanks a lot! Awesome answer, got it now.
    – Cyb3rKo
    Oct 30 '21 at 19:41

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