In regards to this question and a comment made by a Stack Exchange Community Manager, perpetual betas exist in the Stack Exchange. Considering that we have been in beta for five years, is there a way that we can confirm whether or not we are classified as a perpetual beta?
Yes, we are now in perpetual beta.
The fact that we haven't been closed down is pretty good evidence that we are now in perpetual beta.
Our stats suggest that we would have to grow substantially as a site in order to qualify for graduation, and given our niche appeal, I doubt that will happen. As the linked articles say though, graduation is neither expected not desired in every case.
Having said that, our stats are pretty good for a niche site. We are well up at the top of the bottom quartile of many stats, beating several graduated sites in some measures. Our %age answered is excellent, we have healthy number of visits per day, and a reasonable number of questions per day for a small site.
See the full Stack exchange site stats for more details.
As long as people are asking questions here, and we are answering them, we are in no danger of being closed down.
If you are looking for an official answer from Stack Exchange, specific to our situation, then that's unlikely. Stack Exchange themselves only get involved when there are serious problems with a site, and thankfully we are not in that position. Other than appointing new protem moderators, the last time we had an intervention by a Stack Exchange employee on our site was in 2013, and this in itself is a good sign.
We don't generally close sites for lack of question counts or traffic. Just about the only criteria we have to keep a site open is three people willing to step up and say, "yeah, we'll help watch over our site". See Site closure and the health of SE sites.
This was re-iterated more recently in this answer by Robert Cartaino♦ to What is the usual process for closure of a site in public beta?
As discussed in: Graduation, site closure, and a clearer outlook on the health of SE sites, just about the only criteria remaining to keep a site open is having enough users (three minimum) willing to step up to Moderate♦ it.
If a site is in decline and doesn't have enough users willing to take care of it, we will typically issue several calls (through a featured meta post) to try and recruit more interest. Failing that, we will post a final site-closure notice announcing the closure date. This will appear as a 'system message' plastered across the top of every page on the site — all the time referring users back to a meta post where they can express interest in keeping it going.
I can't imagine a site going from functional to "final notice" in less than a month; more typically the effort would persist three to five months… or more.
Given these statements, and the fact that we have four moderators, three of whom are active most days, I don't see that we are in any danger of closure.