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Akyuu was born in 1994. Find the average life span of a XIXth century Japanese girl and start adding it to that year until you reach 2258. There's your answer.

Eh... Juuni?

What about in 2277?

The life span of the Child of Miare shortens every incarnation. Every incarnation will live a shorter time than the previous. Assuming Akyuu was around 10 years old at the time of her introduction, she may very well die in her teens.

A 12th Child may not live to be able to speak, depending on how much time is lost from the lifespan.

Also: The figuring for life expectancy is off. 19th Century is 1801-1900. You'd want 20th century life expectancy, not that it would matter for someone who would be living on a heavily degraded version of life expectancy of 800 AD.

No. It gets shorter, period.

Assuming there's supposed to be a connection to the real Hieda no Are (an actual historical person Akyuu gets her name from), all that's known is that Are was tasked to memorize what eventually became the Kojiki, at 28. It was completed in 712, though there are no other real details about the age. Hypothetically, if Are lived until say, 70, and died in or close to the start of the 9th century (unlikely, as Are was 28 in the 7th century when they were tasked with this job) and 5 years was lost off the life span for every incarnation, Akyuu would live until 25, at best. The 12th incarnation then wouldn't make it past 10.

Of course, this is just with hypothetical numbers with no basis. I'd imagine that Are probably didn't live until 70, and in that case Akyuu's lifespan drops further. Of course, the life span could just be shortened by like a year for each incarnation or something silly like that to ruin all sense of drama.

That's not a given.

Reading the ending monologue of PMiSS, Akyuu says that she no longer thinks that the Gensokyo Chronicle is necessary and that it no longer would require updating, since it's no longer an essential source of information for survival against youkai, it's original purpose. She suggests she may not need to reincarnate any more.

She also writes that she likely will not live long past 30, so I guess my rough estimations were pretty close in the end, and she must lose a fairly significant portion of her lifespan with each incarnation. It's also written that she has to start preparing for the ritual for her reincarnation years in advance, so that makes it more difficult as well, if her life ever reaches a point where she won't live long enough to be able to contribute to the Gensokyo Chronicle and still prepare her next incarnation, that'll likely be the end of it.

As well, apparently 100 years or so pass between reincarnations, so if her life expectancy is 30 years, Akyuu would be dead by 2025, and the Tenth would be born around 2125-ish. Assuming another 30 year life span (for convenience), that incarnation would die by 2155, with the Eleventh born in 2255, making her 22 by 2277.

Seems it's more a personal obligation than any kind of compulsion.