The date can be calculated from the orbital positions of various planets, moons, asteroids, etc. with great accuracy for tens and hundreds of thosuands, possibly milliond, of years in the future.
So for a lost centuries theory to work, all records of the orbital parameters in our solar system would have to be lost.
There is a nutty theory that several false centuries were added to the history of Europe during the middle ages as part of some plot. That ignores the fact that the history of Europe in the Middle Ages is connected with and mentioned in the histories of, for example, the Muslim world. And it also ignores that it would mess up the calculations of the proper date to celebrate Easter, and so would be consideded sinful by churchmen.
And the suggested theory about forgotten centuries in
Star Trek seems just as as nutty as that real life theory.
I suggest once again that you go to the Wikipedia article Calendar Era:
A
calendar era is the period of time elapsed since one
epoch of a
calendar and, if it exists, before the next one.
[1] For example, the
Gregorian calendar numbers its years in the Western
Christian era (the
Coptic Orthodox and
Ethiopian Orthodox churches have their own Christian eras).
https://en.wikipedia.org/wiki/Calendar_era
And take a look at the different years numbers 2021 has in different calendar eras.
Of all the hundreds of year numbers mentioned in
Star Trek, only about half a dozen are specified as being years AD or BC. All the other
Star Trek dates could be in the
Anno Domini Calendar era or in different calendar eras.
Numerous historical errors in various
Star Trek episodes make me suspect that
Star Trek is almost certainly in an alternate universe which diverged from ours sometime before the first
Star Trek episodes were broadcast in 1966. If the creators of
Star Trek didn't want me to deduce that, they should have taken more care to include only accurate historical references in their episodes.
So those are my explainations for the diffrences between Earth's history from 1966 to 2021 in our universe and in the fictional universe of
Star Trek.