The third millennium does not start on either January 1, 2000 or January 1, 2001.

Let's look at what we mean by a millennium, in the first place. A millennium is a set of 1000 years. A year is, on average, 31,556,926 seconds. The year varies, but the 400-year cycle does not; there are 12,622,770,432 seconds in this cycle, give or take a few leap seconds. So a millennium is something like 31,556,926,080 seconds. A second is 9,192,631,770 periods of the radiation cycle of cesium-133. Multiply that all together, and you get an astoundingly huge number that I won't bother to figure out, so let's call it Q. So now we know what a millennium is. Q cycles of a certain atom.

Now, all we have to do to figure out when the millennium is is determine what time is 2Q cycles later than the exact time B.C. became A.D., right?

Oh wait. B.C. never became A.D. There was nobody counting off "Okay, this kid's going to be born in about 5 seconds... 4... 3... 2... 1... yay, we can count time forwards from now on!" The Romans were the dominant empire in the world, and so the world essentially used the Roman calendar.

Okay, but the "zero point" was defined later, right? Sometime in the 500's, this Roman guy decided to figure out when Jesus was born. After some astounding calculations, he arrived at an exact date that was unfortunately about 4 years off. So counting Q cesium-cycles after Christ's birth would put the Millennium sometime between 1995 and 1997.

Okay, you say, so leave Jesus out of this. It's the 90's1 and we call it C.E. so we can be politically correct, and pretend that we've simply decided that the time the Roman guy came up with is a good place to start counting. Back in the 500's, Julius Caesar uses that time as the basis of his new Julian calendar. And if we follow that calendar, we eventually get to January 1, 1000. Or is it January 1, 1001 we should be looking at? Or did they call it 999 because the new year didn't start until April? By this point, it doesn't even matter.

So you count Q cesium-cycles ahead from there, going off by two weeks in the process because of the switch from the Julian to the Gregorian calendar, and add in a fudge factor for all the Julian leap years that didn't quite work. The result: you know with absolute certainty that the Millennium changes within about a year and a half of the year 2000.