The Calendar Machine was a Lanthian artifact responsible for periodically "resetting" the world. Its original purpose was to halt the aging of some Lanthian VIPs (including Mayahuel its designer), forcing a time anomaly onto everyone who was in the room when it was activated and making them ageless; it actually did this. Unfortunately, it had a problem similar to the Y2K bug which gave it a finite operating period of 1450 years - and when it hit its rollover time the entire world reset. Every time the Mayan Long Count ended the world jumped back from 2012 to 562; this cycle repeated 56 times, with the cycle being broken on the 57th iteration.
The Calendar Machine
Fixing the problem was tricky. Both Jin and Brandi kept iterative journals, with the help of Phix, which should have made it a lot easier than it was. As things turned out there was not a single problem but three of them, and all three solutions needed to be executed together.
Obviously, the Calendar Machine needed to be fixed or destroyed. But...
There was a nominally unconnected problem with demons using Monica as a living gateway to the human world. A plan to permanently close off this path had been devised, which was great except...
Jin's mother Mayahuel had been sucked into the Demon Realm back in 562, when the Calendar Machine first malfunctioned, and needed to be returned to Earth; prematurely locking Monica's access would trap her there.
Getting all this to come together at once was tricky, particularly since Monica's demons would hear anything she did and so she could only be briefed on the Calendar Machine part of the plan. In the end it went down like this: Monica, the golem girls, and Shelly all teleported down to the hidden chamber in Teotihuacan where the Calendar Machine was hidden. They surrounded the machine and used the four figurines, which froze everyone but Shelly in place. Monica had been getting more suspicious for a while, exactly as she needed to be - when she was frozen and the spiked blade came out, so did a demon; the gateway was fully open. But the key had two purposes; it converted the two-way door into a one-way demon eviction system[1], removing renegade demons from Earth. Next through the portal was the Calendar Machine[2], which could not manipulate time in the timeless Demon Realm. The last thing through the portal was Bud[3], then Shelly locked the door and broke the key. Bud wasn't marooned, though - Shelly still had the teleport tapestry design as a tattoo, so as soon as Brandi unlocked it Bud could return[4] from the Demon Realm, with Mayahuel. Bud destroyed the tapestry design to prevent anything else from coming through, and all three problems were wrapped up neatly. History could continue, for the first time reaching 2013, and now heads out into an unknown future.
As incidental notes, the Calendar Machine's appearance is modeled on the Antikythera Mechanism[5], although it also looked like the first Macintosh computers[6]. Also, there's no reason the carvings on the walls of the Calendar Machine chamber[7] look like Dietzel; we're assured those are jaguars, okay?
See also Time Loop Chronology and the general Chronology, which extends before and after the time loops.