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Lets say I had activated chimeric mass and it was the only creature on the battlefield.
There is also a pestilence in play.

If the CM changes first, then the Pestilence leaves play. If Pestilence checks first, then it gets to stay.

Do I have that right?

In either case, how is it determined which EoT abilities are first on the stack?

Pestilence triggers in the "end step"
Chimeric Mass stops being a creature in the "cleanup step", which is after the end step

Chimeric Mass is a creature "until end of turn" when you use its activated ability. This means it stops being a creature during the cleanup step, which is after the end step.  Pestilence 's triggered ability goes on the stack at the beginning of the end step, but only if there are no creatures on the battlefield at that precise point.  So, in the scenario you presented, Pestilence will remain on the battlefield, as its triggered ability doesn't actually trigger at all.

"Until end of turn" is a duration. It ends during the cleanup step, the turn's last step. It's the same step where damage on creatures wears off and the active player discard down to the maximum hand size.

"At the beginning of the end step" (current wording for Pestilence 's "at end of turn"; check its Oracle text) is a triggered ability. It triggers when the End Step starts. The End Step is the second-to-last step of the turn.

So, first your end step starts, Pestilence's ability would trigger, but Chimeric Mass is still a creature so its "if-clause" ("if no creatures are on the battlefield") is not true, preventing the ability from triggering.

Later, in the cleanup step, Chimeric Mass stops being a creature. Pestilence stays on the battlefield, and your turn ends.

Rules:

512. Ending Phase
512.1. The ending phase consists of two steps: end and cleanup.

513. End Step
513.1. First, all abilities that trigger "at the beginning of the end step" or "at the beginning of the next end step" go on the stack. (See rule 603, "Handling Triggered Abilities.")
513.1a. Previously, abilities that trigger at the beginning of the end step were printed with the trigger condition "at end of turn." Cards that were printed with that text have received errata in the Oracle card reference to say "at the beginning of the end step" or "at the beginning of the next end step."
513.2. Second, the active player gets priority. Players may cast spells and activate abilities.
(...)
514. Cleanup Step
514.1. First, if the active player's hand contains more cards than his or her maximum hand size (normally seven), he or she discards enough cards to reduce his or her hand size to that number. This turn-based action doesn't use the stack.
514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all "until end of turn" and "this turn" effects end. This turn-based action doesn't use the stack.
(...)

Even if it were not for the EndStep/CleanUp timing, Pestilence would never even trigger, because of its intervening if clause.

Looking at them again, it seems obvious....  thanks.


Is my assumption correct that Pestilence checks at every End Step, not just mine?

Yes, each player's End Step.