I checked on the compost again this afternoon, the temp was 39 now. On the left side, which I shall from now on refer to as batch 1, I found some strange white mushrooms growing. Didn't take a picture of them, just turned them back in there. But I'm taking this as a sign of the compost reaching a new stage in the process of breaking down.
In the middle of writing this, I realized I should take temperature measurements of the insides of both piles. I measured about 15 cm in from the middle. Batch 1 showed 46 deg C, so it really is cooling down. Batch 2 showed 63 deg C, so it's in the midst of being consumed by thermophilic bacteria, while batch one is at the late stages of the same bacteria and will soon return to being consumed by mesophilic bacteria. Batch 2 wasn't nearly as hot as batch 1 has been at it's peak, but I will continue taking temperature readings in the future. I'm also starting to scare myself.