This thesis appears to be related or the underlying research:
https://d.lib.msu.edu/etd/51312
Indicates 2mg and 5mg were given along with the placebo. While these are common dosages available in the US, these are higher dosages than recommended by at least some sleep experts. The thesis doesn't seem to differentiate between the dosages when evaluating cognitive processes.
Now I want to evaluate my sleep stack against morning cognition.
Hard to discuss this with only a short abstract. They don't even give a dose or formulation of the melatonin, and details like "We found no significant differences between the melatonin and sleep groups on any of our sleep measures." raise more questions than they answer.
Could it just be dosage? Commonly found dosage is 10-20x what you'd actually secrete and IIRC absorption is actually quite good.
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Not sleeping impairs my morning cognition - a lot. Melatonin is part of the rotating stacks I use to mitigate insomnia.
My mornings have always been impaired, going back to pre-school. Meds don't make my mornings any worse than my baseline.