The twelve branches were divisions of time before they were anything else: two hours each, twelve to a day, twelve months to a year, twelve years to a cycle. 子 comes first and holds midnight to one, around the winter solstice — the darkest point of both the day and the year, and the moment a new cycle formally begins. In the tradition those are one fact: the darkest place is the hinge.
So 子's nature isn't gloom, it's beginning and depth. It belongs to water, sits due north, and carries the rat — all pointing at one image: moving before anyone else does, and moving quietly.
Where it appears in your pillars, that part of life tends to run inward and deep: not showy, accumulating first and being seen later. It's also the most misread of the twelve, because almost nothing it does shows up at the time.
