Behind the observance sits a piece of arithmetic. A tithi is the time the Moon takes to open a further twelve degrees of separation from the Sun. Chaturthi occupies the arc from 36 to 48 degrees of that separation. Because the Moon travels an elliptical orbit its angular speed is not constant, so that twelve degree arc is crossed in anywhere from roughly nineteen to roughly twenty six hours. The convention that resolves this is simple and consequential: the civil day takes the name of the tithi that is running at sunrise. A tithi shorter than the interval between two sunrises can therefore be skipped entirely, which the tradition calls a kshaya tithi, and one longer than that interval can carry its name over two civil days, an adhika or vriddhi tithi. Both occur in the ordinary course of a year, and both are why an observance can appear on two dates in the same almanac.
Each tithi is halved again, and each half carries a karana, so the first half of Chaturthi carries Vanija, the merchant's karana, and the second Vishti, the karana also called Bhadra. Vishti is the one karana the muhurta literature marks as unfit for beginnings, and its presence in the second half is why this tithi is read with more care than its neighbours. Seven of the eleven karanas repeat, cycling eight times through the month, while four are fixed to particular half tithis at the month's opening and close.
The Moon is waxing through this arc, setting later each evening and carrying about 13 per cent of the disc lit. That figure is a consequence of the definition rather than an observation added to it: elongation fixes the phase, so naming the tithi already names how much of the disc is lit.