The motif originates in the rock-cut chaitya halls carved into cliff faces across western India from around the second century BCE onward, most famously at sites such as Karla, Bhaja and Ajanta, where an entire barrel-vaulted hall, columns, ribbed ceiling and stupa included, was excavated directly out of solid rock rather than built up from separate stone blocks. At the entrance to these halls a large horseshoe-shaped window, the original chaitya arch, let filtered daylight fall down the length of the hall onto the stupa carved at its far end, and early examples of these facade windows sometimes still show carved wooden ribs set into the stone, evidence that the rock-cut form was deliberately copying an earlier tradition of freestanding wooden chaitya halls that have not themselves survived.
As stone and later brick temple architecture developed across India in subsequent centuries, largely replacing the rock-cut hall as the dominant building method, the distinctive arched profile of that original facade window was carried forward as a purely decorative motif long after freestanding chaitya halls of that specific rock-cut type had become rare, so the chaitya arch as seen on temple towers and stupa facades many centuries later, and thousands of miles from western India, is best understood as a decorative memory of this specific early architectural form rather than evidence of a real hall behind it.