The hypothesis that evolution occurs at an irregular rate through geologic time is known as: directional evolution directional equilibrium punctuated equilibrium punctuated evolution

Punctuated equilibrium

Explanation:

Punctuated Equilibrium is a theory that describe the process of evolution based on different factors such as: fossil records and the history of the species, the sequence of the appearance of the specie from first to latter.

According to this theory, species did not originate at a defined rate, but it was irregular. According to it, they originated rapidly in start, in such as way that paleontologists were able to trace back their origin ancestors. This is called punctuation.

Secondly, the fact that organisms which were the product of mutations remained unchanged for a long period of geological time shows stasis or equilibrium.

This theory shows that the process of this punctuation followed by stasis helped in the regulation of morphology as well as physiology of organisms.  However, when this equilibrium was broken, it lead to the initiation of the continuous process of evolution.

This theory therefore seems very difficult from Darwin's theory of natural selection which focuses on a continuous and regular process of natural selection that causes the process of speciation to happen. Based on these facts, Punctuated equilibrium is the right option,

Hope it helps!

thermophiles, notice the prefix thermo.