The Circlet of Rhaelyx is one of the two component pieces of the Crown of Rhaelyx (the other being the Jewel of Rhaelyx) which provides a 10% chance to preserve resources when training non-combat skills.
The base drop rate of the Circlet is [math]\displaystyle{ \frac{1}{10,000,000} }[/math] per action taken. The drop rate increases by [math]\displaystyle{ \frac{1}{2,500,000,000} }[/math] for each Mastery level you have in any skill, to a max of [math]\displaystyle{ \frac{1}{100,000} }[/math] which is reached at 24,750 total mastery level. After that point, additional Mastery levels will not increase the chance further.
This graph shows the relation between the total Mastery level, and the effective drop rate (in percentage).