Learn English – fraction prefix for “(one-)third”

I am a mathematician, working with things called ¹⁄_k-regular polytopes, dubbed thus by Conway. For the case of k = 2, as in ½-regular, it is naturally pronounced and written half-regular.

However, I am now dealing with the ⅓‑regular case, and I’m surprised to find that saying third-regular doesn’t sound right at all!

After considering for a while, I realized that using ordinals to denote fractions here always sounds wrong, but that when there is a different word, it works. For instance, saying quarter-regular for ¼-regular sounds fine, but fourth-regular does not.

This leads to a couple of related questions. In my US-based usage, I would have assumed that quarter and fourth were perfect synonyms for describing fractions. For instance, I can say:

• The project is three-quarters complete.
• The project is three-fourths complete.
• A quarter of the people left.
• A fourth of the people left.

But evidently, they are not as perfect as one might wish. Does the difference lie solely in whether or not the word can be used as a prefix?

For instance, saying a quarter-full glass sounds fine, while saying a fourth-full glass does not, because when pronounced out loud it sounds like you have four full glasses and are referencing the fourth of these. Are there other differences?

Is there a way to form such a prefix form for ⅓ or other cases?

I have imagined something like tertiarily regular,
but I don't think that really means what I want. Consider a tertiarily full glass: that doesn’t work for me. This answer
suggests tertile, but I don’t think that works here.
Tertile-regular certainly doesn’t sound very good.

Metric prefixes will work when available, like deciregular for ¹⁄₁₀ and centiregular for ¹⁄₁₀₀. Can these be extended beyond powers of ten?

This question
asks for a “quarter” equivalent for ¹⁄₉, with the answer that no such term exists. But it wasn’t focused on prefix forms.