This is an AI-generated Evidence Press audio briefing.

How many different multiplication systems can live on a set with a prime-cubed number of elements? For groups, the answer is five. Groups obey a strong rule: changing the brackets never changes a product.

A loop keeps an identity and unique division, while allowing the brackets to matter. Right Bol loops obey a weaker consistency rule. This paper studies a particular family: centrally nilpotent loops. Roughly, these can be simplified in stages by removing central parts, whose elements commute and associate with everything.

The candidate gives a complete count within that boundary. At order eight there are eleven classes. For every odd prime p, at order p cubed there are p plus ten. Both totals include the five groups.

The proof writes down how to multiply, identifies a representative of every class, and proves that different representatives cannot describe the same structure under a relabelling.

The key reduction replaces multiplication tables with a few parameters. Every nonassociative loop in this family has a central part with p elements and a two-dimensional quotient over a finite field. Five parameters describe its multiplication; changes of generators reduce the classification to three.

Small primes need separate treatment. At three, cubing either designated generator can give the identity, while cubing their product does not. The larger-prime rule would miss this distinction, even though it can accidentally give the right class count.

The package adds structural information and a recognition program for normalised parameters. These help researchers generate and compare examples. They do not classify every Bol loop or address isotopy, a broader equivalence relation.

The methods build on earlier central-extension theory. The all-prime conclusion rests on a written proof, supported by exact finite tests. This remains an unrefereed candidate, without formal verification or an absolute priority guarantee. Read the paper, code and review response for the precise scope.