Classification of stringent hyperfields

Statement

Every stringent hyperfield $R$ is a tropical extension of one of the following kinds of residue hyperfield:

1. the Krasner hyperfield;
2. the sign hyperfield;
3. an ordinary field.

More invariantly, there is a totally ordered abelian group $\Gamma$ and an exact sequence

where $F$ is one of the three kinds above. The order on $\Gamma$, the hyperaddition of $F$, and the extension data determine the hyperaddition on $R$: between unequal layers the larger layer is the unique sum, while cancellation in one layer exposes lower layers.

Conversely, the compatible tropical-extension construction from such data produces a stringent hyperfield. The extension need not split, so the theorem does not assert that $R^\times$ is a direct product.