A torsor is a set on which a group acts freely and transitively — every pair of points is connected by exactly one group element — but no point is designated as the identity. The group structure is present in the transitions between points, not in any point itself. Remove the origin from a vector space and you have the intuition: directions and distances exist, but “zero” is gone.
The absence of a canonical origin is the defining feature, not a deficiency (arXiv:2603.11833). In a group, the identity element provides a reference point — every element is “distance g from the identity.” In a torsor, no such reference exists. You can measure the displacement between any two points, but you cannot assign an absolute position to any single point.
This structure underlies sigma protocols in cryptography. A sigma protocol proves knowledge of a secret without revealing it. The proof works because the verifier sees transitions — commitments, challenges, responses — but never the secret itself. The secret is the absent origin. The protocol's security depends precisely on the fact that the transitions (group elements connecting points) are visible while the base point (the secret) is not.
Torsors also arise by gluing. Local trivializations — regions where a canonical origin is chosen — are connected by transition functions satisfying cocycle conditions. The global object has no origin even though each local piece does. The cocycle condition ensures consistency: traversing any closed loop of transitions returns to the starting point, even though no starting point is preferred.
The concept formalizes a pattern that appears wherever structure exists without a preferred reference: gauge theory, principal bundles, affine spaces. The group tells you what transformations are possible. The torsor tells you that none of them is the identity.