Abstract

We prove a full completeness theorem for multiplicative–additive linear logic using a double gluing construction applied to Ehrhard’s *-autonomous category of hypercoherences. This is the first non-game-theoretic full completeness theorem for this fragment. Our main result is that every dinatural transformation between definable functors arises from the denotation of a cut-free proof. Our proof consists of three steps. We show:• Dinatural transformations on this category satisfy Joyal’s softness property for products and coproducts.• Softness, together with multiplicative full completeness, guarantees that every dinatural transformation corresponds to a Girard proof-structure.• The proof-structure associated with any dinatural transformation is a proof-net, hence a denotation of a proof. This last step involves a detailed study of cycles in additive proof-structures.The second step is a completely general result, while the third step relies on the concrete structure of a double gluing construction over hypercoherences.