A fragment is a finitely generated modal-temporal Heyting subalgebra of the semantic domain H.
Concretely, a fragment F ⊆ H is the smallest subset of H that contains some finite seed set S and is closed under all the structure of the semiotic universe: meets (∧), joins (∨), implication (⇒), top (⊤), bottom (⊥), the modality j, and the trace comonad G. Starting from finitely many elements and closing under all operations produces a fragment — a self-contained local reasoning context.
Fragments formalize the idea that reasoning is always local. No agent operates on the entire Heyting algebra at once. Instead, each context of inquiry — a proof, a measurement, a design decision — generates a fragment from the finite set of distinctions it actually uses. The fragment contains everything those distinctions entail under the algebra’s operations, but nothing more.
This locality is what makes fragment-preserving operations meaningful: an operation that respects fragments does not drag in commitments from outside the current reasoning context. It is also what makes fusion possible — two syntactic operators can be identified when they agree on all fragments, because fragmentwise agreement is the strongest operational equality available.
The set of all fragments F(H) is directed under inclusion and covers H: every element of H belongs to some fragment. This makes fragments the compactness tool of the semiotic universe, playing a role analogous to compact opens in topology or finitely presentable objects in category theory.