The thing fragment F_{τ,X} of a thing τ in state X is the least fragment containing the semantic interpretations of τ’s interaction surface: F_{τ,X} = the least fragment containing {⟦t⟧ : t ∈ Ix_X(τ)}. It is the smallest modal-temporal Heyting subalgebra of H generated by the thing’s interaction denotations.
The thing fragment captures everything that can be inferred locally from the thing’s interactions, before applying the full closure operator. It is monotone in the interaction surface: adding interaction terms can only enlarge the thing fragment. The thing fragment sits inside H and is subject to all the fragment-level reasoning tools — fragmentwise equality, hereditary extensionality, and the sheaf condition.
Closing the thing fragment under the state’s operators produces the footprint: Foot_X(τ) = S_X(F_{τ,X}). The thing fragment is the raw local data; the footprint is what remains after closure stabilizes it. The footprint is always bounded by the core semioverse state X*: nothing escapes the least fixed point.