The semiotic footprint of a thing τ in state X is the closed partial state obtained by applying the composite closure operator S to the thing fragment: Foot_X(τ) = S_X(F_{τ,X}). The footprint is what the thing “means” in the Interactive Semioverse — the full semantic, syntactic, and fusion closure of everything generated by the thing’s interactions.
The footprint is always bounded by the core semioverse state X*: for any X ≤ X* and any τ, Foot_X(τ) ≤ X*. This boundedness ensures that no single thing’s interactions can produce semantic material beyond what the semioverse’s least fixed point contains. The footprint is also monotone in the interaction surface: more interactions produce a larger or equal footprint.
Footprints are the objects compared by the delta operation (Δ_foot(τ, τ’; X) = S_X(Δ_frag(F_{τ,X}, F_{τ’,X}))) and diminished by subtraction. They can be quantified by an evaluation valuation μ(τ; X) = μ(Foot_X(τ)), allowing quantitative comparison of how much semantic material different things contribute to the semioverse.