Lemma: Closed finite-experiment constraints

The law of any fixed finite experiment is a continuous functional of $\tau \in {\mathcal{K}}_{N,\nu }$. Consequently, each fixed instance of composability, nullity, no-signaling, and record equality is closed.

Proof. At a positive branch of weight $p$, the rank-one operator $\tau /p$ determines the residual ray continuously. A gate or allocation is a continuous map on that ray. Evaluation of a subsequent branch map is continuous by the common modulus.

At a branch whose weight tends to zero, every downstream terminal contribution is bounded by that first weight. Such a contribution therefore tends to zero independently of the normalized residual. Induction on the finite history tree proves continuity of every reported law. Equality of continuous laws and exact zero conditions are closed.