Answer
A Heyting pseudo-complement, fixed by the base category. This is the part of QUESTION-055 that its question “does the context-indexed form of the communicative object settle which?” asks, split off on 2026-10-21 (ADR-049). It is conditional: CLAIM-038, which makes the communicative object a presheaf over contexts, is Active since 2026-10-29, and that part is QUESTION-332, Answered.
- The general case. THEORY-516 (Active), from Butterfield and Isham (LIT-381, LIT-389). For any presheaf of propositions over a small category of contexts, a sieve-valued valuation obeys functional composition exactly when it is a subobject of the presheaf. Truth values are sieves on the base. Their negation is the pseudo-complement, under which only S ∨ ¬S ≤ 1 holds. The base category fixes this logic, and global sections do not change it: classical physics has them and still gets sieves.
- The quantum instance. THEORY-037 (Active), from Döring and Isham (LIT-335, LIT-343). Over the von Neumann subalgebras, propositions form a distributive Heyting algebra, not an orthocomplemented lattice. The negation of δ(P) is not δ(1 − P), and excluded middle fails already for ℂ². The record re-derives the Heyting laws in THEORY-037’s note of 2026-10-19.
So the context-indexed form does settle it, given the base: the negation is not an orthocomplement. Whether excluded middle holds depends on the base. A base with exactly one arrow between any two objects gives two-valued logic (THEORY-516), and the quantum base does not (THEORY-037).
What it leaves to QUESTION-055
THEORY-516 says the choice of base remains free, and that sieve values are “among the most natural” valuations, not forced. So this answer gives the negation once a base is chosen. It does not say whether the communicative object is a presheaf, which base it has, or whether some probe family splits it into sectors that no probe connects (THEORY-042). The first is QUESTION-332, Answered by CLAIM-038 on 2026-10-29. The other two stay with the parent.