The question
CLAIM-119 holds that communicative categories form a concept lattice and are told apart by “a few social-relational constraints”. To say that affectionate teasing is not sarcastic condemnation, the lattice needs a negation. Non-Boolean lattices of propositions offer three: an orthocomplement, a pseudo-complement under which excluded middle fails, or a product of Boolean sectors. The question is which one the record’s own model gives.
Why the record needs it
The mechanism is structural identity, not analogy. CLAIM-038 makes a communicative object a system of constraints indexed by a cover of contexts, which is a presheaf. The topos programme builds the logic of propositions about a presheaf over contexts. So if CLAIM-038 holds, its result applies to the same kind of object. Superselection applies whenever no observable connects two blocks. In the record’s terms, that is when no probe in the family has cross terms between two categories (QUESTION-031). A blend across them is then a mixture, and that is CLAIM-044’s ambiguity. QUESTION-025 asks whether embeddings carry a non-Boolean lattice. It cannot be scored until the lattice’s operations are fixed.
What the record holds
- THEORY-037: over contexts, propositions form “a distributive Heyting algebra, not an orthocomplemented lattice”, and “the negation of δ(P) is not δ(1 − P)”. The source sketches the Heyting laws, and the record re-derives them (THEORY-037, note of 2026-10-19).
- THEORY-041: the order does not fix the complement. Birkhoff and von Neumann proposed modularity from a probability-like dimension function, and it fails in infinite dimension. Limit: it is mainly an attribution, and orthomodularity is unsourced.
- THEORY-042: under a superselection rule “a coherent superposition across sectors is operationally a mixture”, and the lattice becomes a product of sector lattices. Limit: proved for one case only, and it is an idealisation.
- THEORY-516: for any presheaf of propositions over a category of contexts, valuations obeying functional composition are sieve-valued subobjects, and the logic of the truth values, negation included, is fixed by the base category, not by global sections. Limit: it says which negation once a base is chosen, not which base the communicative object has.
What would answer it
A derivation of the sub-object logic of the communicative object as CLAIM-038 defines it, with its negation written out. Or a probe family shown to be block-diagonal across a pair of categories, which would make their lattice a product of sectors.
Split, 2026-10-21 (ADR-049). THEORY-516 and THEORY-037 answer the conditional part: over a presheaf of contexts, the negation is the Heyting pseudo-complement the base fixes, not an orthocomplement. That is QUESTION-247, Answered. This question keeps what they leave: which base category the communicative object has, and whether some probe family splits it into sectors (THEORY-042).
Split, 2026-10-29 (ADR-049). Whether the communicative object is such a presheaf is QUESTION-332, Answered by CLAIM-038, which went Active at the fourteenth status review.