Compound Cognitive Quasicrystals: Collaborative Superspace and the Anti-Venn Theorem

We extend the cognitive quasicrystal formalism (Elliott, 2026a) from single-agent witness arrays to multi- agent collaborative systems. When N agents with partially overlapping witness arrays cooperate, their individual Stone spaces combine into a compound Stone space whose embedding dimension equals the cardinality of the union of witness dimensions, not the intersection. This produces a compound cogni- tive quasicrystal—a composite aperiodic tiling whose structural properties exceed those of any individual component. The central result is the Anti-Venn Theorem: the value of cooperation is located in the orthogonal com- plement of the shared witness dimensions, not in the intersection. The intersection provides the com- munication channel required for guard alignment; every orthogonal dimension contributed by any agent provides a new basis vector in the collaborative superspace. The dimensionality of the compound crys- tal is therefore bounded below by max(|Γi|) and above by | i Γi|, with the gap between these bounds determining the super-additive capacity of the collaboration. We prove that: (1) insights transferred between agents with orthogonal dimensions manifest as phasons— discontinuous tile rearrangements driven by motion in internal dimensions invisible to the recipient; (2) the compound diffraction spectrum contains satellite peaks not present in any individual spectrum, whose intensity measures the super-additive information capacity; (3) some cohomological obstructions present in individual tilings dissolve in the compound tiling while new collaborative obstructions emerge; (4) Round-Trip Fairness (Elliott, 2025) is the matching rule that prevents parasitic phason extraction and is a necessary condition for compound crystal formation; (5) the magic team sizes n = 2k − 1 from the In- teragent Dynamics framework (Elliott, 2026b) correspond to complete Stone spaces of k-witness Boolean arrays. The gap structure of the compound crystal’s energy spectrum is organized by the Farey sequence, with κ∗ = 2/3 occupying the first nontrivial spectral gap—the minimum cooperation threshold at which the compound crystal becomes more than the sum of its parts. Keywords: compound quasicrystal, collaborative superspace, Anti-Venn Theorem, phason, satellite peak, cohomological resolution, Round-Trip Fairness, matching rule, spectral gap, Farey sequence, κ∗, magic team size, Stone space, witness array, Interagent Dynamics