Establish conditions implying pairwise dominance

Establish a hypothesis on labeled point clouds under which the joint entanglement of every class triple or higher-order class collection is bounded by the strongest constituent pairwise entanglement.

Background

The experiments find that triple entanglement is usually no greater than the maximum entanglement of its three constituent pairs, but the paper constructs counterexamples showing that this pairwise-dominance pattern is not forced by the geometry of arbitrary class clouds. The authors therefore leave unresolved the identification of structural assumptions on labeled clouds that would make pairwise dominance a theorem.

Such a result would explain when the empirically observed near-pairwise interaction structure follows from geometric or topological properties rather than from the particular representations examined in the experiments.

References

The open theory is an analytic null for the profile, which would calibrate the tests without permutations, and a hypothesis on labeled clouds under which pairwise dominance becomes a theorem; the guard-certified regime alone does not suffice (E2).

— Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise  (2609.08561 - Majhi, 8 Sep 2026) in Experiment E2, subsection “Why dominance is the default, and why it is not a theorem”; Section 6, “Conclusion and limitations”