Cohomological Foundations of Triadic Irreducibility

This work lays the formal foundations of a theory of triadic irreducibility—the property of certain structures to carry information about three elements that cannot be reconstructed from pairwise relations. The framework rests on the cohomology group H3 (X; A) of a discrete cochain complex truncated at degree 3, which we show measures precisely the obstruction to dyadic reconstruction. We establish: (1) structural results, including the explicit computation of dim H3 (both normalized and non-normalized cases) and an analysis of behavior under products showing that entanglement arises entirely from truncation; (2) separation theorems demonstrating that the degree of entanglement depends on the class of operations considered (Cₐlg, Cᵣad, Celem) ; (3) conceptual correspondences with classical objects—the Milnor invariant for links and GHZ state correlations—which illustrate the relevance of the formalism.