BEYOND BELL'S BIVECTOR Trivector Closure, GHZ Correlations, and Future-Sufficient Relational Grade From Premature Vectorization to Multipartite Factorization

Bell’s theorem constrains locally causal or factorizable hidden-variable completions of quantum correlations, while GHZ and Mermin extend the incompatibility with context-independent local assignments into multipartite systems [2,3,6,7]. A preceding paper in this program proposed a geometric interpretation of bipartite entanglement in which the source is represented as a nonseparable bivector relation rather than as a preassigned table of local vector outcomes [1]. The present paper develops the higher-order extension of that construction by replacing the heuristic progression from to with an explicit relational exterior representation. For qubits, each subsystem is assigned a local Bloch space , and the Pauli coefficient hierarchy is represented in the restricted cross-system exterior space whose real dimension is . Pauli correlation terms supported on distinct subsystems map to relational exterior -forms. The bipartite singlet correlation becomes while the phase-sensitive GHZ correlation becomes obtained by contracting a full-support relational form with the corresponding measurement blade. The GHZ family then yields a closure result. Every proper reduced density operator of is independent of , while suitable full--party measurements recover that phase. Consequently, the future disclosure map does not factor through the proper-marginal quotient. We call this a closure obstruction and define a future-relative closure rank as the minimum relational order required by a sufficient representation. For the phase-sensitive GHZ future, . Relational grade is kept distinct from connected-correlation order, quantum entanglement, Bell violation, and closure rank. Classical higher-order correlations can possess high closure rank, while GHZ and W states can share tripartite relational order yet belong to different multipartite entanglement classes [11,12]. The framework therefore does not identify higher exterior grade with greater quantumness. Instead, relational grade records correlation support, separability theory records quantum decomposability, and closure theory asks which relational distinctions must survive for a specified future. Keywords: Bell theorem; GHZ states; multipartite entanglement; relational geometry; exterior algebra; bivectors; trivectors; closure mathematics; future sufficiency; quantum foundations; Mermin inequality; quantum information.