Entropy hierarchies on finite quantum-register hypergraphs: data processing, barrier-regularized descent, and connected information

We develop a self-contained finite-dimensional theory of relational-information functionals on hypergraphs whose hyperedges carry quantum registers. The central object is a weighted hierarchy of total correlations evaluated after local product compression. We prove an exact data-processing inequality under compression-compatible block-local product channels and obtain the corresponding monotonicity along local product GKSL semigroups. For the variational layer we separate the continuous base functional on the compact configuration space from its faithful-state dynamics: an extended-valued log-determinant regularization yields interior minimizers, and every smooth objective with compact sublevel sets generates a unique global Riemannian gradient flow whose omega-limit set is critical; isolated critical points give single-point convergence and nondegenerate minima are exponentially stable. We then study the maximum-entropy decomposition of total correlation. Every maximum-entropy level is locally continuous on faithful states with an explicit modulus, invariant under local unitaries, and nondecreasing under local unital channels, while the sum of the connected-information layers is nonincreasing under all local product channels. Individual layers, however, are not data-processing monotones: a known three-qubit counterexample creates irreducible three-party correlation by a local random-unitary channel. The paper therefore gives a rigorous finite-dimensional foundation and an explicit claim boundary; it does not assume global consistency of overlapping hyperedge states, a continuum limit, or a physical realization of the variational flow.