Graphical models

When local consistency forces a global structure, over acyclic bases both classical and quantum.

• MTH.C-2026-6019HiddenChannelCapacity.acyclic_iff_forall_consistent_glues

  The discrete Vorob'ev theorem: a cover's base structure is acyclic exactly when local pairwise consistency always forces a global structure, i.e. every pairwise-consistent family of nonempty local relations on it glues.

• MTH.C-2026-6020HiddenChannelCapacity.dirac_strong

  Dirac's lemma, strong form: in a chordless-cycle-free adjacency structure, every vertex set is a clique or contains two nonadjacent simplicial vertices.

• MTH.C-2026-6021HiddenChannelCapacity.lz_theorem_3_1

  The two-clique quantum junction tree theorem (Lauritzen–Zwiernik, arXiv:2605.19453, Theorem 3.1): over the acyclic two-clique base, the base's compatibility datum decides base-side reconstruction. For strictly positive consistent marginals, a quantum Markov completion exists iff Tr T(R) = 1; the completion is then unique and equals the normalized logarithmic candidate σ(R) = T(R)/Tr T(R). Together with trC_le_one (the trace bound) this is the full theorem.