Global entropic reconciliation of compositional probability systems: mixed-state common separators and finite junction trees

Local probability models on overlapping domains often disagree on shared variables. Standard Markov or compositional combination requires compatible separator marginals, whereas Markov melding first replaces them by a selected pool. We formulate reconciliation as a global information projection. For submodels sharing one separator, the logarithmic separator pool and preserved conditional kernels determine the unique optimal reconstructed marginals over all joint laws; their conditionally independent product is the unique Markov representative. For a finite junction tree, building on earlier Markov-constrained KL alignment, we formulate a weighted version over all global laws and characterise the complete set of joint minimisers; the clique – separator product is their unique Markov and maximum-entropy member. A smooth concave dual yields separator-discrepancy gradients and computable edgewise conflict bounds. Examples include a mixed atomic – continuous closed-form pool, a binary stress-test chain exhibiting sequential order dependence, a four-state distributed-perception chain, and a scaling experiment up to 500 cliques using the separator-consensus dual.