Classical Extensions and Strongly Positive Realizations of Decoherent History Probabilities

Probabilities on several decoherent families of histories may agree on shared events yet admit no classical extension to the Boolean algebra that the families generate. We characterize the strongly positive decoherence functionals that realize given local probabilities on a finite algebra. A covariance decomposition allows any signed extension to be prescribed at any attainable rank and gives the resulting increase in minimum trace. For two contexts, agreement on their intersection suffices for a strongly positive realization. On the bipartite graph of context atoms, the minimum negative mass of a signed extension is $(W_1-1)/2$, where $W_1$ is the transport cost for graph distance. A graph Laplacian formula gives the minimum trace. Its minimizers form an affine copy of a rectangular contraction ball whose extreme points have the smallest realization rank. For a pure-state implementation with one intermediate projective decomposition and a final postselection, the optimal success probability of a realization is the reciprocal of the least trace of a diagonal matrix dominating it. In a family of classical examples, this probability tends to zero for the unique trace minimizer even though another realization succeeds with probability one.