Seven-Factor Stochastic Transition Layer

Integrate this as a transition operator in a small RNN, a neural Markov model, a graph message-passing layer with three node types, or an MoE router over exactly three experts. Use seven fixed factor slots. Each slot has an ordered pair (i_l,j_l) selected from the six possibilities and one trainable scalar alpha_l. For a first MVP, choose the ordered pairs by enumerating or randomly sampling candidate sequences, then optimize only the alpha values. Compute s_l=(1-epsilon)*sigmoid(alpha_l), initialize P=I, and for l=1,...,7 construct K=I, set K[i_l,i_l]=1-s_l and K[i_l,j_l]=s_l, then update P=K@P. Apply y=x@P to row distributions or use P as the recurrent transition. A differentiable variant can learn pair logits beta_l and use K_l=sum_{i != j} softmax(beta_l)_{ij}K_{ij}(s_l); this convex combination remains row-stochastic. The mathematics gives exact nonnegativity, unit row sums, and the bounded seven-factor representation; the implementation should estimate no theorem-specific quantities. Monitor row-sum error, minimum matrix entry, singular values, gradient norms, and long-horizon rollout stability. First test on a synthetic 3-state sequence-prediction task and a tiny GRU whose explicit 3-state transition is replaced by this layer. Compare with a directly learned row-softmax 3x3 matrix, a single-factor model, and a dense unconstrained matrix. Success means equal or better validation negative log-likelihood, zero stochasticity violations, and improved long-horizon calibration or stability at comparable parameter count. Also report whether seven factors actually improve over fewer factors, since the paper's bound is worst-case rather than an assertion that all matrices need seven.