Bidirectional Inference Counts on Forests and Cycles

For a binary restricted Boltzmann machine on a fixed labeled bipartite graph, consider the pair of conditional maximum-a-posteriori maps in both directions, with the same weights used in the two maps and no ties. We count such pairs exactly on forests and on even cycles. On a forest, local threshold functions can share symmetric weights if and only if their monotonicity directions agree on every input edge essential at both endpoints. Positive rescaling gives a constructive proof and an incidence partition function evaluable by a tree recurrence. For paths, the counts satisfy p_1=2, p_2=14, and p_n=10p_(n-1)+6p_(n-2). For even cycles of length n>=4, the count is (5+sqrt(31))^n+(5-sqrt(31))^n-2^(n+1); the subtraction removes exactly two cyclic families of impossible magnitude comparisons. Two trees with identical degree lists in each bipartition class have different bidirectional counts. This invariant is distinct from the classical one-way inference count, which factors over output degrees. Scope: complete theorems for the explicitly bidirectional extension on forests and even cycles, not a claim that the entire AIM-PROBABILITY-0009 record or all higher-degree cyclic graph classes are solved. The standard one-way count and general symmetrization and hyperplane-arrangement frameworks are credited prior work. No absolute-priority claim is made. Unrefereed preprint prepared with AI assistance and originating-workflow self-audit. No independent peer review or formal certification is claimed. The author remains responsible for all claims and the final text. Author: Alper Ferudun, Mercury Software GmbH. Corpus identifier: AIM-PROBABILITY-0009, UnsolvedMath v1.6.0. Paper page: https://eulersolve.org/papers/aim-probability-0009/