Determinacy and Word Reconstruction from Simple Paths in Class-Two Networks: Structural characterization, exact audit design, and a discrete-gauge interpretation

We study selective identification from simple directed path products when standardized edge transports take values in groups nilpotent of class at most two. We distinguish universal determination, represented by dominion membership, from word reconstruction, represented by subgroup membership in the relatively free class-two group. Let H be the subgroup generated by the baseline observations in the relatively free class-two group, and let D be its dominion in the class-two variety. For a connected bouquet of directed cycles, cycle a has length k_a, d_a=k_a-1, with every cycle length at least three, and the baseline disclosure contains every simple path obtained by omitting one edge. We prove D/H≅⨁_(a