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

Abstract

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<c) Z/gcd(d_a,d_c )Z, so a cross-cycle commutator is determined at exponent lcm(d_a,d_c ) but requires exponent d_a d_c for observation-word reconstruction. After one optional direct edge observation is added per cycle, the main theorem characterizes every disclosure subset: determination holds exactly when each active cycle supplies either its direct observation or its full long-path bundle; word reconstruction holds exactly when, in addition, directly observed cycles form a vertex cover of the arithmetic conflict graph. This yields exact ex-ante minimum-cost policies. A separate formula gives the incremental cost of upgrading an already collected route-bundle baseline, so sunk-cost augmentation is not conflated with redesign from scratch. The discrete-gauge interpretation is secondary and kinematic. For each realization, externally framed, gauge-standardized link variables take values in a class-two group K; the universal statements quantify over all K∈N_2. The quotient F(E)/γ_3 (F(E)) is the universal symbolic carrier for that category, not an information-preserving reduction of an arbitrary non-Abelian system. The proposed novelty is the simple-path realization, exact gcd/lcm arithmetic, complete feasible-set characterization, and its audit-design consequences.