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

We develop an exact theory of selective target identification from simple directed path observations in networks whose standardized edge transports take values in groups nilpotent of class at most two. The paper distinguishes two fundamentally different information requirements: universal determination, represented by dominion membership, and explicit word reconstruction, represented by subgroup membership in the relatively free class-two group. The main construction uses a connected bouquet of directed cycles. On each cycle, the baseline disclosure contains every simple directed path obtained by omitting exactly one edge. For this observation family, we derive the complete arithmetic structure of the gap between determination and reconstruction. If two cycles have effective lengths da and dc, where da and dc are the corresponding cycle lengths minus one, then the cross-cycle determinacy threshold is governed by the least common multiple of da and dc, while explicit reconstruction by a uniform word in the disclosed observations requires the full product da times dc. The resulting gap is controlled exactly by gcd(da, dc). Across the full network, the quotient between the dominion and the observation subgroup decomposes into cyclic factors whose orders are these pairwise greatest common divisors. The principal theorem strengthens this arithmetic result into a complete structural characterization of every possible disclosure subset. After adding one optional direct edge observation per cycle, a prescribed target family is universally determined exactly when every active cycle contributes either its direct observation or its complete long-path bundle. Explicit word reconstruction holds exactly when this local completeness condition is satisfied and, in addition, the directly observed cycles form a vertex cover of the arithmetic conflict graph. This necessary-and-sufficient characterization identifies the entire feasible disclosure family rather than only one optimal policy. It then yields exact minimum-cost disclosure designs under arbitrary nonnegative observation costs. The paper also separates redesign from scratch from incremental audit augmentation after route bundles have already been collected, so sunk costs are not conflated with future disclosure costs. The results are constructive. The manuscript provides explicit reconstruction certificates whenever word reconstruction is feasible, torsion-free witnesses when universal determination fails, finite-group obstructions when determination holds but word reconstruction fails, exact lattice algorithms, and reproducible computational checks. A discrete-gauge interpretation is developed as a secondary kinematic layer. Standardized edge variables are treated as discrete link transports, route products as path holonomies, and local reporting frames as gauge choices. The exact theorem domain is the universal two-step nilpotent category, represented symbolically by the free edge-transport group modulo triple and higher commutators. This is not claimed to be an information-preserving reduction of an arbitrary non-Abelian system, nor does the paper address Yang–Mills dynamics, confinement, or the mass-gap problem. The proposed contribution is the exact simple-path realization of the determinacy-versus-reconstruction gap, its gcd/lcm arithmetic, the complete necessary-and-sufficient characterization of feasible disclosure sets, and the resulting exact audit-design theory.