Finite-Budget Structural Identifiability under Bounded Observation

Structural identification is often discussed as if richer observation were automatically more informative. This paper asks a different question: when observation, storage, and computation all consume finite resources, which observational scale supports the most reliable distinction between competing structural regimes? The proposed framework treats observational scale as part of the inference problem. Each scale determines both what aspects of a process become observable and how long the process can be monitored under a fixed total budget. The paper first provides an exact information-theoretic characterization of distinguishability based on the complete probability laws of the observed trajectories. This establishes the fundamental benchmark achievable by an unrestricted statistical test, independently of computational implementation. Because complete trajectory laws are rarely available in practice, the paper also develops an operational sufficient certificate based on observable novelty: the number of distinct observable states encountered after a scale-dependent structural latency. The resulting analysis makes explicit how reliable identification depends on four interacting factors: accumulated separation between regimes, temporal dependence, structural latency, and the cost of observation at a given scale. A central distinction is maintained throughout: failure of the novelty-based certificate does not establish general indistinguishability. It only shows that this particular observable statistic is insufficient under the available budget. Genuine impossibility claims require analysis of the complete observed laws. The paper also proves a refinement safeguard: if one representation contains all the information available at a coarser scale and can be observed for the same horizon at no additional cost, optimal identification cannot become worse. Consequently, a finite optimal observational scale cannot arise from resolution alone. It must be generated by explicit resource costs, non-nested representations, or restriction to a lossy statistic. The framework yields both exact and novelty-certified identifiable scale sets, together with a resource-aware criterion for practical scale selection. Potential applications include search, learning, adaptive control, scientific measurement, and bounded AI evaluation. In AI settings, higher-resolution behavioral traces may improve local visibility while reducing the number or duration of evaluated trajectories because instrumentation, simulation, storage, and review are costly. The paper does not claim a general impossibility theorem for scalable oversight; it isolates a prior component of that problem: resource-aware observability. Version 4.0 is the first substantive revision of the work. It preserves the original research question while correcting the earlier concentration argument, replacing a fixed novelty-gap condition with cumulative window-dependent separation, clarifying the relation between exact and certificate-based identifiability, and qualifying the conditions under which a maximal identifiable scale exists.