Telescoping Under Resource-Bounded Observation

Many approximation, estimation, and verification procedures across mathematics, physics,and computation rely on hierarchical refinement: a coarse baseline is progressively correctedby increasingly fine contributions. Such telescoping structures appear in multilevel numericalmethods, operator approximations, quantum simulation algorithms, and learning systemstrained from finite data.In this work, we argue that telescoping is not merely a convenient design pattern, but astructural consequence of resource-bounded observation. We introduce a general framework”Telescoping Under Resource-Bounded Observation (TURBO)”, in which systems are accessedonly through constrained observation models such as noisy queries, finite sampling, or limitedmeasurements. Under mild and broadly applicable assumptions, we establish fundamental lowerbounds and achievability results showing that telescoping strategies are asymptotically optimal.Our results formalize a universal tradeoff between refinement depth, information gain, andobservational cost. We prove that error reduction beyond a characteristic noise floor is impossiblewithout exponentially increasing observational resources, leading to provable saturation andphase-transition phenomena. This framework unifies classical and quantum approximation limits,clarifies the origin of multilevel methods, and provides a foundation for principled stoppingrules and impossibility results in learning, simulation, and verification under partial access.