Projection Insufficiency and Trajectory Realization - A Unified Constraint-Based Framework for Bounded Systems

This paper isolates a structural limitation shared by bounded information systems, lossy records, forwardlocal decision procedures, and closed-interval dynamical models: any property that depends on a completetrajectory cannot, in general, be recovered from a bounded projection of that trajectory. The obstruction isindistinguishability. When distinct trajectories produce the same representation, any property that differsacross those trajectories is not a function of the representation alone. I formalize this as the Projection Insufficiency Theorem. The theorem is direction-agnostic and yields,as temporal specializations, three consequences: incomplete reconstruction of the past from lossy records,non-locality of future extendability from bounded present state, and the impossibility of guaranteeing globallyconsistent action selection by policies defined only on bounded local projections. The unification is explicitrather than analogical: each case asks whether a trajectory-dependent property can be determined from anon-injective representation, and in each case the answer is no for the same reason. I then place the theorem in a closed-interval dynamical setting as a concrete instance of the same projectionproblem rather than as a separate add-on. In that setting, admissibility depends on compatibility with acondition imposed over an entire interval, so admissibility is itself trajectory-dependent and therefore notinferable from local state alone. The conclusion is structural rather than mechanistic: any system that exhibitsbounded inconsistency over extended horizons must behave as if globally inconsistent trajectories are excludedprior to realization. Under a contractive condition, this exclusion admits a constructive fixed-point realization;outside that regime, the need for global admissibility remains, but uniqueness and geometric convergenceare no longer guaranteed. The paper therefore identifies a common projection-theoretic obstruction beneathreconstruction, prediction, and sequential decision-making under bounded information.