Projection Insufficiency and Trajectory Realization - A Unified Constraint-Based Framework for Bounded Systems
Shawn Kevin Jason · Zenodo (CERN European Organization for Nuclear Research) · 2026
Foundational paper of the Global Admissibility Filtering (GAF) research program. The Projection Insufficiency Theorem isolates a structural obstruction shared by bounded information systems, lossy records, forward-local decision procedures, and closed-interval dynamical models — and is the result that the remainder of the program specializes, extends, and applies. Abstract This paper isolates a structural limitation shared by bounded information systems, lossy records, forward-local decision procedures, and closed-interval dynamical models: any property that depends on a complete trajectory cannot, in general, be recovered from a non-injective projection of that trajectory. The obstruction is indistinguishability. When distinct trajectories produce the same representation, any property that differs across those trajectories is not a function of the representation alone. We 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 globally consistent action selection by policies defined only on bounded local projections. The unification is explicit rather than analogical: each case asks whether a trajectory-dependent property can be determined from a non-injective representation, and in each case the answer is no for the same reason. We then place the theorem in a closed-interval dynamical setting as a concrete instance of the same projection problem rather than as a separate add-on. In that setting, admissibility depends on compatibility with a condition imposed over an entire interval, so admissibility is itself trajectory-dependent and therefore not inferable from local state alone. The conclusion is structural rather than mechanistic: any system that exhibits bounded inconsistency over extended horizons must behave as if globally inconsistent trajectories are excluded prior 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 convergence are no longer guaranteed. A worked example in discrete sequential decision-making makes the theorem explicit and demonstrates that the failure mode it predicts corresponds directly to empirically observed phenomena in planning and language generation. The paper therefore identifies a common projection-theoretic obstruction beneath reconstruction, prediction, and sequential decision-making under bounded information. Companion Lean 4 formalization: https://doi.org/10.5281/zenodo.19687628 GitHub repository: https://github.com/shawnjason/Projection-Insufficiency Related papers in the program: NEO (forward-case specialization): https://doi.org/10.5281/zenodo.19688367IA (stochastic extension): https://doi.org/10.5281/zenodo.19688628HAL (language-model specialization): https://doi.org/10.5281/zenodo.19715059RLM (recursive language models via admissibility dynamics): https://doi.org/10.5281/zenodo.19753549OOL (OOLONG-Pairs empirical companion to RLM): https://doi.org/10.5281/zenodo.20277804SUD (Sudoku-Microscope empirical validation): https://doi.org/10.5281/zenodo.20277939HAM (Hamiltonian-Microscope cross-provider pilot): https://doi.org/10.5281/zenodo.20278073