The Non-Locality of Extendability: An Impossibility Theorem for Bounded Information Systems, with Applications to Generative Sequential Systems

Forward-case specialization of the Projection Insufficiency Theorem (PIT) for generative sequential systems. Introduces the Non-Observability of Extendability (NEO) theorem and the explicit adversarial divergence-kernel construction — the arithmetic-witness machinery on which the remainder of the program's policy-level impossibility arguments build. Abstract Generative sequential systems — language models, planning agents, and reinforcement learning policies — construct trajectories step by step. At each step, an action is selected based on locally available information, under the implicit assumption that locally valid choices can be assembled into globally valid outcomes. In environments governed by non-local constraints — constraints whose satisfaction depends on the complete trajectory rather than any bounded prefix — this assumption fails structurally: there exist states that are locally valid but admit no globally consistent continuation. This paper identifies and formally proves this structural failure mode for forward-local systems with fixed bounded evaluation horizons under non-local constraints, shows that it persists in general at every finite horizon under bounded-window evaluation while clarifying the constraint classes on which it disappears, and unifies the treatment of the corresponding independently named failure modes across automated planning, reinforcement learning, and natural-language generation. We develop a domain-independent formal framework for generative sequential systems, introducing precise definitions of extendability, horizon projections, and non-extendable commitment. We prove the central result — the Non-Observability of Extendability (NEO) theorem — by explicit adversarial construction: for every finite horizon h we exhibit two prefixes sharing an identical horizon projection but differing in extendability, demonstrating that no function of the bounded projection alone can determine global extendability. Policy-level consequences and domain instantiations follow by corollary. A constructive instantiation is conducted over 30,000 episodes per configuration across 16 configurations (four constraint families × four model architectures), with admissible sets computed exactly at every step by exhaustive enumeration. The NEO theorem establishes that for every finite horizon h, no function of the bounded horizon projection Πh can recover extendability for all prefixes in the adversarial environment 𝓔h. The defeating construction scales with h, so horizon extension shifts the failure boundary but does not empty the class of defeating environments uniformly across unrestricted constraint classes. A complementary certification-depth corollary identifies when the obstruction disappears: if extendability is determined by a bounded local certificate, then a sufficiently large operative window recovers it. The resulting constructive instantiation exhibits the expected pattern across the full constraint-localizability spectrum: the global admissibility model achieves 1.000 task success in every environment, while forward-local policies achieve 0.000–0.007 on the hardest non-local constraint families and 0.000–0.407 across the full constraint spectrum. Failure-time distributions concentrate at terminal steps (88.1% of failures in the final three steps), consistent with the deferred-inadmissibility signature predicted by the construction. The constraint requirement — any system guaranteeing global consistency must incorporate a mechanism whose functional effect is to exclude non-extendable selections prior to commitment — is a necessary condition on information use, not a design preference. Dead-end states in planning, absorbing failure states in reinforcement learning, and delayed-constraint-failure hallucinations in language models instantiate a common abstract form of non-extendable commitment. Progress on these structural failure modes requires architectural change toward constraint filtering, not only training improvement or horizon extension; the complementary question is which constraint classes admit bounded local certification and which do not. Companion Lean 4 formalization: https://doi.org/10.5281/zenodo.19687798 GitHub repository: https://github.com/shawnjason/Non-Locality Related papers in the program: PIT (foundational projection-theoretic result): https://doi.org/10.5281/zenodo.19633241IA (stochastic extension of NEO): https://doi.org/10.5281/zenodo.19688628HAL (language-model specialization using NEO's divergence kernel): https://doi.org/10.5281/zenodo.19715059RLM (admissibility-dynamics framework): 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