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

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. The central result — the Non-Observability of Extendability (NEO) theorem — is proved by explicit adversarial construction: for every finite horizon h, two prefixes share an identical horizon projection but differ in extendability, demonstrating that no function of the bounded projection alone can determine global extendability. A complementary certification-depth corollary identifies when the obstruction disappears: if extendability is determined by a bounded local certificate, a sufficiently large operative window recovers it. 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. Machine-checked Lean 4 proofs of the principal results are archived at https://doi.org/10.5281/zenodo.19687799. The foundational projection-theoretic result is established in the companion paper at https://doi.org/10.5281/zenodo.19633241.