Inconsistency Accumulation in Forward-Local Sequential Policies: A Lower Bound under Delayed Constraints
Shawn Kevin Jason · Zenodo (CERN European Organization for Nuclear Research) · 2026
Stochastic extension of the Non-Locality of Extendability (NEO) theorem with a quantitative inconsistency-accumulation lower bound E[I_N] ≥ N/|U| and a matching positive representational result — the Summary Sufficiency proposition — that closes the impossibility gap. Introduces the admissibility-dynamics framework on which the program's subsequent recursive-scaffolding and empirical-validation results build. Abstract We characterize an architectural fault-line in sequential decision-making under delayed constraints, separating forward-local policies — policies that select actions using only a bounded trailing observation window — from policies equipped with a sequentially updatable extendability-preserving summary state. The separation has two halves. On the negative side, for every finite evaluation horizon h and every forward-local policy class Πh, we construct a family of policy-indexed delayed-constraint environments in which non-extendable commitments accumulate without any policy-independent finite bound. Deterministic policies are forced into one non-extendable commitment per block; stochastic policies incur such commitments at a rate bounded below by N/|U| where |U| is the action-space cardinality, uniformly over the class. The construction uses an explicit reset–choice–silent–certify block that isolates a locally indistinguishable choice point whose consistency consequences lie strictly beyond the policy's evaluation window. In particular, no forward-local class admits uniformly bounded inconsistency propagation across all admissible delayed-constraint environments. On the positive side, any environment family admitting a sequentially updatable extendability-preserving summary state admits a policy that achieves zero accumulated inconsistency. The architectural content of the paper is therefore not the impossibility alone but the identification of the precise representational primitive — an extendability-preserving summary — that closes the gap. Any system that seeks a uniform bound must incorporate a mechanism whose functional effect is to exclude non-extendable continuations prior to commitment. We discuss, as a single bracketed application, the structural relevance of retrieval augmentation, chain-of-thought, constrained decoding, and process supervision. These mechanisms move the operative information state in the direction identified by the theorem, though none by itself guarantees the full extendability-preserving property defined here, and real large-language-model architectures are richer than the abstract policy class to which the theorem strictly applies. A separately reported simulation suite corroborates the structural separation on synthetic constraint families beyond the minimal witness: bounded local policies collapse on non-local, geometric, and temporal constraint families while a constraint-aware summary policy saturates. All discrete-case policy-level results of the paper — Lemmas 1–2, both clauses of Theorem 1, and Proposition 1 — are formalized and machine-checked in Lean 4, including the full measure-theoretic construction of the trajectory probability space, the bridge lemma identifying the conditional expectation of the per-block failure indicator with the policy's per-window commitment probability, and the main integration theorem yielding E_π[I_N] ≥ N/|U| via the tower property. Companion Lean 4 formalization: https://doi.org/10.5281/zenodo.19687093 GitHub repository: https://github.com/shawnjason/Inconsistency-Accumulation Related papers in the program: PIT (foundational projection-theoretic result): https://doi.org/10.5281/zenodo.19633241NEO (forward-case impossibility theorem extended stochastically here): https://doi.org/10.5281/zenodo.19688367HAL (language-model specialization): https://doi.org/10.5281/zenodo.19715059RLM (recursive language models via the admissibility-dynamics framework introduced here): 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