The Need for an External Observer Formalizing the Sufficiency Gap: A Mathematical Extension of Mixture Identifiability and Contextual Grounding in Sequence Models

Language models trained on observed sequences are often described as learning the conditional distribution of the next token given previous tokens. This description is only conditionally correct. A model trained on realized token trajectories does not observe full conditional laws; it receives sampled continuations. Moreover, real language generation is conditioned not only on previous words but also on non-textual circumstances: facts, events, intentions, goals, beliefs, social context, and task-specific constraints. This paper distinguishes three objects that are often conflated: the full conditional language process conditioned on latent circumstances, the marginal text-only process obtained by integrating those circumstances out, and the model-induced distribution learned from finite observed corpora. The paper argues that interpreting model training as estimating the marginal text-only law requires strong assumptions of stationarity, representativeness, and ergodicity, assumptions that are standard in statistical estimation but problematic when applied to heterogeneous language corpora. Even if these assumptions hold, the marginal text-only law is useful only when the observed prefix is an approximately sufficient statistic for the latent circumstances relevant to continuation. In information-theoretic terms, usefulness requires that the residual conditional mutual information between the next token and the omitted circumstances, given the observed text, be small. The paper then extends this argument to heterogeneous training corpora. Finally, the paper interprets Retrieval Augmented Generation (RAG) and tool use as conditional sufficiency devices.

Recent benchmarks reveal that despite strong reasoning capabilities, large language models (LLMs) still struggle to faithfully apply complex contextual knowledge. These failures are often not wholesale reasoning collapses: in context-rich tasks, models may follow the central reasoning path while missing peripheral, persistent, or format-sensitive requirements.

Retrieval-augmented generation promises to ground language model outputs in external evidence, yet the field has no reliable way to verify whether retrieved context actually governs generation -- a prerequisite for any high-stakes deployment. The standard assumption, that context-consistent output implies context-governed output, breaks when the retrieved document overlaps with the model's pretraining data: the model can produce faithful-looking text entirely from parametric memory, and both pathways yield indistinguishable output. We name this failure the attribution blind spot and introduce Computational Reality Monitoring (CRM) to address it. CRM operationalizes a principle adapted from cognitive science's reality monitoring framework: comparing internal representations with and without context reveals membership-conditioned representational divergence that output-level monitors systematically miss. CRM does not certify which source an individual generation used; it detects whether pretraining exposure leaves a measurable internal trajectory signature, establishing a necessary substrate for source attribution. Across nine model variants spanning three families, this divergence concentrates in architecture-specific layer patterns, receives converging support from block-level noise intervention, and generalizes across tasks and datasets while collapsing on domain-confounded benchmarks. The attribution blind spot is measurable and partially addressable: internal representations carry a diagnostic signal invisible at the output level, establishing a foundation for systems whose internal awareness of evidence provenance governs their external behavior.