# The Information Shadow: Measuring Structural Limits on What Language Models Can Learn

arXiv: https://arxiv.org/abs/2607.18305 · HTML: https://mlsanity.com/paper/2607.18305/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 33 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 21 of 21)

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Right for the Wrong Reasons: Diagnosing Syntactic Heuristics in Natural Language Inference](https://mlsanity.com/paper/1902.01007/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Underspecification Presents Challenges for Credibility in Modern Machine Learning](https://mlsanity.com/paper/2011.03395/) (2020)
- [Assessing the Ability of LSTMs to Learn Syntax-Sensitive Dependencies](https://mlsanity.com/paper/1611.01368/) (2016)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Theoretical Limitations of Self-Attention in Neural Sequence Models](https://mlsanity.com/paper/1906.06755/) (2019)
- [Neural Networks and the Chomsky Hierarchy](https://mlsanity.com/paper/2207.02098/) (2022)
- [Deep learning generalizes because the parameter-function map is biased towards simple functions](https://mlsanity.com/paper/1805.08522/) (2018)
- [Predictive Multiplicity in Classification](https://mlsanity.com/paper/1909.06677/) (2019)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [What Formal Languages Can Transformers Express? A Survey](https://mlsanity.com/paper/2311.00208/) (2023)
- [Overcoming a Theoretical Limitation of Self-Attention](https://mlsanity.com/paper/2202.12172/) (2022)
- [On Linear Identifiability of Learned Representations](https://mlsanity.com/paper/2007.00810/) (2020)
- [BERTs of a feather do not generalize together: Large variability in generalization across models with similar test set performance](https://mlsanity.com/paper/1911.02969/) (2019)
- [Failures of Gradient-Based Deep Learning](https://mlsanity.com/paper/1703.07950/) (2017)
- [Does syntax need to grow on trees? Sources of hierarchical inductive bias in sequence-to-sequence networks](https://mlsanity.com/paper/2001.03632/) (2020)
- [Provable Limitations of Acquiring Meaning from Ungrounded Form: What Will Future Language Models Understand?](https://mlsanity.com/paper/2104.10809/) (2021)
- [Inoculation by Fine-Tuning: A Method for Analyzing Challenge Datasets](https://mlsanity.com/paper/1904.02668/) (2019)
- [Provable Advantage of Curriculum Learning on Parity Targets with Mixed Inputs](https://mlsanity.com/paper/2306.16921/) (2023)

## Abstract (arXiv)

Some limits on what language models know are not gaps in data coverage but structural properties of learning from text. We introduce the information shadow: the region of phenomena that a text-trained learner cannot acquire regardless of scale, comprising (I) structures language cannot express, (II) functions that are statistically non-identifiable from the training distribution, and (III) functions that are representable but unreachable by gradient-based training. We give each type a probe that is decisive because the premise of the shadow is, in that setting, provable. For Type I, Language Compression Residuals compare a text learner, which sees only a lossy text-like encoding of the signal, against a full-signal learner, which sees the underlying signal directly. The text learner sits at a computable expressibility ceiling while the full-signal learner pulls away by a gap that stays flat across 300x more data, so the deficit is a property of the channel, not of training. For Type II, the Counterfactual Distinction Test trains models on data exactly consistent with two incompatible rules. Across a provable string task and a language-like agreement task, behavior on counterfactuals is set by the model's inductive bias, while 5% disambiguating data steers the learned rule bidirectionally to either target (r = +/-1.0, p < 1e-10). For Type III, Basin Escape Mapping exhibits a function that is representable at 100% (by hand construction) yet reached 0% of the time by standard training and instantly from a nearby initialization, with width scaling providing no help (p = 1.6 x 10^-14). Each effect is isolated by a control that rules out a capacity or modality artifact. We release the probe suite and discuss implications for benchmark design, capability auditing, and shadow-aware uncertainty.