A Mathematical Theory of Understanding

Large language models are trained primarily on human-generated data and feedback, yet they exhibit persistent errors arising from annotation noise, subjective preferences, and the limited expressive bandwidth of natural language. We argue that these limitations reflect structural properties of the supervision channel rather than model scale or optimization. We develop a unified theory showing that whenever the human supervision channel is not sufficient for a latent evaluation target, it acts as an information-reducing channel that induces a strictly positive excess-risk floor for any learner dominated by it. We formalize this Human-Bounded Intelligence limit and show that across six complementary frameworks (operator theory, PAC-Bayes, information theory, causal inference, category theory, and game-theoretic analyses of reinforcement learning from human feedback), non-sufficiency yields strictly positive lower bounds arising from the same structural decomposition into annotation noise, preference distortion, and semantic compression. The theory explains why scaling alone cannot eliminate persistent human-aligned errors and characterizes conditions under which auxiliary non-human signals (e.g., retrieval, program execution, tools) increase effective supervision capacity and collapse the floor by restoring information about the latent target. Experiments on real preference data, synthetic known-target tasks, and externally verifiable benchmarks confirm the predicted structural signatures: human-only supervision exhibits a persistent floor, while sufficiently informative auxiliary channels strictly reduce or eliminate excess error.

Machine learning is usually evaluated after an evidence interface has been fixed. A dataset, sensor suite, query language, action set, or experimental protocol determines which observations can be obtained, and learning is judged by what it extracts from them. We study a complementary capability. A learner's state can determine which evidence-generating experiments it can reliably realise under bounded resources, even when primitive affordances remain fixed. We call this learner-relative experiment family its effective epistemic reach, and use Wide Learning for task-relevant learning-induced changes in that family.We formalise effective reach relative to learner state, deployment budget, reliability threshold, and evaluation distribution. In a controlled construction, two hidden worlds have exactly the same public observation law. An informative diagnostic exists in a fixed five-primitive substrate. Before calibration, one address attempt realises it with probability at most $2^{-10} = 1/1024$, below a pre-specified 0.95 threshold; after calibration, held-out realisation is 1. Public-channel total variation is 0, whereas the realised diagnostic has total variation 1, and sealed binary risk moves from approximately 1/2 to 0. The construction establishes that learning can change effective epistemic reach even when primitive affordances and deployment resources are held fixed. It opens a complementary evaluation question for learning systems: not only what they infer from available evidence, but what informative evidence experience teaches them to bring within reach.

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.