Information-Space Dimensionality and the Generative Limits of Large Language Models
Year: 2026
Type: preprint
Abstract: A recurring informal claim holds that a language model cannot generate information that was not already present in its training data. This paper gives the claim a precise form, and finds that the strongest true version of it needs none of the hypotheses usually attached to it. The central result: generation cannot exceed the source. Let Z be any state of affairs in the world, T the training corpus, θ the trained parameters, X the input and Y the output. Because Z → T → θ → Y is a Markov chain, the data processing inequality gives I(Y;Z) ≤ I(T;Z) + H(X) A model's information about the world is bounded by what its corpus carried about the world, plus what its prompt supplied. This holds with no assumption about parameter count, numerical precision, architecture, training procedure, or inference compute. It holds for infinite-precision real weights and for a model trained to global optimality by an oracle. Applying Fano's inequality converts it into the separation statement the informal argument reaches for: a system holding N bits about a domain that requires N + k bits to specify is wrong with probability at least 1 − (N+1)/(N+k). This is the rigorous form of the two-dimensional ant that cannot perceive a third dimension — and it constrains the source, not the model. Unlimited computation does not help. The usual objection is that a system with unbounded compute can derive anything. That is true only of a closure which does not distinguish one system from another, or any from none, and is therefore empty. For a fixed system the claim is false by counting: a deterministic map from m bits reaches at most 2^m outputs however long it runs, so a fraction 1 − 2^(m−n) of an n-bit target space is produced by no input whatsoever. Running longer does not enlarge the image of a function. Enumeration does not escape this either — producing a list of all candidates converts identification into selection of identical size — and randomisation buys coverage of the target space at exactly the rate it loses concentration, leaving the probability of being right unchanged. Two refinements, where the source is not the binding constraint. Capacity Under finite precision, with P parameters of b bits, I(Y;Z) ≤ Pb + H(X), and any target with K(y|x) > Pb + O(1) is unreachable for every parameter setting whatsoever. Dimension With no precision assumption but a locally Lipschitz decoder, the reach has Hausdorff dimension at most P + q and is Lebesgue-null and meagre when P + q < M. A metric-entropy argument shows the two refinements are one bound at different resolutions. What the parameter count actually measures. It bounds retention — the pipe between corpus and weights — and neither what the corpus knew nor what the world contains. So Pb cannot represent the information dimensionality of a training set, because it is not a property of the training set at all. The quantity that can is I(T;Z), and it is a function of the domain being asked about rather than a single number: there is no one N for a corpus. The paper then prices the channels that genuinely raise these bounds — retrieval by its novelty relative to the weights, world-coupled tools by their measurements, human selection by a few hundred bits — and proves that closed-loop self-training raises them by nothing beyond the bits its own selection policy injects. What this does not claim. Every bound here constrains algorithmic information K, never time-bounded complexity K^t. The deductive closure of a small axiom set already contains every theorem of mathematics, and the entire difficulty lies in finding them. Since essentially all mathematical and scientific value is a K^t phenomenon, none of this implies that language models cannot produce novel results. It implies something narrower and more defensible: they cannot manufacture bits about a world that neither their corpus nor their input contained. A closing section adjudicates the formulations of the claim one meets in the wild: six are theorems as stated, five require correction before they can be believed, and none is simply false — which is why the claim is so persistent. Every correction but one turns out to be downstream of a single substitution, the parameter count, which is a bridge to familiar machine-learning vocabulary and is not part of the claim. 29 pages. Self-contained: background boxes cover the borrowed machinery (entropy and the data processing inequality, Fano's inequality, Kolmogorov complexity, Lipschitz maps, Hausdorff dimension, covering numbers) for readers without those definitions to hand. LaTeX source included.
Author: Kaplan, Mehmet
Language: English
Topic: Topic Modeling
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204670048