What Can Language Models Inherit from Solomonoff Induction? Limits, Attainable Properties, and What Served Conditionals Reveal

Abstract

A familiar line of reasoning runs as follows: language models are flexible approximators trained under a strictly proper loss, so scale brings them closer to Solomonoff induction, and their predictive success is evidence of that approach. We apply known results from algorithmic information theory, universal prediction and Bayesian consistency. In its uniform form the conclusion holds for no computable model: each has cheap environments it never learns, and every uniformly computable family of total predictors shares one on which each member fails beyond a horizon growing with its index, so ``closer'' can hold only pointwise, one environment at a time. Nor could the offered routes certify even that: finite evaluation cannot certify agreement on rarely reached contexts, perfect corpus fit selects the corpus law as prior, and asymptotic criteria cannot see the constants that carry Solomonoff's guarantee. Our lens is the Kraft inequality read for predictors. Every model has at most the same budget of predictive success, so Solomonoff's predictor, which leaves part of it unspent, differs from a language model not in having more of it but in where it spends it; on mutually singular families this is exact, and only full-mass Bayes mixtures reach the frontier. We then discuss what is inheritable, including in-context learning as amortised Bayesian inference; what survives within a context window; which Solomonoff-like properties a deployed system should have or avoid; and what served conditionals determine. A closing table states what common practices certify and what to report. Tests are developed in a companion paper.