When the Canonical Completion Is Wrong: Formalizing and Measuring the Jump in Large Language Models

-cross Abstract: Whether large language models (LLMs) can perform the abductive leap from evidence to a new system of axioms, commonly referred to as a jump, has recently attracted considerable debate. A prominent position holds that LLMs are structurally incapable of such jumps, while recent studies challenge both its mechanism and empirical evidence. One of the main reasons why the debate remains open is the difficulty of defining the jump precisely enough to test it. In this paper, we attempt to develop a formal account of the jump in four steps and measure the second. These steps ask what the default completion of partial data is, when the constraints exclude it, whether the new structure agrees with later observations, and how successive jumps compound. We define a \emph{jump instance} as a finite extension problem whose constraints exclude the canonical completions given by the Kan extensions and leave one correct completion up to renaming. In this setting, a model with a canonical default performs the second step by producing the correct completion under the constraints. We evaluate fourteen models across three certified families. The canonical completion returns once in $13{,}300$ constrained answers across all runs. Several calibrated models also give the correct completion reliably, including three API models that solve $159$ of $162$ primary chain trials, suggesting that they can jump at this step. We further formalize the third and fourth steps, whose empirical evaluation remains future work. We hope our work paves the path for formalizing and measuring the full jump in the future. The code of the paper is available at https://github.com/EEthanShi/kan-jump-test.