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.
Dai Shi, Xiao-Yu Li, José Miguel Hernández-Lobato· 0 citations
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 its evidence. However, the debate remains difficult to settle, since the field still lacks a formal definition of the jump and a measure to test either side. In this paper, we develop a formal account of the jump in four steps and measure the second. The steps ask what the default completion of partial data is, when abandoning it is forced, when the abandonment is correct, and how successive jumps compound. Specifically, we define a jump instance as a finite extension problem with a machine-checked certificate that a correct completion exists, is unique up to renaming, and differs from the canonical completion of the data. The canonical completion is given by the left and right Kan extensions and is also what models produce without constraints, so it serves as the default. We prove that jump instances are well-posed and establish a family theorem that certifies instances of unbounded difficulty without enumeration. We further formalize when a jump is correct and how successive jumps compound. Finally, we run the measurement on nine certified instances and four frontier models. The Kan-default rate is zero in all 248 constrained trials, so the models do jump at this step and abandon the excluded default every time. Failures at higher difficulty stem from exhausted reasoning budgets or constraint errors, never from reverting to the default. These results indicate that the second step is not the bottleneck. If the disputed incapacity is real, it lies in generating the constraints or inventing the framework. Code can be found at: https://github.com/EEthanShi/kan-jump-test.
Dai Shi, Xiaoyu Li, José Miguel Hernández-Lobato· 0 citations
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