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当规范补全出错时：大语言模型中“跳跃”的形式化与度量

文章背景与核心概要

大语言模型（LLM）是否能够执行“溯因跳跃”（即从经验证据过渡到全新的公理系统）一直是学界热烈争论的焦点。一些观点认为大语言模型在结构上不具备这种跳跃能力，而另一些研究则对其机制和证据提出了挑战。由于此前缺乏对“跳跃”的严格形式化定义以及衡量双方观点的量化指标，这一争论迟迟难以定论。

为了填补这一空白，本文提出了一套四步法形式化框架，对大语言模型中的“跳跃”进行了严格定义和度量。通过引入Kan扩展定义默认补全，并结合机器检查的证书，作者在四个前沿大模型上对9个经认证的实例进行了测试。研究发现，在所有受约束的试验中，模型成功抛弃错误默认补全的比例达到100%。这表明大语言模型推理的瓶颈并不在于“跳跃”动作本身，而在于生成必要的约束条件或发明底层框架。

当规范补全出错时：大语言模型中“跳跃”的形式化与度量 (When the Canonical Completion Is Wrong: Formalizing and Measuring the Jump in Large Language Models)

摘要 (Summary)

Whether Large Language Models (LLMs) can execute an "abductive jump"—transitioning from empirical evidence to a novel system of axioms—has been a subject of intense debate. This paper provides a formal framework to settle the discussion by defining the "jump" through a four-step process: determining default completions (via Kan extensions), identifying when abandoning the default is forced and correct, and evaluating how successive jumps compound. Evaluating nine certified instances across four frontier models, the authors discovered that models successfully abandon incorrect default completions every time. Consequently, the bottleneck in LLM reasoning does not lie in making the jump itself, but rather in generating the necessary constraints or inventing the underlying framework.

大语言模型（LLM）是否能够执行“溯因跳跃”——即从经验证据过渡到一个新的公理系统（通常被称为“跳跃”）——近期引发了激烈的争论。一种突出的观点认为，LLM 在结构上不具备这种跳跃能力，而近期的研究则对这一机制及其证据提出了质疑。然而，由于该领域仍然缺乏对跳跃的形式化定义以及用于检验双方观点的度量标准，这一争论至今难以解决。

元数据 (Metadata)

  • arXiv ID: arXiv:2608.26187 [cs.CL]
  • 作者 (Authors): Dai Shi, Xiaoyu Li, José Miguel Hernández-Lobato
  • 提交时间 (Submitted): 2026年8月22日
  • 学科领域 (Subjects): 计算与语言 (cs.CL); 人工智能 (cs.AI); 机器学习 (cs.LG); 计算机科学中的逻辑 (cs.LO)
  • 代码库 (Code Repository): GitHub - kan-jump-test

摘要 (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 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.

在本文中，我们分四个步骤对“跳跃”进行了形式化阐述，并对其中的第二步进行了度量。这几个步骤分别探讨了：部分数据的默认补全是什么、在何时放弃该默认是强制且正确的、这种放弃在何时是正确的，以及连续的跳跃是如何复合的。具体而言，我们将一个跳跃实例定义为一个有限扩展问题，并配有机器检查的证书，该证书证明了正确补全的存在性、在重命名意义下的唯一性，且该补全不同于数据的规范补全（canonical completion）。规范补全由左、右Kan扩展给出，同时这也是模型在无约束条件下产出的结果，因此它充当了默认选项。

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.

我们证明了跳跃实例是适定（well-posed）的，并建立了一个族定理（family theorem），能够在无需枚举的情况下认证具有无限难度的实例。我们进一步将跳跃何时正确以及连续跳跃如何复合进行了形式化。最后，我们在9个经认证的实例和4个前沿模型上进行了测量。在所有248次受约束的试验中，Kan默认率均为零，这表明模型确实在此步骤中完成了跳跃，并且每次都抛弃了被排除的默认选项。在更高难度下的失败源于推理预算的耗尽或约束错误，而绝非回归到默认选项。这些结果表明，第二步并不是瓶颈。如果人们所争论的无能为力确实存在，那么它也存在于生成约束或发明框架的过程之中。

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.

附加资源 (Additional Resources)