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

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.

LLMs Don't Pay for the Jump

Zahavy [2026] argues that Large Language Models, despite their capabilities in induction and deduction, cannot perform the abductive "Jump" that produced Einstein's equivalence principle, and attributes this limitation to the absence of embodied simulation. Zheng-Xin [2026] and Farmer [2026] question whether embodiment is necessary for abduction, pointing to alternative routes to General Relativity and forms of abduction that require no sensorimotor grounding. Max Planck resolved the blackbody radiation problem in 1900. Planck's move to E = hν required no embodied simulation. It was motivated by a mathematical consequence of classical theory, an infinite predicted energy for a finite measured quantity, that could not be physically accepted. We show that neither induction nor deduction could have produced the postulate and argue that its adoption required a coupling between epistemic error and physical cost. We formalize this distinction through thermodynamic coupling and show that fixed-weight transformer inference lacks such coupling, regardless of model scale. This is consistent with empirical results showing that output entropy remains nearly unchanged across tasks with sharply increasing causal difficulty, even as accuracy falls from 100% to 17%. We therefore argue that the missing ingredient in machine abduction may lie deeper than embodiment: a system must have some physical mechanism through which epistemic error becomes costly enough to force revision.

A Minimal $κ$--$τ$ Logic for Risk-Sensitive Abduction

Standard approaches to abductive reasoning can retain multiple candidate explanations, but they do not generally combine explicit compositional cross-hypothesis interaction with an internal, rival-sensitive commitment judgment. This paper argues that in risk-sensitive domains -- where premature commitment carries asymmetric downside costs -- the timing of commitment is itself a governed decision that the inferential apparatus should formally represent. We present a minimal $κ$--$τ$ logical framework built on two primitives: epistemic interaction among hypotheses ($κ$) and a normative commitment threshold ($τ$). Hypotheses may coexist, reinforce or inhibit one another, and form emergent composite explanations, while collapse into committed conclusions is regulated by governance constraints rather than forced by inference alone. The logic is developed in two complementary modes sharing the interaction relation and the governance apparatus: a synthetic mode, in which atomic hypotheses are composed upward into emergent explanations, and an analytic mode, in which complex observed states of affairs are decomposed into causal clusters of latent factors, with commitment governed at both the cluster and the factor level. The framework provides formal machinery for domains in which the distinction between highly likely and commit-worthy is operationally consequential. The $κ$--$τ$ logic is positioned as the symbolic governance layer of a neurosymbolic architecture: its epistemic parameters are naturally estimated by neural components -- semantic embeddings and generative models, as demonstrated in existing computational realizations -- while its normative parameters remain under explicit human governance, yielding transparent and auditable abductive reasoning for deployment in high-stakes settings.

Analytic Abduction: Causal Decomposition and Governed Commitment for Human--AI Coordination

Abductive reasoning operates in two directions. The synthetic mode builds explanations from available hypotheses; the analytic mode, conversely, identifies the latent factors whose interaction accounts for a complex observed state. This paper develops the analytic mode as a non-greedy, risk-sensitive discipline of commitment, in which candidate factors coexist and interact, resolving into committed conclusions only when explicit governance conditions are met. The formal core is the $κ$-$τ$ apparatus: $κ$ encodes the epistemic interaction among hypotheses, and $τ$ sets a commitment threshold calibrated to the decision's stakes. The central contribution is the causal cluster, a structured object recording which latent factors participate in a decomposition, with what weights and interaction structure, together with a two-level architecture (intra-cluster $κ^*$, inter-cluster $κ^{**}$) that guards against causal misattribution. Demonstrated in epidemiological crisis decomposition and adversarial cyber threat analysis, the framework's contribution to human-AI reasoning is the legibility of suspended decomposition as a shared coordination object, providing structural resistance to premature convergence. In practice, the decision-maker is handed not a single imposed answer but the competing explanatory scenarios, weighted by plausibility and paired with the evidence that would resolve between them, so that sound action is possible even before the ambiguity is resolved.

LLMs Can See the Smoke but not the Fire: Evaluating Abductive Reasoning with Elenchos

Large language models (LLMs) excel at pattern recognition and text generation, but their capacity for abductive inference - inferring latent hypotheses that explain observed behavior - remains poorly understood. Here, we introduce Elenchos (named after the Socratic method of cross-examination), a generative evaluation framework that measures abductive reasoning as a structural inverse problem. Given a reference formal system, such as the lambda-calculus, and a potentially mutated counterpart, agents must determine whether a mutation has occurred and infer the rule modifications responsible for the resulting behavioral differences. Evaluating frontier and mid-tier LLMs reveals a consistent detection-attribution dissociation: models often recognize that a system has been altered but struggle to identify the latent mutations causing the observed discrepancies. Performance degrades substantially under interacting mutations, where models frequently recover only a subset of the underlying mutations. Preliminary evidence also suggests diminishing returns from increased inference-time reasoning, with only modest improvements under larger reasoning budgets, though this finding requires further validation.

AGM-like Paraconsistent Partial Meet Abductive Expansion Operation

In his 1996 doctoral thesis, Maurice Pagnucco created the first AGM-like abductive expansion operation. Taking his operation as a basis, as well as a taxonomy -- inspired by Atocha Aliseda -- responsible for highlighting and formalizing the main components of abductive reasoning, the main aim of this paper is to present a new paraconsistent AGM-like abductive expansion operation -- capable of assimilating contradictory explanatory hypotheses without trivialization and the consequent absurd epistemic state -- with its postulates and its transitively relational partial meet construction. To a large extent, the formal development presented in this paper was only made possible by the recent creation of the paraconsistent logic RCbr, an LFI (Logics of Formal Inconsistencies) that establishes properties especially relevant to belief revision contexts, in particular, the ability to be self-extensional -- i.e., to satisfy the replacement property. This is the first of two papers: the paraconsistent abductive expansion operation announced here -- which is part of a new system called AGMpabd -- despite bringing many interesting features, does not assign any relevant epistemic role to the paraconsistent operators of negation and consistency. Only in a second paper will an analogous paraconsistent abductive expansion operation -- which is part of another new system, AGMcircabd -- be enhanced in this direction. Nevertheless, to the best of my knowledge, the operation developed in this paper is the first of its kind in the AGM literature.

Abduction Prover in Isabelle/HOL

Proof assistants based on expressive logics suffer limited automation for proof search, raising the cost of formal verification based on proof assistants. We address this problem by introducing the Abduction Prover for Isabelle/HOL. Given a challenging proof goal, the Abduction Prover constructs a proof script for the goal by identifying useful conjectures using abductive reasoning.