AI has its own proving style: reproving all 465 propositions of Euclid's Elements, AI runs match each other at 0.68 versus 0.37 with humans
Synopsis
Three large language models independently reproved all 465 propositions of Euclid's Elements, and their citation choices differ systematically from Heath's human-proof baseline. Mean pairwise Jaccard similarity among three Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs, so the difference is stable and reproducible. Its concrete signature is that about 40% of AI citations point to foundational elements such as definitions, common notions and postulates, against only 13% in human proofs.
Interpretation
Large language models develop a recognizable citation style in mathematical proving that differs systematically from human mathematicians' choices. Earlier observations of AI-human proof differences stayed at the level of verbosity, dwelling on local details and preferring procedural strategies, without answering whether these reflect stable regularities in how AI draws on mathematical knowledge. Mean pairwise Jaccard similarity among three independent Claude Opus 4.6 runs is 0.68, nearly twice the 0.37 between AI and human proofs; across all categories, 69.5% of propositions receive the same classification in all three runs.
The core of this style is rebuilding arguments from foundational elements rather than directly citing later, more established results. In the human network, 86.8% of citations point to theorems, with definitions, common notions and postulates together at only 13.2%; in AI proofs theorem citations fall to about 60%, while the three foundational types make up about 40%. Opus proofs have a mean citation distance of 111.2 and median 61, versus 86.3 and 38 for human proofs; AI citation distances remain longer in the theorem-only network, and Fable 5.1 and GPT-6 Astra agree with Opus.
Most of AI's extra citations fall within the upstream dependencies of the human proof, indicating both sides often enter the same logical structure at different levels. On path-different D-class propositions, AI-only and human-only citations were examined separately to test whether AI merely retraces the steps needed to derive mature results. Among D-class propositions, a mean 47.0% of AI-only citations appear in the ancestor set of human citations, and 34.5% when restricted to ancestors of human-only citations; Fable 5.1 gives 52.7% and 42.7%, GPT-6 Astra 48.2% and 39.0%.
Perspective
The demonstrated scope is the 465 propositions of Euclid's Elements, with Heath's English proofs as the human baseline and proofs written in natural language; formal proofs are not addressed.
Prompt changes barely affect the ground-up tendency, but whether fine-tuning, citation preferences in training objectives or a modified proof-generation harness can steer this style remains to be determined by further experiments; Fable 5.1's path-difference rate did not show a statistically reliable pattern because it produced only 34 D-class proofs.