LANTERN: Illuminating Hidden Mathematical Knowledge in Language Models
Pavel Tikhonov, Elena Tutubalina, Ivan Oseledets, Dmitry I. Ignatov, Mikhail Seleznyov
cs.CL
2026-09-26
A classifier over Qwen3 hidden states ranks all 50M OEIS sequence pairs; agents then propose and verify relations, yielding 62 verified in 8 hours, 4 absent from literature.
Language models can now close olympiad problems and resolve items from curated open-problem lists, but a human still picks what to work on. The paper treats this as the real bottleneck of AI-assisted mathematics. Existing systems draw candidates from statements already written in papers, from missing edges of a knowledge graph, or from the model's own generation inside a topic. LANTERN uses a different source: the assumption that pretraining left mathematical knowledge encoded in hidden states, and that this can be read out to rank which pairs of objects are worth investigating.
The test bed is the OEIS, chosen for three properties: well-defined objects, a record of relations people noticed (cross-references), and cheap verification, since any exact claim about two sequences can be checked against stored terms.
Four stages:
The main configuration's funnel: 50M pairs to 500, 118, then 44, and all 44 survived verification. Across three configurations: 62 verified statements. A content screen dropped reparametrizations and definitional consequences, leaving 13, of which 9 are informative or insightful; 4 appear neither in the OEIS nor in a targeted literature search. The whole run took under 8 hours, with 8 minutes 20 seconds to score all 50 million pairs.
| Control | Activation classifier | Baseline |
| Topic-matched ROC-AUC (793 new links) | 0.57-0.58 | Surface text 0.51 (chance) |
| 42 substantive new links, AUC | 0.67-0.69 | Surface text 0.61 |
| 2,379 entries created after the cutoff | 0.88 | 0.89 on older entries |
The controls are the strongest part of the paper. The text baseline uses the same pairwise classifier with only the input representation swapped, so surface text explains the topic but not which entries within a topic are related. On entries created after the model's knowledge cutoff the classifier holds at AUC 0.88 versus 0.89, ruling out page memorization. Asking the model directly to rate relation depth on a 0-9 scale recovers the same signal (median percentile 88.5 versus 88.0 for the classifier), so the information sits in the model, not in the training labels; the classifier's advantage is throughput.
Two of the four novel finds: live-cell counts of the 2D Fredkin replicator are reconstructed exactly from the 1D Rule 150 sequence via g(k) = f(k)² − f(k−2)² assembled over binary runs; and the Rogers-Ramanujan continued fraction to the fifth power times a theta quotient squared is the generating function of 5-core partitions.
Problem selection is a long-discussed gap in AI mathematics, and this is a cheap, reproducible template: hidden states as prior, recorded relations as weak supervision, executable verification as referee. Any base with defined objects, recorded relations, and machine-checkable claims is in principle a candidate. For interpretability work it is another data point that residual streams carry structured, exploitable knowledge, put to work rather than merely probed.
The yield deserves honesty: 49 of the 62 verified statements were screened out, and the genuine new knowledge is 4 relations. The metric that matters for a problem-selection system is cost per real discovery, and the paper reports wall-clock only, not API cost.
The authors name two: OEIS is an unusually structured test bed (human-written definitions, formulas, stored terms, explicit cross-references, and a corpus biased toward frequently referenced sequences), so transfer to less formal objects or bases without explicit links is untested; and only one pretrained model family, Qwen, was evaluated.
Close reading adds more. Appendix D concedes the top of the ranking is fragile: two probes differing only in which layers feed the PCA block correlate at 0.5 over 500,000 random pairs yet share a single pair in their top 100, so the head of the funnel is sensitive to implementation detail. Appendix F states that all derivations were produced by a language model and have not been checked by a mathematician. An N2 label means a targeted search found nothing, not priority. The S/N grading itself was done by models, with full re-reading only at the gold-label stage.