{"schema":"postcutoff/event@1","as_of":"2026-10-13T07:38:00+02:00","url":"https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/","md":"https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-27-erdos-993-forest-independence-polynomials-unimodal","date":"2026-09-27","date_precision":"day","short_title":"Erdős Problem #993 (1987)","deck":"Independent-set sequences of all forests claimed unimodal; a second proof and its Lean 4 formalization are attributed entirely to AI agents (Zhang, Li, Vallier)","takeaway":"A 1987 question of Alavi, Malde, Schwenk and Erdős asks whether the numbers of independent sets of each size in a tree or forest always rise and then fall. Tong Zhang and Wei Li claimed a computer-assisted proof on 27 Sept 2026 with help from GPT-6 Astra. A second proof (arXiv, 6 Oct) says AI systems produced all of its new mathematics, its Lean proof and most of its text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.07943: Unimodality of Forest Independence Polynomials (Li, Vallier, Zhang; second proof)","url":"https://arxiv.org/abs/2610.07943","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Zenodo: Zhang & Li, Unimodality of Forest Independence Polynomials (first proof, 27 Sept 2026)","url":"https://doi.org/10.5281/zenodo.22999166","type":"paper","group":"primary","domain":"doi.org"},{"n":3,"title":"Lean 4 formalization (selfreferencing/erdos993-lean)","url":"https://github.com/selfreferencing/erdos993-lean","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Fang, Lu, Nevo, Yao & Zheng, Unimodality of independence polynomials for sufficiently large forests (arXiv 2609.20961)","url":"https://arxiv.org/abs/2609.20961","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Erdős Problem #993 (erdosproblems.com)","url":"https://www.erdosproblems.com/993","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":6,"title":"Erdős Problem #993: proof claims thread","url":"https://www.erdosproblems.com/forum/thread/993/proof-claims","type":"discussion","group":"reactions","domain":"erdosproblems.com"}],"official":4,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Northwestern Polytechnical University","University of Toledo","OpenAI","Anthropic"],"title":"Erdős Problem #993 (1987): independent-set sequences of all forests claimed unimodal; a second proof and its Lean 4 formalization are attributed entirely to AI agents (Zhang, Li, Vallier)","summary":"On 27 Sept 2026 Tong Zhang and Wei Li (Northwestern Polytechnical University, Xi'an) deposited on Zenodo a computer-assisted proof that the independence sequence of every finite forest is unimodal, the 1987 question of Alavi, Malde, Schwenk and Erdős listed as Erdős Problem #993. Their record says \"OpenAI's GPT 6.0 Astra assisted with the computational work; the authors provided the overall proof strategy.\" On 6 Oct 2026 Wei Li, Kevin Vallier (University of Toledo) and Tong Zhang posted arXiv 2610.07943 with a second proof and a Lean 4 formalization. It states: \"AI systems produced the new mathematical arguments and formal proofs reported here and wrote most of the text\", naming Claude Opus 5.5, Claude Fable 5.1 and several OpenAI models. Both are unrefereed; the arXiv paper calls itself a provisional manuscript shared \"for examination by the community\".","key_facts":["Problem: for a graph G let i_k(G) be the number of independent vertex sets of size k. Alavi, Malde, Schwenk and Erdős (1987) showed the sequence need not be unimodal for general graphs and asked: 'For trees (or perhaps forests), is the vertex independence sequence unimodal?' It is Problem #993 on erdosproblems.com","Why it was hard (paper): the stronger property, log-concavity, fails. Levit and Mandrescu conjectured it in 2004; Kadrawi, Levit, Yosef and Mizrachi found two trees on 26 vertices that are not log-concave","17 Sept 2026: Fang, Lu, Nevo, Yao and Zheng (arXiv 2609.20961) proved unimodality for all forests with at least N_0 vertices, for an unspecified constant N_0. Their proof claim on erdosproblems.com says 'using Odin Automatic AI Research Agent'","27 Sept 2026: Zhang and Li, 'Unimodality of Forest Independence Polynomials', Zenodo doi 10.5281/zenodo.22999166. Abstract: 'We reduce the large-order threshold to 59 vertices and extend the finite-order coverage through 60 vertices; the overlapping ranges complete the proof. OpenAI's GPT 6.0 Astra assisted with the computational work; the authors provided the overall proof strategy.'","Zenodo update of 30 Sept: 'The arXiv submission was withdrawn on 28 September 2026, before it was publicly announced. We are revising the manuscript to make its exposition clearer and easier to understand.'","6 Oct 2026: Li, Vallier and Zhang, arXiv 2610.07943 [math.CO], 81 pages, 'Draft, version 2.2': a second proof. Forests with at most 24 vertices by exact counting; forests with at least 25 vertices by one moment argument on the hard-core model. 'No forest is enumerated.'","AI disclosure (abstract): 'AI systems produced the new mathematical arguments and formal proofs reported here and wrote most of the text, under the direction described in Section 9. The paper has not undergone external peer review'","Section 9: 'The project account attributes all of the new mathematics of this paper and all of its formal proofs to AI systems in a project directed by K. Vallier. The systems were Claude agents, running Anthropic's models Claude Opus 5.5 and Claude Fable 5.1; two OpenAI Codex agents, called Sol and Astra in the project, which over the period ran the OpenAI models GPT-5.6 Sol, GPT-6 Astra in Codex mode, GPT-6 Sol and GPT-6.1 Sol; and ChatGPT conversations with the OpenAI model GPT-6 Astra in Pro mode'","Human roles per Section 9: Zhang and Li gave the earlier proof whose decomposition, binomial-mixture reduction and several inequalities the new proof reuses; Zhang wrote the introduction, Li reviewed it, Vallier directed the AI project. The section adds: 'This attribution does not assert that every author has reviewed the complete integrated manuscript.'","Timeline per Section 9: the moment argument was developed on 27–29 Sept 2026, the proofs by hand on 28–30 Sept, the final route fixed on 1 Oct and the last inequality produced on 1–2 Oct","Lean 4 (Mathlib): final theorem erdos993_v22_final : Erdos993Statement, with 'no premise'; code at github.com/selfreferencing/erdos993-lean. The paper notes the use of compiled evaluations; Zhang and Li's Zenodo note on the earlier release v1.0-claim says the large-order part 'uses native_decide and therefore additionally relies on Lean's compiler and runtime'","erdosproblems.com (read 13 Oct 2026) lists two proof claims for #993 (Fang et al., 21 Sept; Tong Zhang 'using GPT6.0 astra', 27 Sept) under the notice 'Appearing on this page is no guarantee of proof correctness'. The site froze new proof claims and removed status labels on 6 Oct"],"key_numbers":[],"tags":["math","combinatorics","graph-theory","erdos-problems","independence-polynomial","unimodality","lean","gpt-6-astra","claude-opus-5-5","claude-fable-5-1","ai-generated-proof"],"science":{"field":"mathematics","subfield":"combinatorics (graph polynomials)","problem":"Erdős Problem #993 (Alavi, Malde, Schwenk, Erdős 1987): is the independent set sequence of every tree or forest unimodal?","result":"Claimed: yes for every finite forest. Two proofs: Zhang–Li (27 Sept 2026, computer-assisted, GPT-6 Astra for computations) and Li–Vallier–Zhang (6 Oct 2026, new arguments and Lean 4 proof attributed to AI agents).","open_since":"1987","ai_system":["Claude Opus 5.5","Claude Fable 5.1","GPT-6 Astra","GPT-5.6 Sol","GPT-6 Sol","GPT-6.1 Sol"],"human_role":"First proof: human strategy with GPT-6 Astra for computations. Second proof: AI-generated per the paper (all new mathematics, the Lean proof and most of the prose), in a project directed by Kevin Vallier and built on Zhang and Li's reductions","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 formal proof of the final statement (uses compiled evaluations, so the Lean compiler is trusted as well as the kernel); unrefereed provisional manuscript; no independent expert review found"},"status":"pending","shock":"A 39-year-old conjecture got a partial proof, a full proof and an AI-written second proof with a Lean formalization within three weeks.","short":"Erdős #993: forest independence sequences are unimodal"},"body_md":"## What happened\n\nAn independent set in a graph is a set of vertices with no edge between any two of them. Count the independent sets of size 0, 1,\n2, … and you get the graph's independence sequence. In 1987 Alavi, Malde, Schwenk and Erdős showed that for general graphs this\nsequence can go up and down in any pattern, and asked whether for trees and forests it always rises to a peak and then falls\n(unimodality). The question became one of the best-known open problems about graph polynomials. A natural route, proving the\nstronger log-concavity, was closed when non-log-concave trees on 26 vertices were found.\n\nIn September 2026 the problem moved quickly. On 17 Sept, Fang, Lu, Nevo, Yao and Zheng proved the statement for all sufficiently\nlarge forests. On 27 Sept, Tong Zhang and Wei Li claimed the full result: large forests (59 vertices and up) by a probabilistic\nreduction, small ones (up to 60) by structural inequalities with exact computer certificates. They credit GPT-6 Astra with the\ncomputational work and themselves with the strategy. They withdrew the arXiv submission the next day to rewrite the exposition\nand left the manuscript on Zenodo.\n\nKevin Vallier (University of Toledo) then ran a project in which Claude and OpenAI Codex agents built a different argument on top\nof Zhang and Li's reductions and formalized it in Lean 4. The resulting paper, posted on 6 Oct with all three as authors, is\nunusually explicit: it attributes \"all of the new mathematics of this paper and all of its formal proofs\" to AI systems, lists the\nmodels, dates each step between 27 Sept and 2 Oct, and says that most of the prose was written by Claude agents. It also says the\nauthors are rewriting the manuscript before journal submission and that not every author has necessarily reviewed the whole text.\n\n## Why it matters\n\nIt is an Erdős problem with a long literature, not an obscure one, and the paper describes a mode of work that goes beyond\n\"AI-assisted\": agents of two companies' models produced the arguments, checked one another's certificates and wrote the formal\nproof, with a human directing. The Lean theorem gives the claim a firm anchor that most AI-written proofs of 2026 lack, with the\ncaveat that parts of it rely on compiled evaluation. The human-readable proof is 81 pages and by the authors' own account still\nprovisional.","disputed":[{"label":"Verification","text":"Lean 4 formal proof of the final statement (uses compiled evaluations, so the Lean compiler is trusted as well as the kernel); unrefereed provisional manuscript; no independent expert review found","sources":[]}],"related":[{"id":"2026-10-06-erdos-problems-site-freezes-proof-claims","url":"https://postcutoff.com/e/2026-10-06-erdos-problems-site-freezes-proof-claims/","date":"2026-10-06","date_precision":"day","short_title":"erdosproblems.com freezes proof claims and drops 'open/solved' labels and solver credits after a wave of unexplained AI proofs","deck":null,"takeaway":"erdosproblems.com was where AI-for-math claims were counted and disputed, from the GPT-5 controversy of October 2025 to the 2026 waves of GPT-6 Astra and Claude results.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-12","orgs":["erdosproblems.com"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-13","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument","deck":"Mathematicians race to simplify and extend it","takeaway":"Erdős–Sós is one of the best-known conjectures in extremal graph theory, and this is among the clearest cases of an AI finding a genuinely new, short idea that experts call \"ingenious\" and \"surprising\".","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":9,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Epoch AI"]},{"id":"2026-07-24-rota-flats-unimodality-disproved","url":"https://postcutoff.com/e/2026-07-24-rota-flats-unimodality-disproved/","date":"2026-07-24","date_precision":"day","short_title":"Rota's 1970 unimodality conjecture for matroid flats disproved, with ChatGPT 5.6 Pro suggesting a key construction","deck":null,"takeaway":"Rota's conjecture was one of the best-known open problems about matroids.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-13","type":"filed","text":"Created from the arXiv PDF, the Zenodo record and the erdosproblems.com proof-claims thread"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-13","run":null,"sources_read":"The arXiv PDF, the Zenodo record and the erdosproblems.com proof-claims thread","updated":"2026-10-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":150,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":89,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":180,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":119,"in_training_data":false}],"short_url":"https://postcutoff.com/s/independent-set-sequences-all"}