{"schema":"postcutoff/event@1","as_of":"2026-10-18T22:10:00+02:00","url":"https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/","md":"https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean","date":"2026-09-30","date_precision":"day","short_title":"mt0-svg: AI-made Tammes N=15, Robbins V>2, frog model (Lean)","deck":null,"takeaway":"Between Sept 28 and Oct 18, 2026 one pseudonymous GitHub account released six papers whose 'results, proofs and code were found and written by AI', each with a Lean 4 proof and comparator check: the first new case of the Tammes problem since N=14 (2015), a 1996 question on Robbins' problem, and both open cases of a 2017 frog-model conjecture. No model is named and no expert review is known.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"GitHub mt0-svg/tammes-15 (paper, Lean proofs, certificates)","url":"https://github.com/mt0-svg/tammes-15","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"Zenodo: The Tammes problem for fifteen points (v1.4.0)","url":"https://zenodo.org/records/23450670","type":"paper","group":"primary","domain":"zenodo.org"},{"n":3,"title":"GitHub mt0-svg/robbins-problem: The value of Robbins' problem is larger than 2","url":"https://github.com/mt0-svg/robbins-problem","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Zenodo: The value of Robbins' problem is larger than 2","url":"https://zenodo.org/records/23124501","type":"paper","group":"primary","domain":"zenodo.org"},{"n":5,"title":"GitHub mt0-svg/frog-model-tree: frog model recurrent on the 3-ary tree, transient on the 4-ary tree","url":"https://github.com/mt0-svg/frog-model-tree","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"GitHub mt0-svg/keith-zanello-73: Keith and Zanello's Conjecture B is false","url":"https://github.com/mt0-svg/keith-zanello-73","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"GitHub mt0-svg/furio-lombardo-quartic","url":"https://github.com/mt0-svg/furio-lombardo-quartic","type":"code","group":"primary","domain":"github.com"},{"n":8,"title":"GitHub mt0-svg/regret-kappa","url":"https://github.com/mt0-svg/regret-kappa","type":"code","group":"primary","domain":"github.com"},{"n":9,"title":"Musin & Tarasov: The Tammes problem for N = 14 (arXiv 1410.2536)","url":"https://arxiv.org/abs/1410.2536","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"tammes-15 issue #1 by Satvik-patil: the two optimal configurations are the only ones (Claude-assisted)","url":"https://github.com/mt0-svg/tammes-15/issues/1","type":"discussion","group":"reactions","domain":"github.com"}],"official":9,"filed":"2026-10-18","updated":"2026-10-18","orgs":["mt0-svg (pseudonymous)"],"title":"Anonymous 'AI-generated research' account mt0-svg posts Lean-checked solutions to the Tammes problem for 15 points, Robbins' V > 2 and the Hoffman–Johnson–Junge frog-model conjecture","summary":"The GitHub account mt0-svg (created Aug 1, 2026, no name) published, with Zenodo DOIs, six results labelled \"AI-generated research: the results, proofs and code were found and written by AI\". (1) Tammes problem for N = 15: the largest minimal angular distance of 15 points on the sphere is arccos u ≈ 53.65785°, attained by exactly two configurations (Buddenhagen–Kottwitz) whose contact graphs differ, which answers Musin–Tarasov's uniqueness question negatively for N = 15 (v1.0.0 Sept 30, v1.4.0 Oct 18). (2) Robbins' problem: V(n) > 2.0038 for n ≥ 300, so V > 2, answering Bruss and Ferguson (1996). (3) The frog model on rooted d-ary trees is recurrent for d = 3 and transient for d = 4, the open cases of Hoffman–Johnson–Junge's conjecture (Ann. Probab. 2017). Also: Keith–Zanello Conjecture B is false (p = 73), the Furio–Lombardo quartic has exactly four rational points, and the minimax regret constant of 1D online linear regression is 3. Each has a Lean 4 proof with only standard axioms, comparator-checked; Tammes and the frog model also rely on certified computations entered as hypotheses or replayed by Lean checkers. Not peer reviewed; no expert reaction found by Oct 18.","key_facts":["Label on every repo and paper: 'This is AI-generated research: the results, proofs and code were found and written by AI. Credit goes to all the humans whose work it builds on.' No AI model or company is named","Tammes N = 15: optimum arccos u ≈ 53.65785°, u the root in (1/2, 7/10) of 13u⁵ − u⁴ + 6u³ + 2u² − 3u − 1; method follows Musin–Tarasov (N = 13 in 2012, N = 14 in 2015): written reduction to plane graphs with ≤ 15 vertices, interval computation with replayed search-tree certificates, local rigidity near the two optima","Tammes Lean scope (quoted): 'Lean 4 proves every step except the enumeration of the plane graphs and their refutation by the program, which enter the formal proof as hypotheses' (plantri enumeration and a few thousand lines of Rust)","Tammes Theorem 1.3 (exactly two optimal configurations) is credited to GitHub user Satvik-patil, who posted it as issue #1 on Oct 6–7, 'Produced with an AI system (Claude)', 'not reviewed by a specialist'","Robbins' problem: Lean theorem Robbins.main_so gives V(n) ≥ 137704521264/2^36 (≈ 2.0038) for all n ≥ 300, no sorry, no native_decide; the kernel checks 300 tables of a certificate by evaluation; question from Bruss & Ferguson (1996) and Bruss, 'What is known about Robbins' problem?' (J. Appl. Probab. 2005)","Frog model: Lean theorems FrogModel.recurrent_three and FrogModel.transient_four, mean visits to the root ≤ 5.756 for d = 4; Lean checkers proved sound run on certificate data; 'Conjecture 2 of Hoffman, Johnson and Junge (Ann. Probab. 2017; arXiv:1404.6238)'","Others: Keith–Zanello Conjecture B (Ann. Comb. 2026) false, exactly twenty odd t work at p = 73, largest t = 203; Furio–Lombardo Conjecture 1.6 (Proc. LMS 2026) proved via descent over a degree-21 field and Selmer group Chabauty; regret constant κ = 3 for doubly uniform 1D online linear regression (question of Gaillard et al., ALT 2019)","Zenodo first versions: Keith–Zanello and Furio–Lombardo Oct 2, regret Oct 2–3, frog model and Robbins Oct 4, Tammes Sept 30 (v1.0.0) to Oct 18 (v1.4.0); all repos have 0 stars on Oct 18"],"key_numbers":[],"tags":["math","discrete-geometry","tammes-problem","spherical-codes","probability","optimal-stopping","robbins-problem","frog-model","number-theory","lean","computer-assisted-proof","zenodo","ai-for-math"],"science":{"field":"mathematics","subfield":"discrete geometry (spherical codes), probability (optimal stopping, interacting particle systems), number theory","problem":"Tammes problem for 15 points (open since Tammes 1930; solved for N ≤ 14 and N = 24); Robbins' problem (is V > 2?, Bruss–Ferguson 1996); Hoffman–Johnson–Junge frog-model conjecture for d = 3, 4 (2017)","result":"Tammes optimum for N = 15 determined, with exactly two optimal configurations; V > 2 for Robbins' problem; frog model recurrent on the 3-ary and transient on the 4-ary tree; plus three smaller conjectures settled.","open_since":"1930","ai_system":["unnamed AI system(s)"],"human_role":"Claimed fully AI-generated ('results, proofs and code were found and written by AI'); the human operator is anonymous. One Tammes theorem came from a third party using Claude","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 with Mathlib (standard axioms, comparator-checked); Tammes leaves graph enumeration and program correctness as explicit hypotheses backed by replayed certificates; unrefereed Zenodo preprints"},"status":"pending","shock":"An anonymous account with no named model settled the first new Tammes case in a decade and a 30-year-old question on Robbins' problem within a week, with machine-checked proofs."},"body_md":"## What happened\n\nOn Oct 18 the sweep's Zenodo check turned up \"The Tammes problem for fifteen points\" (v1.4.0). Its author is listed only as\nmt0-svg, and it opens with: \"This is AI-generated research: the results, proofs and code were found and written by AI.\" The same\nGitHub account had released five more such papers since Sept 28. Each comes with a Lean 4 development, a comparator challenge,\nCI that rebuilds everything, and computation certificates.\n\n- **Tammes, N = 15.** The paper extends the Musin–Tarasov method used for 13 and 14 points: a written reduction to irreducible\n  contact graphs, an interval-arithmetic search with recorded and replayed trees, and a local rigidity estimate near the two known\n  optima. Lean proves everything except that the plantri enumeration is complete and that the Rust program matches its formal model;\n  those enter as stated hypotheses. Exactly two non-isomorphic optimal configurations exist. That answers Musin and Tarasov's\n  question whether the maximal contact graph is unique (it is not, for N = 15). The two-configuration classification (Theorem 1.3)\n  was contributed by GitHub user Satvik-patil in an issue that says it was \"produced with an AI system (Claude)\".\n- **Robbins' problem.** It asks for the least expected rank of a value chosen by a stopping rule from n i.i.d. uniform values with full\n  information. The paper relaxes it to a memory of the five smallest values seen and certifies V(n) > 2.0038 for n ≥ 300. That\n  answers Bruss and Ferguson's 1996 question whether V > 2. The Lean proof has no hypotheses beyond the definitions.\n- **Frog model.** Recurrence on the rooted 3-ary tree and transience on the 4-ary tree, the cases Hoffman, Johnson and Junge left as a\n  conjecture, proved through exact recursions on subtree response laws, with certificate checkers verified in Lean.\n- Smaller items: a counterexample to Keith–Zanello's Conjecture B at p = 73, a proof of Furio–Lombardo's conjecture that a twist of\n  the Klein quartic has exactly four rational points, and κ = 3 for doubly uniform regret in 1D online linear regression.\n\nWeb searches on Oct 18 found no news coverage, no posts by experts and no Hacker News thread. The repos had no stars.\n\n## Why it matters\n\nIf they hold up, these are real advances on named problems. The Tammes problem is a classical problem in discrete geometry, and each new case so far has\nneeded a major computer-assisted effort. Robbins' problem is one of the best-known open questions in optimal stopping. The account\nnames no model and no person, so we cannot tell which system made them or how much human direction there was. The Lean proofs and\ncertificates are public and can be checked, which is not true of many AI maths claims. The Tammes proof's trust boundary\n(enumeration and program correctness) is stated explicitly.","disputed":[{"label":"Verification","text":"Lean 4 with Mathlib (standard axioms, comparator-checked); Tammes leaves graph enumeration and program correctness as explicit hypotheses backed by replayed certificates; unrefereed Zenodo preprints","sources":[]}],"related":[{"id":"2026-10-18-ai-assisted-results-october-17-listing","url":"https://postcutoff.com/e/2026-10-18-ai-assisted-results-october-17-listing/","date":"2026-10-17","date_precision":"day","short_title":"AI-assisted results, 16–17 Oct 2026 (Zenodo and arXiv leftovers)","deck":"Ramanujan's prime-counting inequality settled unconditionally with Claude, a Claude Code paper mill on the Hodge conjecture, and LLM agents in materials science","takeaway":"With no new arXiv announcement over the weekend, the notable AI-assisted items came from Zenodo, led by Padhi's Claude-assisted proof that Ramanujan's inequality holds unconditionally for all x ≥ 38,358,837,683.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","various mathematicians"]},{"id":"2026-10-06-eleven-square-packing-optimality-lean","url":"https://postcutoff.com/e/2026-10-06-eleven-square-packing-optimality-lean/","date":"2026-10-06","date_precision":"day","short_title":"Walter Trump's 1979 packing of 11 unit squares proved optimal, with a full Lean verification (AI involvement reported, unconfirmed)","deck":null,"takeaway":"It settles a well-known open case in packing theory, widely known from xkcd.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-07","updated":"2026-10-13","orgs":["Open-source community"]},{"id":"2026-09-06-collatz-positive-density-log-time-mazur-codex","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["ProofAtlas","OpenAI","Axiom Math"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-18","type":"filed","text":"Created (found via Zenodo search during the evening science check; Zenodo API, repo READMEs, the Tammes PDF via pdftotext and tammes-15 issue #1 read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-18","run":null,"sources_read":"Found via Zenodo search during the evening science check; Zenodo API, repo READMEs, the Tammes PDF via pdftotext and tammes-15 issue #1 read","updated":"2026-10-18","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":153,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":92,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":183,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":122,"in_training_data":false}],"short_url":"https://postcutoff.com/s/mt0-svg-ai-made-tammes-n"}