{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai/","md":"https://postcutoff.com/e/2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai/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":"2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai","date":"2026-10-29","date_precision":"day","short_title":"q-colourability threshold of G(n, d/n) exists, says a machine-written paper","deck":null,"takeaway":"Achlioptas and Friedgut proved in 1999 that q-colourability of the random graph G(n, d/n) has a sharp threshold, but only along a sequence that might depend on n. Whether that sequence converges to a constant d_q was recorded as an open problem and is widely called well known. On 29 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted on Hexagon a proof that it converges for every q ≥ 3, with a window of O(n(log n)^(−3/2)) steps for the hitting time. The authors' note says the whole document except that note is machine-written. The provenance names Opus 5.5, GPT 6 Astra, GPT Luna 6 and GPT 6.1 Sol, and says the results were formalised in Lean, but the paper links no Lean repository. Human understanding: 'some parts'. Two companion papers on the dynamics of the spherical Sherrington–Kirkpatrick model came the same day.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hexagon 2610.00608: Existence of the q-Colorability Threshold for Sparse Random Graphs (Aronow, Lopatto)","url":"https://hexagonmath.org/abs/2610.00608v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":2,"title":"Hexagon 2610.00609: Dynamics of the spherical SK model, Part I","url":"https://hexagonmath.org/abs/2610.00609v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":3,"title":"Hexagon 2610.00610: Dynamics of the spherical SK model, Part II","url":"https://hexagonmath.org/abs/2610.00610v1","type":"paper","group":"primary","domain":"hexagonmath.org"}],"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"],"title":"Existence of the q-colourability threshold of sparse random graphs, open since Achlioptas–Friedgut (1999), claimed in a machine-written Hexagon paper by P. M. Aronow and Patrick Lopatto (Claude Opus 5.5 and GPT models, Lean claimed)","summary":"\"Existence of the q-Colorability Threshold for Sparse Random Graphs\" (Hexagon 2610.00608, 29 Oct 2026; P. M. Aronow and Patrick Lopatto) claims that for every fixed q ≥ 3 there is d_q > 0 such that G(n, d/n) is q-colourable with probability tending to 1 for d < d_q and to 0 for d > d_q, and similarly for G(n, ⌊cn⌋) at c = d_q/2. This answers the question Achlioptas and Friedgut left open in 1999. The proof uses interpolation for convergence and controls the last colourable graph by identifying random vertices. The paper is machine-written; the provenance says Lean formalization was done by Opus 5.5, GPT Luna 6 and GPT 6.1 Sol, without a public link.","key_facts":["Claim (abstract, verbatim): 'We prove that there exists d_q > 0 such that the random graph on n vertices with independent edge probability d/n is q-colorable with probability tending to one for fixed 0 ≤ d < d_q and to zero for fixed d > d_q, as n → ∞.' Also: the first loss of colourability under random edge additions lies, with probability ≥ 1 − ε, in a deterministic window of O_{q,ε}(n(log n)^(−3/2)) steps","Background (paper): Achlioptas and Friedgut proved a sharp threshold along a sequence d̂_n and 'recorded its convergence as an open problem'; Achlioptas and Molloy noted it was 'widely conjectured'; later works call it 'a well-known open problem'. Known 3-colouring bounds: colourable for d ≤ 4.03, not for d > 4.697. The proof does not need the Achlioptas–Friedgut theorem","Authors' note (verbatim): 'With the exception of these notes, the entire document is machine-written. We have carefully supervised its preparation to ensure that the proofs are readable and that previous literature is appropriately cited.' A 'more carefully written version' is in preparation for journal submission","Provenance (Hexagon, verbatim): 'Proof discovery and verification were done with a combination of Opus 5.5, GPT 6 Astra, GPT Luna 6, and GPT 6.1 Sol. The results were formalized in Lean 4 by Opus 5.5, Luna 6, and GPT Sol 6.1. The writing was done by Opus 5.5, GPT 6.1 Sol, and Astra 6.' Text 'primarily AI-generated'; human understanding 'some parts'","Builds on OpenAI release preprints cited in the paper: 'A limiting satisfiability threshold for every fixed clause size' (its deletion argument) and 'A sharp threshold bound for monotone graph properties'; compare the random k-SAT entry","Same day, same pair: 'Dynamics of the spherical Sherrington–Kirkpatrick model' Parts I–II (Hexagon 2610.00609, 2610.00610), the asymptotic total-variation distance to equilibrium of Langevin dynamics, with cutoff and a Gaussian profile for β < β_c = 1/√2; also machine-written with the same models and Lean claims","Verification status: unrefereed Hexagon preprint; Lean formalization claimed in the provenance but no repository or Palomar entry linked in the paper; no expert reaction or HN thread found on 29 Oct. The pair's 26 Oct five-paper 3D Yang–Mills claim is likewise unchecked"],"key_numbers":[],"tags":["math","probability","random-graphs","graph-coloring","sharp-threshold","hexagon","claude-opus-5-5","gpt-6-astra","gpt-6-1-sol","lean","unverified"],"science":{"field":"mathematics","subfield":"probability and combinatorics (random graphs)","problem":"Existence of the q-colourability threshold of G(n, d/n) (Achlioptas–Friedgut, 1999)","result":"Claimed: for every q ≥ 3 a constant threshold d_q exists for G(n, d/n) (d_q/2 for G(n, ⌊cn⌋)), with an O(n(log n)^(−3/2)) hitting-time window","open_since":"1999","ai_system":["Claude Opus 5.5","GPT-6 Astra","GPT-6 Luna","GPT-6.1 Sol"],"human_role":"Authors supervised; the models did proof discovery, verification, Lean formalization and all writing; human understanding 'some parts'","verification":{"key":"lean","label":"Lean-verified","text":"Unrefereed Hexagon preprint; Lean formalization claimed, not linked"},"status":"pending","shock":""},"body_md":"## What happened\n\nThe chromatic number of sparse random graphs goes back to Erdős and Rényi. For q-colourability, Achlioptas and Friedgut showed in 1999\nthat the transition is sharp, but their threshold was a sequence d̂_n that could in principle oscillate. Whether it converges has been\nan open problem since then. It is the colouring analogue of the random k-SAT threshold question, which Ding, Sly and Sun settled for large k.\n\nOn 29 Oct Aronow and Lopatto posted a proof of convergence for every q ≥ 3. They say the document is machine-written and that they\nsupervised it for readability and citations. The provenance names four models for discovery, verification and writing, and three for a\nLean formalization that the paper does not link. The argument cites OpenAI's September release preprints on the random k-SAT threshold.\n\n## Why it matters\n\nThe question is classic in probabilistic combinatorics and appears in surveys as open. The claim joins the AI-generated random k-SAT\nthreshold proofs of October and the same pair's Yang–Mills papers. Until the Lean files are public or a specialist reads the proof, it is a\nclaim.","disputed":[{"label":"Verification","text":"Unrefereed Hexagon preprint; Lean formalization claimed, not linked","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing-3","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-3/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct (part 3): Hexagon's evening batch","deck":null,"takeaway":"Hexagon published 58 new works on 28–29 Oct that we had not cited. Four have their own entries: the claimed sharp Berry–Esseen constant (Kołodziejek–Rodziewicz), the existence of the q-colourability threshold (Aronow–Lopatto, machine-written), deterministic factoring in N^o(1) (Rudich–Rousseau, Lean, ideas by Claude) and the refutation of Moschovakis's uniform-process lower bounds for gcd (Davies, Lean). Others claim the quantum Jellium = uniform-electron-gas energy (Csirik), C₂-cofiniteness of all finite-group orbifolds (Riesen), analytic floating bodies of equilibrium in every dimension ≥ 3 (Price–de Groot, GPT-6 Pro) and Cowling's conjecture for almost connected groups (Arhancet). From the backlog: S(13) = 45 comparators for sorting 13 inputs and an area law for 3D U(1) Wilson lattice gauge theory (Tomar, Lean), Enomoto's settlement of the third Brauer–Thrall conjecture over algebraically closed fields (GPT-6 Astra) and Soberón's counterexample to Matschke's table conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai","url":"https://postcutoff.com/e/2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai/","date":"2026-10-26","date_precision":"day","short_title":"3D SU(2) Yang–Mills continuum limit claimed in five AI-written papers","deck":null,"takeaway":"Building Yang–Mills theory as a rigorous continuum measure is open even in three dimensions, where the theory is superrenormalizable. On 26 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted five Hexagon papers. They claim that flowed Wilson loops of 3D SU(2) lattice Yang–Mills on a torus converge as the lattice spacing goes to zero, to a unique measure on the Cao–Chatterjee state space. The proof is a Bałaban-style multiscale renormalization-group analysis. The provenance statement says proof discovery, verification and writing were done by Opus 5.5, GPT 6 Astra and GPT 6.1 Sol, with only 'limited background formalizations' in Lean. Hexagon records human understanding as 'no parts'. It is not the four-dimensional Clay problem. We found no expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"]},{"id":"2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai","url":"https://postcutoff.com/e/2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai/","date":"2026-10-05","date_precision":"day","short_title":"Random k-SAT threshold exists for every k ≥ 3: a human proof and an OpenAI Lean-checked proof","deck":null,"takeaway":"The satisfiability conjecture says that random k-SAT switches from satisfiable to unsatisfiable at one limiting clause density α_k. Friedgut proved a sharp but possibly n-dependent threshold in 1999. Ding, Sly and Sun proved the conjecture for large k (Annals, 2022). On 5 Oct 2026 Cambridge PhD student Gaia Carenini posted on ECCC a proof for every fixed k ≥ 3. Her report states that none of its arguments was generated by a language model. The next day OpenAI's maths release included a separate proof, dated 25 Sept, which credits Carenini with priority. Its Lean statement was accepted in the community Comparator run, and it adds Θ(n) hitting-time variance and computability of the 3-SAT threshold. Neither proof has had a published expert assessment as of 29 Oct. Neither determines the value of α_3.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["University of Cambridge","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (Hexagon API records and PDF of 2610.00608, records of 2610.00609–00610; HN search found no thread)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":182,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":121,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":212,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":151,"in_training_data":false}],"short_url":"https://postcutoff.com/s/q-colourability-threshold-g-n"}