{"schema":"postcutoff/event@1","as_of":"2026-10-21T14:07:00+02:00","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","md":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-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":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"Two independent claims settle the linear MDS conjecture (from Segre's 1955 question on arcs) for every finite field of even order: an [n, k] MDS code over F_q, q even, has n ≤ q + 1, except n ≤ q + 2 for k = 3 and k = q − 1.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.28208: On the MDS conjecture in even characteristic (Chen, Huang)","url":"https://arxiv.org/abs/2610.28208","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Palomar registry API: PALOMAR-2026-10-15-000014 (Kumar)","url":"https://data.palomar-registry.org/api/v1/results?id=PALOMAR-2026-10-15-000014","type":"docs","group":"primary","domain":"data.palomar-registry.org"},{"n":3,"title":"GitHub: PranavaKCode/EvenOrderMDS (Lean 4 proof)","url":"https://github.com/PranavaKCode/EvenOrderMDS","type":"code","group":"primary","domain":"github.com"}],"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"],"title":"MDS conjecture settled in even characteristic: two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","summary":"Two independent claims settle the linear MDS conjecture (from Segre's 1955 question on arcs) for every finite field of even order: an [n, k] MDS code over F_q, q even, has n ≤ q + 1, except n ≤ q + 2 for k = 3 and k = q − 1. Pranava Kumar registered a Lean 4 proof on the Palomar registry on 15 Oct (Claude helped translate it into Lean; Claude Opus 5.5 ported and packaged it); Bocong Chen and Jing Huang posted a different proof on arXiv on 20 Oct, written with ChatGPT's help on proof strategies and calculations.","key_facts":["Statement (Chen–Huang, arXiv 2610.28208): 'every nontrivial linear [n, k] maximum distance separable (MDS) code over a finite field Fq of characteristic two satisfies n ≤ q + 1, except in dimensions k = 3 and k = q − 1, where n ≤ q + 2. This resolves the linear MDS conjecture in even characteristic.'","History: Segre's 1955 problems on arcs in finite projective spaces; Ball (JEMS 2012) proved the conjecture for k ≤ p, settling prime fields; odd characteristic remains open in general","Chen–Huang AI statement: 'The authors used ChatGPT to assist with exploring proof strategies, checking intermediate calculations, and editing the exposition. The authors are responsible for all mathematical arguments and for the final exposition.'","Kumar's Lean proof: Palomar record PALOMAR-2026-10-15-000014 (registered 15 Oct 2026, trust level 'high'), theorem EvenOrderMDS.maximum_length_of_MDS_code over Mathlib, standard axioms only, checked by Comparator against Challenge.lean and by the Lean kernel plus the NanoDa and con-ron kernels","Kumar repo authorship note: 'The mathematical argument was devised by Pranava Kumar … Claude (Anthropic) assisted him in translating the proof into Lean 4' and 'The present repository was assembled with the help of an AI agent (Claude Opus 5.5, in Claude Code)', which wrote Challenge.lean and Solution.lean; 'No independent human expert has yet reviewed the mathematics.'","Chen–Huang note the overlap: Kumar's report 'also claims to establish Theorem 1.1'; both use a Frobenius obstruction, a ternary colon ideal and Gorenstein Hilbert-function unimodality, with different local constructions"],"key_numbers":[],"tags":["math","coding-theory","finite-geometry","mds-conjecture","segre","lean","palomar","chatgpt","claude","claude-opus-5-5"],"science":{"field":"mathematics","subfield":"coding theory / finite geometry","problem":"Linear MDS conjecture (Segre 1955; maximum length of MDS codes / arcs), even characteristic","result":"For q even and 2 ≤ k ≤ q, the maximum length of a k-dimensional linear MDS code over F_q is q + 1, or q + 2 when k ∈ {3, q − 1}","open_since":"1955","ai_system":["ChatGPT","Claude","Claude Opus 5.5"],"human_role":"Human-led with AI tools: Kumar devised the argument and Claude helped formalize it in Lean; Chen and Huang used ChatGPT to explore proof strategies and check calculations","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 proof (Mathlib, Comparator, three kernels) registered on Palomar; the Challenge statement adds two definitions (HasMinimumDistance, IsMDS) and was written by the AI agent; arXiv preprint unrefereed"},"status":"pending","shock":"A 70-year-old central problem of finite geometry closed for half of all fields by two groups within a week, one of them with a machine-checked proof"},"body_md":"## What happened\n\nThe MDS conjecture asks how long a maximum distance separable code (one that meets the Singleton bound) can be. Equivalently, it\nasks for the largest arc in a finite projective space, a question Beniamino Segre posed in 1955. The expected answer is q + 1,\nwith exceptions in even characteristic. Simeon Ball proved it over prime fields in 2012.\n\nOn 15 Oct 2026 Pranava Kumar registered a Lean 4 proof of the full even-characteristic case on Palomar, the registry of\nLean-verified mathematics. The repository's README says the argument is Kumar's: Claude helped him translate it into Lean, and\nClaude Opus 5.5 in Claude Code later ported it, wrote the Challenge and Solution files, the README and the report. It also says\nthat no independent human expert has yet reviewed the mathematics. Palomar lists the record with trust level \"high\".\n\nOn 20 Oct Bocong Chen (South China University of Technology) and Jing Huang (Guangzhou University) posted a 33-page proof on\narXiv. Their AI statement says ChatGPT helped explore proof strategies, check intermediate calculations and edit the text. They\ncite Kumar's report as an independent claim and compare the two arguments.\n\nNo expert reaction or press coverage was found on 21 Oct.\n\n## Why it matters\n\nIf it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem. It is also an early case of a major\nclassical result arriving with a kernel-checked Lean proof from the start, built with an AI coding agent. Two caveats remain. The\nLean statement was written by the AI agent; it was audited against the standard formulations but uses its own definitions of\nminimum distance and MDS, not Mathlib's. And neither proof has been refereed yet.","disputed":[{"label":"Verification","text":"Lean 4 proof (Mathlib, Comparator, three kernels) registered on Palomar; the Challenge statement adds two definitions (HasMinimumDistance, IsMDS) and was written by the AI agent; arXiv preprint unrefereed","sources":[]}],"related":[{"id":"2026-10-21-ai-assisted-results-october-21-listing","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"A ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-08-18-palomar-lean-registry","url":"https://postcutoff.com/e/2026-08-18-palomar-lean-registry/","date":"2026-08-18","date_precision":"day","short_title":"Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims","deck":null,"takeaway":"Formal verification became the main way to trust AI mathematics in 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":5,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Lean FRO","ICARM"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-21","type":"filed","text":"Created (PDF, Palomar record and repository README read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-21","run":null,"sources_read":"PDF, Palomar record and repository README read","updated":"2026-10-21","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":173,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":112,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":203,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":142,"in_training_data":false}],"short_url":"https://postcutoff.com/s/mds-conjecture-settled"}