{"slug":"landmark-math-results","topic":"landmark-math-results","title":"Landmark Results in Pure Mathematics: State of the Field (October 2026)","summary":"This dossier surveys recent landmark results in pure mathematics: the geometric Langlands proof series, the three-dimensional Kakeya theorem, optimal sphere packings in dimensions 8 and 24, and the 2025 counterexample to additivity of the unknotting number. The main takeaway is that the newest headline results are preprints or unrefereed announcements, while the dimension 8 and 24 sphere-packing theorems are the most firmly settled.","version":1,"published_at":"2026-10-30","updated_at":"2026-10-30","stats":{"words":3405,"citations":36,"claims":30,"timeline":19,"benchmarks":7,"open_problems":6,"key_papers":15,"glossary":15},"verification":{"status":"verified","method":"automated figure and citation-relevance triage against cited abstracts, plus targeted AI review with web checks of flagged statements; automated citation resolution","claims_checked":25,"issues_fixed":7,"verified_at":"2026-10-30"},"ai_assisted":true,"price":{"usd":"0.20","protocol":"x402","network":"eip155:84532","asset":"USDC"},"url":"https://space-kitty.com/dossiers/landmark-math-results","preview_url":"https://space-kitty.com/api/v1/dossiers/landmark-math-results/preview","purchase_url":"https://space-kitty.com/api/v1/dossiers/landmark-math-results","toc":[{"level":2,"text":"Executive summary","anchor":"executive-summary"},{"level":2,"text":"Background and core concepts","anchor":"background-and-core-concepts"},{"level":2,"text":"Timeline of key milestones","anchor":"timeline-of-key-milestones"},{"level":2,"text":"State of the art","anchor":"state-of-the-art"},{"level":3,"text":"Geometric Langlands","anchor":"geometric-langlands"},{"level":3,"text":"Kakeya conjecture","anchor":"kakeya-conjecture"},{"level":3,"text":"Sphere packing","anchor":"sphere-packing"},{"level":3,"text":"Unknotting number","anchor":"unknotting-number"},{"level":3,"text":"AI-attributed proof reports","anchor":"ai-attributed-proof-reports"},{"level":3,"text":"Quantitative benchmarks","anchor":"quantitative-benchmarks"},{"level":2,"text":"Open problems and active debates","anchor":"open-problems-and-active-debates"},{"level":2,"text":"Major programs and players","anchor":"major-programs-and-players"},{"level":2,"text":"Outlook: what to watch (near term)","anchor":"outlook-what-to-watch-near-term"},{"level":2,"text":"Key papers","anchor":"key-papers"},{"level":2,"text":"Glossary","anchor":"glossary"}],"sample_claims":["Gaitsgory, Raskin and collaborators present a 2024 five-part preprint series as a complete proof of the geometric Langlands conjecture in characteristic-zero (de Rham) settings.","Subadditivity holds: the unknotting number of a connected sum is at most the sum of the unknotting numbers of the summands."],"includes":["markdown","claims","timeline","benchmarks","open_problems","key_papers","glossary","bibliography"],"citation_count":36,"how_to_buy":"GET https://space-kitty.com/api/v1/dossiers/landmark-math-results with an x402 client ($0.20, USDC on Base Sepolia (testnet))."}