{
  "id": 12644234,
  "title": "AI-assisted proof of optimal packing for 11 squares",
  "url": "https://urgent.news/2026/10/07/ai-assisted-proof-of-optimal-packing-for-11-squares",
  "topic": "science",
  "section": "Science",
  "published": "2026-10-07T14:10:55.000Z",
  "source": {
    "name": "Hacker News",
    "slug": "hacker-news",
    "url": "https://github.com/Queuingtheorydotcom/11SquaresFormalized"
  },
  "original_language": "en",
  "account": "The verification of optimal packing for 11 squares has been successfully completed using AI assistance. The proof was subjected to rigorous verification with native numerical certificates and passed all checks. The optimization process reached an approximate value of 3.8770835900228141773, which is the maximum achievable under the given conditions. The construction allows for arbitrary orientations, legal boundary contact, and disjoint open interiors. The Lean 4.34.1 programming language and Mathlib revision d13f23b723b8a846827a245b89c10fc7d3f11612 were used to formalize the proof, along with the specific commit 1bf942a7af1ea330e95489d8997deebd4227ca71. The verification process involved checking all 7,920 local Lean modules and ensuring a zero admission rate. The final theorem relies on the Lean kernel and native compiler, not just the kernel alone. The numerical declarations and their source hashes are recorded in verification/native-certificates.json. The public statements in ElevenSquare/Optimality.lean and the complete T03 source tree remain unchanged from the previous main branch. The project maintains pinning to specific versions of Lean and Mathlib, and the build configuration remains consistent. The verification process can be resumed at any point, and existing valid receipts are reusable with the appropriate flags. The successful source run utilized EvolvingPrograms' larger runner, but this does not guarantee a cold-build runtime or a 2-3 hour macOS guarantee. The source materials and acknowledgments for the formalization and verification work are provided in the accompanying documentation.",
  "summary": null,
  "key_points": [],
  "editors_take": null,
  "illustration": null,
  "coverage": {
    "outlets": 1,
    "also_reported_by": []
  },
  "ai_generated": true,
  "disclaimer": "Summaries, key points and the editor’s take are written by software from other outlets’ reporting and may contain errors — always check the linked original."
}