This work gives a machine-checked, Lean 4 formal proof that the best way to pack eleven equal squares into a larger square has side length T = (6u+4)/(1+2u-u^2), where u is the unique root in (9/25,37/100) of 5u^8 − 10u^7 − 2u^6 + 14u^5 + 12u^4 − 6u^3 + 2u^2 + 2u − 1. The construction attains T ≈ 3.8770835900228141773, allows arbitrary square orientations and legal boundary contacts, and requires disjoint open interiors. The formal argument combines geometric analysis, an exact endpoint construction, a closed-cell cover and a finite case reduction to check all configurations; key formal sources include ElevenSquare/Foundations.lean and ElevenSquare/Optimality.lean.
Verification completed a full Lean run that accepted 7,920 local modules with zero admissions. Selected high-cost, exact numeric checks use native_decide and are recorded with source hashes in verification/native-certificates.json; geometric reasoning, checker soundness, and proof assembly remain standard Lean proofs. The stated trust model is Lean’s kernel plus the native compiler for those numeric certificates. The project pins Lean 4.34.1 and a specific Mathlib revision and provides scripts (scripts/run_verification.sh) and directories (Sqpack, ElevenSquare/Tasks, interop with wand125) to reproduce the verified build and to audit the native-certificate evidence.
Summary generated by AI from the linked article. hn.today is not affiliated with Hacker News or Y Combinator.
Margaret Hamilton, who led the software development for the Apollo moon missions, died at age 90. She spent over two decades at MIT and was awarded the Presidential Medal of Freedom for her pioneering work in computer science and engineering. (news.mit.edu)
Emerging economies are rapidly adopting solar power, bypassing fossil fuels at a faster rate than Europe or the US. This shift is significantly transforming their energy landscapes and development trajectories. (dw.com)
Rosalind Franklin was the first to understand the structure of DNA, contrary to some claims made by Watson. Her work was crucial in the discovery of DNA's double-helix structure. (link.springer.com)
OpenAI released a proof of the longstanding Unique Games Conjecture, a major breakthrough in mathematics. The complexity of the proofs has made AI assistance essential for understanding and verification. (scottaaronson.blog)
OpenAI announced a proof of the Unique Games Conjecture, a major breakthrough in mathematics that many experts are still trying to understand. The complexity of the proofs has made AI assistance essential for analyzing and verifying these results. (scottaaronson.blog)
A 5.3-million-year-old whale necropolis has been discovered in the Diamantina Zone, covering a vast area of the southeastern Indian Ocean. The site contains fossilized and modern whale remains, revealing deep-sea ecosystems and cetacean evolution over millions of years. (nature.com)
Today's best Hacker News stories, summarized and screenshotted, one email a day.