Lean 4

Mathlib is a formal mathematics library for use with Lean proofs.

An OpenAI repository containing Lean 4 formalizations of ten results presented in its paper on advances in mathematics and theoretical computer science. The formalizations cover high-dimensional sphere packing; binary and spherical codes; non-sofic groups; Connes’s rigidity conjecture; arithmetic circuit complexity; quantum parallel repetition; the closest vector problem; Ehrhart’s volume conjecture; multicolor Ramsey numbers; and compactness and degeneracy conjectures in extremal graph theory. Each result is implemented as a separate Lean module, including `SpherePacking.lean`, `MetricCodes.lean`, `NonSoficGroup.lean`, `ConnesRigidity.lean`, `Permanent.lean`, `QuantumParallelRepetition.lean`, `GapCVP.lean`, `EhrhartVolumeInequality.lean`, `MulticolorTriangleRamsey.lean`, and `CompactnessAndDegeneracy.lean`. The project uses Lean 4.32.0, mathlib, and Lake. With the mathlib cache installed, all formalizations can be built with `lake build All`, or individual modules can be built by name; the repository also provides reasoning walkthroughs and documents independent checking with Comparator.

A research artifact containing a complete machine-checked Lean 4 proof of Fermat's Last Theorem, following the Frey–Serre–Ribet–Wiles and Taylor–Wiles argument and built on Mathlib. The theorem states that for natural numbers n, a, b, and c with n ≥ 3 and a, b, and c positive, a^n + b^n ≠ c^n. Lean's kernel checks the proof, which uses only Lean's three standard axioms—propext, Classical.choice, and Quot.sound—and the repository excludes sorry statements, added axioms, native_decide, unsafe code, and related escape mechanisms. A comparator independently replays the environment against a Mathlib formulation, while Nanoda, an independent Lean kernel implemented in Rust, checks an exported version of the environment. The repository also includes PROOF-PATH.md and an offline HTML presentation of the proof's theorems, definitions, dependencies, and generated documentation. It is released under the Apache License 2.0 and is described as not maintained and not accepting contributions.

LLM Space v4 is a local-first desktop workbench for prototyping and developing AI agents. It lets users version prompts, system messages, tools, and model settings; trace each model call and tool run in an agent loop; replay historical runs to debug failures; and evaluate agent performance across runs. It can generate prompts and tools with AI and turn a thread into a runnable LangGraph agent. Threads, project files, and API keys remain on the user's computer, while the project describes the app as cloud-ready for managed agents. The desktop application uses Electrobun with a React, Tailwind CSS, and shadcn/ui interface, and is built as a Bun monorepo around Pi Agent Core.

Bend 2 is a parallel programming language created by Victor Taelin and contributors, designed for AI-written code that must preserve declared rules. Its Python-like syntax includes dependent types, purity, linearity, and explicit proofs. A project can define invariants in a `LAWS.bend` file and provide proofs in `PROOF.bend`; the compiler checks those proofs when code changes, blocking implementations that violate the declared laws. Bend uses divide-and-conquer parallelism rather than manually managed threads, locks, or GPU kernels, and can target CPUs and GPUs through C, CUDA, and Metal, with JavaScript also supported. It is distributed through an install script and is positioned primarily for Linux and macOS; the repository describes it as a young language with limitations including no Windows support, no proof search, slow native compilation, and experimental areas of the compiler and runtime.