Verified 3D Mesh Intersection

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.

Colite is an experimental C++17 structure-from-motion and multi-view stereo system that reconstructs 3D scenes from overlapping images. Its pipeline performs incremental structure-from-motion with support for multiple models, continuation, and Sim(3) merging; feature extraction and matching; PatchMatch stereo with geometric consistency; stereo fusion; meshing; simplification; and texture-atlas generation. It produces COLMAP-compatible sparse models, RGB and normal dense point clouds, Poisson or Delaunay meshes, and textured OBJ models with MTL and atlas PNG files. CPU and optional CUDA implementations are provided for SIFT, descriptor matching, and PatchMatch Stereo, and resumable stage manifests allow processing to continue from saved stages. The repository describes the current build as alpha-quality, primarily developed and tested on Ubuntu 22.04 and WSL2; native Windows builds are not yet validated as a supported release target.

NavierStokesAndEuler is an open-source Lean 4 formalization of results concerning finite-time breakdown in the Navier–Stokes and incompressible Euler equations, developed by OpenAI. For Navier–Stokes with positive viscosity, it formalizes constructions on both three-dimensional Euclidean space and the periodic torus using smooth initial data and external forcing; the results rule out global smooth solutions under the stated kinetic-energy or existence conditions. For the unforced Euler equations, it formalizes a smooth, compactly supported, divergence-free initial velocity whose solution develops a finite-time singularity, with the velocity’s C1 norm becoming unbounded and the time integral of the vorticity’s L-infinity norm diverging. The repository uses Lean 4, Mathlib, and Lake. Its formalizations can be built after fetching the Mathlib cache, and the repository provides separate instructions for independently checking them with Comparator.

Open CASCADE Technology (OCCT) is an open-source software development platform and CAD geometry kernel for 3D surface and solid modeling, CAD data exchange, and visualization. Its functionality is provided mainly as C++ libraries for applications in computer-aided design, manufacturing, and engineering analysis; the videos describe it as compiled to WebAssembly for browser-based parametric CAD. OCCT is distributed under the GNU Lesser General Public License version 2.1 with a special exception, and may alternatively be used under an Open CASCADE commercial license or contractual agreement.

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.