[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"YxPluS4YHd":3},"# Leray–Hopf weak existence in Lean\n\n[![Release-candidate build attestation](https://github.com/uda-lab/leray-hopf/actions/workflows/release-attestation.yml/badge.svg?event=workflow_dispatch)](https://github.com/uda-lab/leray-hopf/actions/workflows/release-attestation.yml)\n\n> The badge links to on-demand build attestations for individual commits, not a\n> per-commit CI signal — see [`docs/build-and-checks.md`](docs/build-and-checks.md)\n> for what it does and does not certify.\n\nA Lean 4 + mathlib formalization of **Leray–Hopf weak existence** for the\nincompressible Navier–Stokes equations, on the periodic 3-torus 𝕋³ and on whole\nspace ℝ³. Two capstone existence theorems are proved and machine-checked, and both\nare **kernel-only**: `#print axioms` returns only the standard kernel axioms\n(`propext`, `Classical.choice`, `Quot.sound`), with zero project axioms and no\n`sorryAx`. This repository does **not** claim smoothness or higher regularity\nbeyond the stated energy-class properties, nor uniqueness or non-uniqueness of the\nsolutions it constructs.\n\n**Start here:** [**uda-lab.github.io/leray-hopf-notes**](https://uda-lab.github.io/leray-hopf-notes/)\nis an interactive, browsable companion to this repository — it presents the Lean\ndeclarations, their dependency graph, Japanese-language mathematical exposition, and\nper-declaration proof status.\n\n## Explore the formalization\n\n- **Interactive notes (Japanese exposition):** \u003Chttps://uda-lab.github.io/leray-hopf-notes/>\n- **Lean source:** [`LerayHopf/`](LerayHopf/)\n- **Architecture / module map:** [`docs/architecture.md`](docs/architecture.md)\n- **Exact claims and scope:** [`docs/claims-and-scope.md`](docs/claims-and-scope.md)\n- **Citation:** [`CITATION.cff`](CITATION.cff)\n\n## Main results\n\n```lean\n-- LerayHopf/Torus/GalerkinODECapstone.lean   (𝕋³)\ntheorem exists_lerayHopf_torus3 (u₀ : L2Sigma) (ν : ℝ) (hν : 0 \u003C ν)\n    (T : ℝ) (hT : 0 \u003C T) :\n    ∃ F : Torus3NSForms, Nonempty (LerayHopfSolutionFull F ν T u₀)\n\n-- LerayHopf/R3/GalerkinODECapstone.lean      (ℝ³)\ntheorem exists_lerayHopf_r3 (u₀ : L2Sigma_R3) (ν : ℝ) (hν : 0 \u003C ν)\n    (T : ℝ) (hT : 0 \u003C T) :\n    ∃ (𝔊 : R3GalerkinScheme) (F : R3NSForms 𝔊),\n      Nonempty (LerayHopfSolutionFull_R3 𝔊 F ν T u₀)\n```\n\nFor any divergence-free initial data `u₀` in `L²`, viscosity `ν > 0`, and time\nhorizon `T > 0`, a Leray–Hopf weak solution exists on `𝕋³` (unit-period torus) and\non `ℝ³` alike — each proof-carrying, with the divergence-free property, the weak\nNavier–Stokes identity, the energy inequality, and the initial-value trace all\nexposed as fields of the returned solution structure, not just asserted in prose.\n\nThe precise field-by-field statement of what each part of the solution structure\nguarantees — and, just as importantly, what it does not — is\n[`docs/claims-and-scope.md`](docs/claims-and-scope.md).\n\n## Scope and limitations\n\n- **No external force** — the homogeneous Navier–Stokes equation only.\n- **Finite time horizon `[0, T]`** for an arbitrary given `T > 0`, not a single\n  solution simultaneously valid on `[0, ∞)`.\n- **𝕋³ is the unit torus** (period 1 in each coordinate), with no periodicity\n  assumption on ℝ³.\n- **Separated-variable weak formulation**: test functions are `ψ(t)·w(x)`, not a\n  general space-time test function.\n- **No smoothness or higher regularity beyond the stated energy-class\n  properties** (a.e.-in-time H¹ membership plus integrable viscous dissipation);\n  **uniqueness and non-uniqueness are not claimed.**\n- **`LerayHopf.Experimental`** isolates incomplete, opt-in additional work; it is not\n  reachable from the release surface below and not needed by either capstone.\n\nSee [`docs/claims-and-scope.md`](docs/claims-and-scope.md) for the exact, field-level\nversion of every bullet above.\n\n## Verification status\n\n- `import LerayHopf` is the release surface: it is `sorry`-free and project-axiom-free,\n  enforced in CI (`scripts/check-release-cone.sh`).\n- Build/toolchain-exact verification of a specific release-candidate commit is done by\n  manual attestation (the badge above); see\n  [`docs/build-and-checks.md`](docs/build-and-checks.md).\n- The badge itself does not expire, but the workflow artifact and run log behind it are\n  retention-limited. Durable copies of that evidence are stored as\n  [Release assets](https://github.com/uda-lab/leray-hopf/releases) (e.g.\n  [`v0.1.0-rc1`](https://github.com/uda-lab/leray-hopf/releases/tag/v0.1.0-rc1)), which\n  certify one exact SHA and are not subject to that Actions retention deadline — see\n  \"Durability caveat\" in\n  [`docs/build-and-checks.md`](docs/build-and-checks.md#durability-caveat).\n- Incomplete additional work is isolated behind the explicit opt-in\n  `import LerayHopf.Experimental`, never pulled in by `import LerayHopf`.\n\n## Getting started\n\n```lean\nimport LerayHopf\n```\n\n```bash\nlake build\n```\n\nFor the full import matrix (which import brings in what, and its exact\nsorry/axiom status), the CI policy, and how to reproduce a release attestation, see\n[`docs/build-and-checks.md`](docs/build-and-checks.md) and\n[`docs/claims-and-scope.md`](docs/claims-and-scope.md).\n\n## Repository map\n\nThe area of each rectangle is proportional to the number of non-comment,\nnon-blank lines (code LOC) in the corresponding Lean source file, covering\n[`LerayHopf.lean`](LerayHopf.lean) and every file under [`LerayHopf/`](LerayHopf/);\ncolor marks the top-level module a file belongs to.\n\n[![Lean source code treemap](docs/assets/code-loc-treemap.svg)](docs/assets/code-loc-treemap.svg)\n\nSee [`docs/architecture.md`](docs/architecture.md#visual-overview-code-loc-treemap)\nfor the full-size figure, the measurement method, and how to regenerate it.\n\n## Star History\n\n\u003Ca href=\"https://www.star-history.com/?repos=uda-lab/leray-hopf&type=date&legend=top-left\">\n \u003Cpicture>\n   \u003Csource media=\"(prefers-color-scheme: dark)\" srcset=\"https://api.star-history.com/chart?repos=uda-lab/leray-hopf&type=date&theme=dark&legend=top-left&sealed_token=seblTJCpa7k-WwWrbWgEIEcfB7J8ZiCwjxREO4qohU0rD65saqGYQJIVkXGjXTtu7E6rBeWSoLwCjFiCdMLs1XiPeYYnIjDcsUpyvM9KXZSjIxqCX5D5jngEDa25wroBaFIgnv8hY75VJLwU-swtQAnAvrGMmjYKIlJW0wOBpLuSRw6Aa1--x4mqRdoG\" />\n   \u003Csource media=\"(prefers-color-scheme: light)\" srcset=\"https://api.star-history.com/chart?repos=uda-lab/leray-hopf&type=date&legend=top-left&sealed_token=seblTJCpa7k-WwWrbWgEIEcfB7J8ZiCwjxREO4qohU0rD65saqGYQJIVkXGjXTtu7E6rBeWSoLwCjFiCdMLs1XiPeYYnIjDcsUpyvM9KXZSjIxqCX5D5jngEDa25wroBaFIgnv8hY75VJLwU-swtQAnAvrGMmjYKIlJW0wOBpLuSRw6Aa1--x4mqRdoG\" />\n   \u003Cimg alt=\"Star History chart for uda-lab/leray-hopf\" src=\"https://api.star-history.com/chart?repos=uda-lab/leray-hopf&type=date&legend=top-left&sealed_token=seblTJCpa7k-WwWrbWgEIEcfB7J8ZiCwjxREO4qohU0rD65saqGYQJIVkXGjXTtu7E6rBeWSoLwCjFiCdMLs1XiPeYYnIjDcsUpyvM9KXZSjIxqCX5D5jngEDa25wroBaFIgnv8hY75VJLwU-swtQAnAvrGMmjYKIlJW0wOBpLuSRw6Aa1--x4mqRdoG\" />\n \u003C/picture>\n\u003C/a>\n\n## Documentation, contributing, citation, license\n\n- [`docs/architecture.md`](docs/architecture.md) — module map.\n- [`docs/claims-and-scope.md`](docs/claims-and-scope.md) — exact claims table and\n  import guide.\n- [`docs/STATUS.md`](docs/STATUS.md) — axiom/`sorry` ledger and integrity backstop.\n- [`docs/build-and-checks.md`](docs/build-and-checks.md) — build, discipline checks,\n  CI policy, and the Star History embed's token-rotation runbook.\n- [`CONTRIBUTING.md`](CONTRIBUTING.md) — build-cost policy, statement-changing PR\n  review requirement, and issue/PR conventions.\n- [`SECURITY.md`](SECURITY.md) — soundness issues, guard bypasses, and supply-chain\n  concerns.\n- [`CITATION.cff`](CITATION.cff) — citation metadata; cite via GitHub's \"Cite this\n  repository\" or this file directly.\n- [`LICENSE`](LICENSE) — Apache License 2.0. Copyright 2026 Tomoki Uda. The license\n  covers the Lean formalization code and repository materials; it does not purport to\n  license mathematical facts or theorems themselves.\n",1784995007256]