[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"0HBjpy0Gcj":3},"# goldbach-lean\n\nA Lean 4 project formalizing progress toward **Goldbach's conjecture** and\nbuilding reusable analytic number theory, with **Chen's 1+2 theorem** and\n**Li–Liu's 1+1.9 theorem** formalized. Stronger results remain a research direction.\n\nThe Li–Liu result proves:\n\n> Every sufficiently large even natural number has a representation\n> `N = p + r*q`, where `p` and `q` are prime, `r = 1` or `r` is prime, and\n> `r^10 ≤ q^9`.\n\nIt also proves the strict `0.0004` lower bound for the number of distinct prime\nfirst summands satisfying this condition, in the Liu singular-series normalization.\nThe earlier **1+2** result remains available through its original import and\npublic names: every sufficiently large even natural number is a prime plus\nanother prime or a product of two primes, whose factors may be equal.\n\nThe [release notes](docs/RELEASE_NOTES.md) describe the new results and retain the\nhistorical v1.0.0 record. The [recorded build benchmarks](docs/BUILD_BENCHMARK.md)\nmeasure the v1.0.0-rc1 and v1.0.0 **1+2** developments.\n\n**[Project homepage](https://subfish-zhou.github.io/goldbach-lean/)**\n· **[Lean API documentation](https://subfish-zhou.github.io/goldbach-lean/docs/)**\n· **[Interactive proof Blueprint](https://subfish-zhou.github.io/goldbach-lean/blueprint/)**\n\nThe Lean API documentation covers all four project libraries, with declaration\nsearch, source links, and import navigation. See [how the website is built](docs/DOCUMENTATION.md).\n\n| Read the mathematics | Explore the proof | Check the result |\n|---|---|---|\n| [Theorems and normalization](docs/THEOREMS.md) | [Architecture and source map](docs/ARCHITECTURE.md) | [Verification and trust boundary](docs/VERIFICATION.md) |\n\n## Proof at a glance\n\nThe following roadmap describes the Chen 1+2 route. The Li–Liu extension has\nits own [public entry and source route](docs/ARCHITECTURE.md#liliu-extension).\n\n```mermaid\nflowchart BT\n  distribution[\"Prime distribution and analytic estimates\"] --> lower[\"Weighted lower sieve\"]\n  comparison[\"Jurkat-Richert functions and Suzuki comparison\"] --> lower\n  distribution --> upper[\"Switched-source Selberg upper sieve\"]\n  lower --> counting[\"Finite counting: lower bound minus triple penalty\"]\n  upper --> counting\n  counting --> quantitative[\"Quantitative representation bound\"]\n  quantitative --> chen[\"Chen's theorem: prime + at most two primes\"]\n```\n\nThis is a mathematical roadmap: arrows point from an input to the result it\nsupports, and may summarize several modules. The [source map](docs/ARCHITECTURE.md)\nseparates this view from direct imports and compiled declaration dependencies.\n\nThe [interactive Blueprint](https://subfish-zhou.github.io/goldbach-lean/blueprint/)\nrenders selected declaration dependencies from LeanArchitect, with links to their\nLean source. Continuous integration (CI) validates the documentation and assembles\nthe project homepage, Lean Doc, and Blueprint into one website, published from `main`. The Blueprint remains available as the `goldbach-blueprint` artifact.\n\n## Main results\n\nThe two developments have separate imports over shared analytic foundations.\nUse `import Goldbach` for the established 1+2 interface,\n`import Goldbach.OnePlusOneNine` for 1+1.9, or `import Goldbach.All` for both.\n\n```lean\nimport Goldbach.OnePlusOneNine\n\n#check Goldbach.one_plus_one_nine\n#check Goldbach.one_plus_one_nine_real\n#check Goldbach.one_plus_one_nine_count\n#check Goldbach.one_plus_one_nine_lower_bound\n```\n\nThe natural-power theorem retains the exact `r^10 ≤ q^9` condition; the real-power\nversion states `r ≤ q^((19/10 : ℝ) - 1)`. The count theorem gives\n`(1/2500) * liuSingularSeries N * N / (Real.log N)^2 \u003C D19 N`, where `D19`\ncounts different eligible primes `p`. The stronger lower-bound family allows\neach fixed real `κ \u003C 515093/800000000`, with a threshold depending on `κ`.\nSee [the precise statements and normalization](docs/THEOREMS.md).\n\nThe original Chen interface is unchanged:\n\n```lean\nimport Goldbach\n\n#check Goldbach.chen_theorem\n#check Goldbach.representation_lower_bound\n```\n\n`Goldbach.chen_theorem` proves `Goldbach.ChenTheorem`. The specification in\n[`Goldbach/Statement.lean`](Goldbach/Statement.lean) is deliberately independent\nof the sieve implementation:\n\n```lean\n∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N → Even N →\n  ∃ p q : ℕ, p.Prime ∧\n    (q.Prime ∨ ∃ r s : ℕ, r.Prime ∧ s.Prime ∧ q = r * s) ∧ N = p + q\n```\n\nThe quantitative theorem gives an eventual lower bound of\n`0.67 * liuSingularSeries N * N / (Real.log N)^2` for the cardinality of the\nactual good-representation set, for even `N`. See\n[`docs/THEOREMS.md`](docs/THEOREMS.md) for definitions and normalization.\n\n## Build and check\n\nInstall [elan](https://github.com/leanprover/elan), then run from this directory:\n\n```sh\nunset LEAN_PATH LEAN_SRC_PATH\nlake exe cache get\nlake --wfail build\npython3 scripts/check.py\nlake env lean Goldbach/Checks.lean\nlake env leanchecker --verbose Goldbach.Theorem\n```\n\nThe toolchain and all Git dependencies are pinned by `lean-toolchain` and\n`lake-manifest.json`. The checkout contains the complete local source closure.\n`lake exe cache get` downloads Mathlib's compiled cache; the project proofs are\nbuilt from source. `scripts/check.py` runs separate statement and axiom probes\nfor Chen 1+2 and Li–Liu 1+1.9. `Goldbach/Checks.lean` retains the Chen checks;\n`Goldbach/OnePlusOneNineChecks.lean` checks the new interface. The `leanchecker`\ncommand above replays the Chen public module against cached imports. The\n[verification guide](docs/VERIFICATION.md) gives the Li–Liu inspection and replay\ncommands as well.\n\nThe checks reject proof placeholders and custom axioms in the shipped source,\nverify the public theorem axiom reports against Lean's standard classical\naxioms (`propext`, `Classical.choice`, `Quot.sound`), and check source hygiene.\nRead [`docs/VERIFICATION.md`](docs/VERIFICATION.md) for the exact coverage and\nlimitations.\n\n### Focused builds and upgrading an existing checkout\n\n```sh\nlake build Goldbach.Theorem          # Chen 1+2\nlake build Goldbach.OnePlusOneNine   # Li–Liu 1+1.9\nlake build Goldbach.All              # both public interfaces\nlake build                          # full project\n```\n\nIf you have already built the project, keep `.lake/` and update your checkout\nwithout cleaning it. To update an existing `main` checkout:\n\n```sh\ngit switch main\ngit pull --ff-only\nlake build Goldbach.OnePlusOneNine\n```\n\nTo use the integration branch instead, run `git fetch origin`, then\n`git switch integrate/liliu19-latest`. If that branch is not yet local, use\n`git switch --track origin/integrate/liliu19-latest` instead. Preserve or commit\nany local work before switching branches.\n\nThe package configuration, toolchain and dependency pins are unchanged. Lake\ncan reuse unchanged dependency artifacts and rebuilds new or changed modules\nand affected consumers as needed. Keep the existing `.lake/` directory; rebuild\nwork depends on your cache state and selected target.\n\n## Organization\n\n- `Goldbach/`: independent statement, public theorems, and acceptance checks.\n- `MathlibNt/`: Chen and Li–Liu sieve and analytic proof implementations.\n- `AnalyticNumberTheory/`: reusable prime-distribution, Mertens, and sieve results.\n- `PrimeNumberTheoremAnd/`: the attributed, adapted prime-number-theorem source closure.\n- `scripts/`: reproducible source and trust checks.\n- `docs/`: theorem map, architecture, provenance, and verification instructions.\n\nThe proof uses a modern combination of Jurkat–Richert/Richert linear sieve,\nSuzuki comparison results, Selberg upper sieve, and proved distribution bounds.\nThe [architecture guide](docs/ARCHITECTURE.md) explains how these ingredients\ncombine to prove the public results.\n\n## Sources and license\n\nDistributed under Apache-2.0. Upstream copyright notices are retained.\nThis project builds on `UyNewNas/chen-theorem-lean`,\n`UyNewNas/analytic-number-theory-lean`, Mathlib, and the adapted source closure\nof `AlexKontorovich/PrimeNumberTheoremAnd`. Attribution is described in\n[`docs/PROVENANCE.md`](docs/PROVENANCE.md).\n",1789154827776]