[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"h8XKDciUKy":3},"# Kakeya in dimension three\n\nThis is a standalone Lean 4 formalization of the three-dimensional Kakeya\nargument following Guth, Wang and Zahl: every compact set in real Euclidean\nthree-space containing a unit segment in every direction has Hausdorff\ndimension three. The statement is reached in two layers. `KakeyaDimensionThree`\nproves it from one explicit mathematical input, and `Unconditional.KakeyaDimensionThree`\ndischarges that input, so the conjecture is obtained with no hypothesis, no\n`sorry` and no project axiom.\n\n`KakeyaDimensionThree` takes one explicit mathematical input as a hypothesis:\n`StickyKakeya.StickyFrostmanHypothesis`, the project's formulation of GWZ\nTheorem 7.3(A), asserted for every ambient real inner-product space satisfying\n`Module.finrank Real E = 3`. That input is *not* proved in the `Kakeya` library,\nso the library on its own establishes an implication. It declares no axiom of\nits own; the remaining axioms are Lean's `propext`, `Classical.choice`, and\n`Quot.sound`. See [PROOF-PATH.md](PROOF-PATH.md) for the exact entry points.\n\nThe hypothesis is discharged by the `Unconditional` library in this repository,\nwhich yields\n\n```lean\ntheorem Unconditional.KakeyaDimensionThree : KakeyaSetConjecture 3\n```\n\nwith no `sorry` and no project axiom. See\n[The unconditional result](#the-unconditional-result) below; that library needs a\nsecond development, which this repository does not vendor and which you download\nyourself.\n\n## Build and check\n\nUse Linux or WSL2. Install [elan](https://github.com/leanprover/elan), Git, Bash, and Python 3.11 or\nlater. The committed `lean-toolchain` selects Lean `v4.32.0-rc1` and\n`lake-manifest.json` fixes all dependency commits, including Mathlib\n`1b0782d8191b03e0001caac10e1601d17f2cd580`.\n\n```sh\nlake exe cache get\npython3 verification/check.py --scope core\n```\n\nThe cache download is optional; `lake build` can compile the pinned dependencies\nfrom source. Keep the manifest when reproducing the build. `lake update` changes\nthe dependency resolution and is not part of these verification commands.\n\n`verification/run.sh` scans every project Lean source outside ignored build\nproducts, builds the entire `Kakeya` library, and checks the expected public\naxiom closures through `FinalCheck.lean`. A source census and compiler check\nserve different purposes: the former finds unfinished proof tokens and\nconfirms that the sources declare no axiom, while the latter verifies\nelaborated declarations and their actual dependencies.\n\nThis command also runs the verifier regressions and writes logs and a receipt\nunder `.verification-results/`. It checks the conditional library. The full\nverification command below adds the linked proof and both comparator targets.\nSee [RELEASING.md](RELEASING.md) for reproduction instructions, receipt contents,\nand resource requirements.\n\n## Source layout\n\n| Path | Content |\n| --- | --- |\n| `Kakeya/` | Definitions, infrastructure and proofs |\n| `Kakeya.lean` | Complete library import root |\n| `FinalCheck.lean` | Public theorem and axiom assertions |\n| `Unconditional/` | The linking layer that discharges the Sticky hypothesis |\n| `Unconditional.lean` | Import root of the linking layer |\n| `verification/` | Source, axiom and comparator checks |\n| `verification/unconditional/` | Build and check tooling for the linking layer |\n| `upstream/3d-sticky-kakeya` | Submodule reference to the second development |\n\nThe repository contains the Kakeya mathematical development and its maintained\nverification tools. See [ATTRIBUTION.md](ATTRIBUTION.md) for mathematical\nattribution and dependency information.\n\nComments cite labels of the form `lem:...`, `def:...` and `note:...`. These name\nthe statements of the informal proof outline that the formalization follows; the\noutline itself is not distributed here, and the labels are kept because they\nrecord which informal statement a Lean declaration corresponds to.\n\n## The unconditional result\n\nGWZ record that their Theorem 7.3(A), the hypothesis of `KakeyaDimensionThree`,\nis Theorem 5.2 of Wang and Zahl, *The Assouad dimension of Kakeya sets in R3*.\nThat theorem has been formalized, unconditionally, by Nankai University and the\nByteDance Seed AI4Math Team in\n[`M32026/3d-sticky-kakeya`](https://github.com/M32026/3d-sticky-kakeya) as\n`Kakeya.Assouad.PureWZ2Theorem5_2Unconditional`.\n\nThe two formulations of the hypothesis are not the same Lean statement: GWZ\nDefinition 7.1(A) imposes a Frostman condition on the classes of a nested cover\nhierarchy, while the Assouad paper's Definition 2.12 asks for Convex Wolff Axioms\non covers at every nearby scale, and GWZ Remark 7.2 relates the two without proof.\nThe `Unconditional/` library closes that gap. Its endpoints are\n\n| Declaration | Module |\n| --- | --- |\n| `stickyFrostmanHypothesis_of_pureWZ2` | `Unconditional/StickyFrostman.lean` |\n| `Unconditional.KakeyaDimensionThree` | `Unconditional/KakeyaConjecture.lean` |\n\n`Unconditional` is not a default build target, because it imports the second\ndevelopment. That development is not vendored here: `upstream/3d-sticky-kakeya`\nis a Git submodule recording only its URL and the reviewed revision. Fetch it and\nrun one command:\n\n```sh\ngit submodule update --init upstream/3d-sticky-kakeya\npython3 verification/check.py --scope full --jobs 2\n```\n\nPass `--upstream DIR` if you keep a checkout elsewhere. The recorded revision must\nmatch `bytedance_commit` in `verification/unconditional/bridge-lock.json`; the\nbuild refuses to proceed otherwise. The command checks the current Numina source\nidentity and the pinned upstream checkout, builds `Unconditional`, and asserts\nthe axiom closure of both endpoints through\n`verification/unconditional/AxiomCheck.lean`, which must report exactly\n\n```\n[propext, Classical.choice, Quot.sound]\n```\n\nBoth sides are compiled against Lean `v4.32.0-rc1` and Mathlib\n`1b0782d8191b03e0001caac10e1601d17f2cd580`. The pinned identities, the five exact\nhash-checked local notation mappings that reconcile renamed Mathlib identifiers,\nand the build driver are in `verification/unconditional/`; its `configure` step\nrecords the Numina commit and source hash, verifies the clean upstream commit\nand every package pin, and checks the exact compiler revision before building.\nThe full command then runs the conditional and unconditional comparator targets;\nthe latter states the conjecture using Mathlib imports only. Add `--require-clean`\nwhen producing evidence for a committed release candidate.\n\nThe first bridge build compiles all 4,706 upstream modules in the endpoint's\nimport closure from source; externally built upstream artifacts are not adopted.\nLater runs can reuse artifacts with matching local build receipts. At the default\n`--jobs 2`, budget more than six hours for a first full run. Use a machine with at\nleast 24 GB RAM and 25 GB free disk space. `--jobs 8` can reduce build time on\nmachines with enough memory for more concurrent compiler processes. See\n[RELEASING.md](RELEASING.md) for resource measurements and receipt details. The\ncore verification command does not require the upstream checkout.\n\n## References and attribution\n\n- Guth, Wang and Zahl, [A streamlined proof of the Kakeya conjecture in\n  R3](https://arxiv.org/abs/2601.14411). The argument formalized here.\n- Wang and Zahl, [The Assouad dimension of Kakeya sets in\n  R3](https://arxiv.org/abs/2401.12337), Invent. Math. 241(1):153-206, 2025.\n  Theorem 5.2 is GWZ Theorem 7.3(A), the estimate used by the conditional endpoint.\n- Wang and Zahl, [Volume estimates for unions of convex sets, and the Kakeya\n  set conjecture in three dimensions](https://arxiv.org/abs/2502.17655).\n- Wang and Zahl, [Sticky Kakeya sets and the sticky Kakeya\n  conjecture](https://arxiv.org/abs/2210.09581).\n- Nankai University and ByteDance Seed AI4Math Team,\n  [3d-sticky-kakeya](https://github.com/M32026/3d-sticky-kakeya). Formalization of\n  Wang-Zahl Theorem 5.2, the input discharged by the linking library.\n\nLean sources originate in Project Numina's Kakeya development. Existing source\nnotices are preserved; see [ATTRIBUTION.md](ATTRIBUTION.md), [NOTICE](NOTICE)\nand [LICENSE](LICENSE).\n\n## Acknowledgements\n\nWe are deeply grateful to Professor [Hong Wang](https://sites.google.com/view/hongwang/home)\nand Professor [Xiao Ma](https://sites.google.com/view/xiaom-homepage) for their generous help\nand guidance.\n\nWe thank the Nankai University and ByteDance Seed AI4Math Team for their\nformalization of Wang-Zahl Theorem 5.2 in\n[3d-sticky-kakeya](https://github.com/M32026/3d-sticky-kakeya), and for their work\non integrating it with this development, which is what makes the unconditional\nresult above possible.\n\n## Contributors and citation\n\nSee [CONTRIBUTORS.md](CONTRIBUTORS.md) for the contributor list and\n[CITATION.cff](CITATION.cff) for citation metadata. Until a release is archived,\ncite the repository with the exact commit used. Upstream attribution and the\npending upstream release terms are recorded in [ATTRIBUTION.md](ATTRIBUTION.md).\n",1789154825079]