[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"BUA1kA6WmC":3},"# The Complete Mathematical Works of Theodore J. Kaczynski, formalized in Lean 4\n\nThis repository formalizes every published mathematical paper of Theodore J. Kaczynski, together\nwith his doctoral dissertation (1964–1969). Most of them are about how functions in the disk and the\nhalf-plane behave at the boundary; the first is a new proof of Wedderburn's theorem. (Not included:\nthree short problems he posed in the *American Mathematical Monthly* and *Mathematics Magazine*.)\nIt contains:\n\n- a **blueprint**: one self-contained mathematical document that proves every result of these works;\n- a **Lean 4 formalization** of all of it (about 22,400 lines, on Mathlib);\n- an **errata** ([`ERRATA.md`](ERRATA.md)) recording where the published proofs needed a correction,\n  relied on outside theorems, or were replaced by different arguments.\n\n## Status\n\nAll **21 pinned headline theorems** are proved, and no declaration in the project uses `sorry`.\nEvery headline theorem depends only on Lean's three standard axioms (`propext`,\n`Classical.choice`, `Quot.sound`). The blueprint has 141 items, all formalized.\n\nThe results the sources quote from the literature are proved here, or the argument that used them\nis replaced:\n- the crosscut theorem (a form of the Jordan curve theorem for arcs across the disk);\n- Tietze's arc theorem;\n- the Poisson integral facts;\n- Banach's theorem on Baire classes;\n- an analytic non-Borel set;\n- Bagemihl's ambiguous-point theorem for continuous functions.\n\nTwo deep theorems the sources use, the full Jordan curve theorem and Kierst's theorem, are avoided by\ndirect arguments (see [`ERRATA.md`](ERRATA.md), §2). Wedderburn's theorem is proved by Kaczynski's\nargument, without Mathlib's own proof of it (Witt's), which is switched off in that file.\n\n## The sources\n\n| Source | Topic | Blueprint | Lean |\n|---|---|---|---|\n| *Amer. Math. Monthly* 71 (1964) | a group-theoretic proof of Wedderburn's theorem | §J | [`Wedderburn.lean`](TJK/Wedderburn.lean) |\n| *Math. Magazine* 41 (1968) | Sutcliffe's digit-reversal conjecture | §A | [`Sutcliffe.lean`](TJK/Sutcliffe.lean) |\n| *J. Math. Mech.* 14 (1965), known only from its review | boundary functions in the disk; homeomorphisms | §§E, G | [`BoundaryFn/`](TJK/BoundaryFn), [`Discont/`](TJK/Discont) |\n| *Michigan Math. J.* 13 (1966) | McMillan's F\u003Csub>σδ\u003C/sub> theorem; cluster sets | §D | [`McMillan/`](TJK/McMillan) |\n| Dissertation, *Boundary functions*, Michigan (1967) | Baire classes of boundary functions | §§B–G, App. A | [`Baire/`](TJK/Baire), [`Discont/`](TJK/Discont), [`Variant.lean`](TJK/Variant.lean) |\n| *Trans. AMS* 141 (1969) | realizing F\u003Csub>σδ\u003C/sub> sets and boundary functions | §§B, F | [`Construct/`](TJK/Construct) |\n| *Trans. AMS* 137 (1969) | boundary functions of bounded harmonic functions | §H | [`Harmonic/`](TJK/Harmonic) |\n| *Proc. AMS* 23 (1969) | a non-Borel set of curvilinear convergence in a cube | §I | [`Cube/`](TJK/Cube) |\n\nShared plane topology (arcs, crosscuts) is in [`Plane/`](TJK/Plane).\n\n## Headline theorems\n\nThe trusted statements are in [`TJK/Defs.lean`](TJK/Defs.lean) (definitions) and\n[`TJK/MainTheorems.lean`](TJK/MainTheorems.lean) (statements). They are frozen by\n[`blueprint/pinned.json`](blueprint/pinned.json), and `scripts/blueprint.py check` fails if any of them\nchanges. They include:\n\n- **McMillan's theorem** and Kaczynski's generalization to cluster sets: the set of curvilinear\n  convergence of a continuous function in the disk is F\u003Csub>σδ\u003C/sub>.\n- **Realization:** every F\u003Csub>σδ\u003C/sub> subset of ℝ is the set of curvilinear convergence of a\n  bounded continuous function in the half-plane. This answers McMillan's problem, with prescribed\n  boundary functions (Trans. AMS 141, (F)).\n- **Baire classes:** boundary functions are of honorary Baire class 2; a boundary function of a\n  Baire class ξ function is of class ξ + 1 (dissertation, Theorem 8).\n- **Homeomorphisms of the disk:** their boundary functions are continuous off a countable set\n  (1965, Theorem 1).\n- **Measure:** there are pairwise disjoint arcs, one at each point of ℝ, whose union is null\n  (dissertation, Theorem 10), with its corollaries.\n- **Bounded harmonic functions:** a bounded harmonic function with a boundary function that is not of\n  Baire class 1, answering Bagemihl–Piranian's Problem 5. Also a first-category theorem for harmonic\n  functions with bounded integral means.\n- **The cube:** a bounded continuous function in a cube whose set of curvilinear convergence is not\n  Borel.\n- **Wedderburn's theorem**, by Kaczynski's proof: every finite division ring is commutative.\n- **Sutcliffe's conjecture:** if base `n` has a 3-digit number that is multiplied by an integer when\n  its digits are reversed, it also has a 2-digit one.\n\n## Repository layout\n\n- [`paper/main.pdf`](paper/main.pdf): the blueprint, typeset (source: `paper/main.tex`, `paper/sections/`). Each numbered item has `\\lean{...}` (the\n  Lean declarations) and `\\uses{...}` (its dependencies). Appendix B maps every result of the\n  sources to its blueprint item.\n- `blueprint/map.md`: the generated dependency graph and per-item status.\n- `blueprint/DECISIONS.md`: the design log, covering how statements were encoded and the repairs.\n- `ERRATA.md`: the divergences from the published texts.\n- `TJK/`: the Lean sources. `TJK.lean` imports everything.\n- `scripts/blueprint.py`: blueprint tooling (`map`, `check`, `pin`).\n\n## Building\n\nYou need [elan](https://github.com/leanprover/elan) (the Lean toolchain manager). The toolchain is\npinned in `lean-toolchain` (Lean v4.33.1) and Mathlib in `lakefile.toml` (v4.33.1).\n\n```sh\ngit clone \u003Cthis repository>\ncd \u003Crepository>\nlake exe cache get     # download prebuilt Mathlib (otherwise Mathlib takes hours to compile)\nlake build             # build the formalization\n```\n\nA full build needs several GB of memory. On a small machine, limit parallelism with\n`LEAN_NUM_THREADS=2`.\n\nTo check the axioms yourself, create a file containing\n\n```lean\nimport TJK\n#print axioms TJK.Harmonic.exists_bounded_harmonic_boundaryFn_not_baireOne\n```\n\nand run `lake env lean \u003Cfile>`. Do the same for any declaration of `TJK/MainTheorems.lean`.\n\nThe blueprint tooling needs Python 3:\n\n```sh\npython scripts/blueprint.py check   # references, cycles, axioms, pinned statements\npython scripts/blueprint.py map     # regenerate blueprint/map.md\n```\n\nTo typeset the blueprint, run `latexmk -pdf main.tex` in `paper/`.\n\n## How this was made\n\nThe blueprint and the formalization were written by Claude, Anthropic's AI model, working as an\norchestrator of Lean-writing agents under the direction of the repository owner. The agent\ninstructions are in [`CLAUDE.md`](CLAUDE.md). Every proof is checked by Lean. The blueprint's prose\nproofs were also reviewed against the sources, and that review found the errors listed in\n[`ERRATA.md`](ERRATA.md).\n\n## License\n\nApache License 2.0 (see [`LICENSE`](LICENSE)), the license used by Mathlib. The original papers are\nnot included in this repository.\n",1792350344007]