Cryptids: Formally Verified Conjecture-Checking & Busy Beaver Turing Machines

This repository contains two Lean 4 (v4.35.0-rc4) formalization libraries for small Turing machines whose halting behavior encodes deep mathematical problems or complex dynamical invariants:

1. Conjecture-Checking Turing Machines (Cryptids/) — Certified state reductions (goldbach_21, zf_380, riemann_104) and end-to-end Lean 4 mathematical semantics proofs (Halts M ↔ ...) for Goldbach's Conjecture, ZF Set Theory Consistency, and the Riemann Hypothesis.
2. Busy Beaver Holdouts (BusyBeaver/) — Kernel-verified behavioral proofs (IsHalt, IsNotHalt, IsCryptid) for Busy Beaver 6 (BB(6)) holdouts (some are potential Cryptids, subject to community acceptance), connected to ConjectureMachines.Machine via BusyBeaver/Bridge.lean.

Overview

Library	Scope	Key Results	Documentation
Cryptids/	(Q, 2) Turing machines for Goldbach, ZF consistency, and the Riemann Hypothesis	Certified state reductions (25 → 21, 393 → 380, 120 → 104) and end-to-end halting equivalences (Halts M ↔ ¬ GoldbachConjecture, ZFInconsistent, ¬ RiemannHypothesis)	Cryptids/README.md, Cryptids/Semantics/README.md
BusyBeaver/	BB(6) holdout (6, 2) Turing machines (some are potential Cryptids, subject to community acceptance)	Shared execution semantics (BusyBeaver/Defs.lean), foundational lemmas (BusyBeaver/Basic.lean), RLE loader and ConjectureMachines bridge (BusyBeaver/Bridge.lean), and certified behavioral proofs	BusyBeaver/README.md

All Lean 4 proofs are checked by the kernel without sorry and depend only on the standard axioms (propext, Classical.choice, Quot.sound).

Building and Verifying

Fetch the precompiled Mathlib cache:

lake exe cache get

Build the conjecture-checking library (Cryptids and Challenge, default targets) and audit with leanprover/comparator:

lake build
lake comparator --config comparator/all.json
python3 verify.py

Build the Busy Beaver library (BusyBeaver and BusyBeaverChallenge) and audit with leanprover/comparator:

lake build BusyBeaver BusyBeaverChallenge
lake comparator --config comparator/busy_beaver.json

Repository Structure

• Cryptids/ — Conjecture-checking Turing machines (Machines.lean, Propositions.lean), state-reduction proofs (Goldbach21.lean, Zf380.lean, Riemann104.lean), and end-to-end mathematical semantics (Cryptids/Semantics/).
• BusyBeaver/ — Busy Beaver holdout specifications (Defs.lean), foundational lemmas (Basic.lean), RLE loader and bridge to ConjectureMachines (Bridge.lean), and machine behavioral proofs.
• Challenge.lean, BusyBeaverChallenge.lean & comparator/ — Self-contained target theorem specifications and configuration files for leanprover/comparator.
• artifacts/ — Canonical machine.txt transition tables and reduction write-ups for goldbach_21, zf_380, and riemann_104.
• baselines/ — Verbatim upstream source files, canonical 5-tuple tables, and PROVENANCE.md.
• RELATED_WORK.md — Literature lineage and (Q, 2) Turing machine conventions.
• verify.py — Standalone Python 3 verification script for the conjecture-checking machines.

Contributing

Pull requests and issues are welcome, especially for smaller Turing machines for these problems, machines for other conjectures, Busy Beaver holdout proofs, or streamlined Lean proofs. Please also open an issue if you notice anything amiss, such as a missing reference or provenance detail. See CONTRIBUTING.md for general guidelines.

Acknowledgements

We thank Eric Wieser for careful review, as well as the Busy Beaver Challenge community, int-y1 (whose BusyLean project pioneered Lean 4 Busy Beaver formalizations and directed-head configuration representations), Yijun Leng, Stefan O'Rear, Yuri Matiyasevich, Scott Aaronson, Adam Yedidia, Andrew Wade, and Joseph Shunia for the baseline Turing machines and compilers, and the Mathlib community.

License and disclaimer

Copyright 2026 Google LLC

All software is licensed under the Apache License, Version 2.0 (Apache 2.0); you may not use this file except in compliance with the Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at: https://www.apache.org/licenses/LICENSE-2.0

All other materials are licensed under the Creative Commons Attribution 4.0 International License (CC-BY). You may obtain a copy of the CC-BY license at: https://creativecommons.org/licenses/by/4.0/legalcode

Unless required by applicable law or agreed to in writing, all software and materials distributed here under the Apache 2.0 or CC-BY licenses are distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the licenses for the specific language governing permissions and limitations under those licenses.

This is not an official Google product.