google-deepmind/cryptids

google-deepmind/cryptids

Language: Lean

License: Apache-2.0

Stars: 2

Forks: 0

Open issues: 0

Created: 2026-10-09T10:27:36Z

Pushed: 2026-10-12T14:31:12Z

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README:

Certified State Reductions and Semantics of Conjecture-Checking Turing Machines

This repository contains formally verified 2-symbol Turing machines and Lean 4 semantics proofs for Goldbach's Conjecture, ZF Set Theory Consistency, and the Riemann Hypothesis, developed by Maxence Faldor and Moritz Firsching using Gemini Argon.

Formalized Machines

| Conjecture | Machine | States | Source / Provenance | Proved Halting Condition (Halts M ↔ ...) | Lean 4 Proofs | | :--- | :--- | :--- | :--- | :--- | :--- | | Goldbach's Conjecture | goldbach_21 | 21 | Reduced from goldbach_25 (25 states, Yao / Leng 2024) | ¬ GoldbachConjecture | [Goldbach21.lean](Cryptids/Goldbach21.lean), [Semantics/Goldbach.lean](Cryptids/Semantics/Goldbach.lean) | | ZF Consistency | zf_380 | 380 | Reduced from zf_393 (393 states, Wade Aug 2026) | ZFInconsistent (¬ Halts candidate proved) | [Zf380.lean](Cryptids/Zf380.lean), [Semantics/Zf393.lean](Cryptids/Semantics/Zf393.lean) | | ZF Consistency | zf_393 | 393 | Andrew Wade (`turing_machine_explorer` 9c5c2e82, Aug 2026) | ZFInconsistent (¬ Halts zf393 proved) | [Zf393.lean](Cryptids/Zf393.lean), [Semantics/Zf393.lean](Cryptids/Semantics/Zf393.lean) | | ZF Consistency | zf_432 | 432 | Andrew Wade (`turing_machine_explorer` 33b30300, Aug 2025) | WadeZFInconsistent | [Semantics/Zf.lean](Cryptids/Semantics/Zf.lean) | | Riemann Hypothesis | riemann_104 | 104 | Reduced from riemann_120 (120 states, Shunia Sept 2026) | ¬ RiemannHypothesis | [Riemann104.lean](Cryptids/Riemann104.lean), [Semantics/Riemann120.lean](Cryptids/Semantics/Riemann120.lean) | | Riemann Hypothesis | riemann_120 | 120 | Joseph M. Shunia (`arXiv:2609.30306`, Sept 2026) | ¬ RiemannHypothesis | [Riemann120.lean](Cryptids/Riemann120.lean), [Semantics/Riemann120.lean](Cryptids/Semantics/Riemann120.lean) | | Riemann Hypothesis | riemann_744 | 744 | Matiyasevich, O'Rear & Aaronson (riemann_744_sorear.tm, 2016) | ¬ RiemannHypothesis | [Semantics/Riemann.lean](Cryptids/Semantics/Riemann.lean) |

See [RELATED_WORK.md](RELATED_WORK.md) for the historical progression across each domain.

What Is Formally Proved

All proofs in [Cryptids/](Cryptids/) are checked by the Lean 4 kernel (v4.35.0-rc4) without sorry and use only the standard axioms (propext, Classical.choice, Quot.sound):

1. **State reduction and halting equivalence (candidate.states 104) splits merged scanner state 32 via a 5-cell window invariant (120 -> 121), replaces the two-phase 89.8M-step bootstrap (-9 states) with a self-bootstrapping header and dynamic on-demand zero padding (padTotal -> padTotal + 1), and consolidates return scanners and binary call-stack frames (mapStack, -8 states).

Building and Verifying

Build all Lean 4 proofs:

lake exe cache get
lake build

Independently audit the proofs against [Challenge.lean](Challenge.lean) with [leanprover/comparator](comparator/README.md):

lake comparator --config comparator/all.json

Run the standalone zero-dependency Python verifier (re-compiles upstream baseline sources, checks SHA-256 digests and transition tables, and simulates 250,000 steps per domain):

python3 verify.py

Repository Structure

• [Cryptids/](Cryptids/) — Shared problem definitions ([Propositions.lean](Cryptids/Propositions.lean)), Turing machine specifications ([Machines.lean](Cryptids/Machines.lean)), halting-equivalence proofs, and end-to-end semantics ([Cryptids/Semantics/README.md](Cryptids/Semantics/README.md)).
• [Challenge.lean](Challenge.lean) & [comparator/](comparator/) — Self-contained target theorem specification and configuration files for [leanprover/comparator](comparator/README.md).
• [artifacts/](artifacts/) — Canonical machine.txt transition tables and reduction write-ups for [goldbach_21](artifacts/goldbach_21/README.md), [zf_380](artifacts/zf_380/README.md), and [riemann_104](artifacts/riemann_104/README.md).
• [baselines/](baselines/) — Verbatim upstream source files, canonical 5-tuple tables, and [PROVENANCE.md](baselines/PROVENANCE.md).
• [RELATED_WORK.md](RELATED_WORK.md) — Literature lineage and (Q, 2) Turing machine conventions.
• [verify.py](verify.py) — Standalone Python 3 verification script.

Contributing

Pull requests and issues are welcome, especially for smaller Turing machines for these problems, machines for other conjectures, 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](CONTRIBUTING.md) for general guidelines.

Acknowledgements

We thank Eric Wieser for careful review, as well as the Busy Beaver Challenge community, 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.