# Certified State Reductions and Semantics of Conjecture-Checking Turing Machines

> Source: <https://github.com/google-deepmind/cryptids>
> Published: 2026-10-15 10:10:29+00:00

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.

| 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`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Goldbach21.lean) ,[`Semantics/Goldbach.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Goldbach.lean) | 
| ZF Consistency | `zf_380` | `380` | Reduced from `zf_393` (`393` states, Wade Aug 2026) | `ZFInconsistent` (`¬ Halts candidate` proved) | [`Zf380.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Zf380.lean) ,[`Semantics/Zf393.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Zf393.lean) | 
| ZF Consistency | `zf_393` | `393` | Andrew Wade ( [`turing_machine_explorer`](https://codeberg.org/ajwade/turing_machine_explorer)` 9c5c2e82` , Aug 2026) | `ZFInconsistent` (`¬ Halts zf393` proved) | [`Zf393.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Zf393.lean) ,[`Semantics/Zf393.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Zf393.lean) | 
| ZF Consistency | `zf_432` | `432` | Andrew Wade ( [`turing_machine_explorer`](https://codeberg.org/ajwade/turing_machine_explorer)` 33b30300` , Aug 2025) | `WadeZFInconsistent` | [`Semantics/Zf.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Zf.lean) | 
| Riemann Hypothesis | `riemann_104` | `104` | Reduced from `riemann_120` (`120` states, Shunia Sept 2026) | `¬ RiemannHypothesis` | [`Riemann104.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Riemann104.lean) ,[`Semantics/Riemann120.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Riemann120.lean) | 
| Riemann Hypothesis | `riemann_120` | `120` | Joseph M. Shunia ( [`arXiv:2609.30306`](https://arxiv.org/abs/2609.30306) , Sept 2026) | `¬ RiemannHypothesis` | [`Riemann120.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Riemann120.lean) ,[`Semantics/Riemann120.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Riemann120.lean) | 
| Riemann Hypothesis | `riemann_744` | `744` | Matiyasevich, O'Rear & Aaronson ( `riemann_744_sorear.tm` , 2016) | `¬ RiemannHypothesis` | [`Semantics/Riemann.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/Riemann.lean) | 

See [`RELATED_WORK.md`](https://github.com/google-deepmind/cryptids/blob/main/RELATED_WORK.md) for the historical progression across each domain.

All proofs in [`Cryptids/`](https://github.com/google-deepmind/cryptids/blob/main/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 < reference.states` and `Halts candidate ↔ Halts reference`):** Each reduced record machine (` goldbach_21` ,`zf_380` ,`riemann_104` ) has fewer states than its reference machine (`goldbach_25` ,`zf_393` ,`riemann_120` ) and halts from the blank tape if and only if the reference machine halts ([`Goldbach21.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Goldbach21.lean) ,[`Zf380.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Zf380.lean) ,[`Riemann104.lean`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Riemann104.lean) ).
2. 
**End-to-end mathematical semantics ([`Cryptids/Semantics/`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/README.md)):**  - 
**Goldbach (`goldbach_21` and `goldbach_25`):** Both` goldbach_21` and`goldbach_25` halt if and only if Goldbach's conjecture is false (`Halts candidate ↔ ¬ GoldbachConjecture` ).
  - 
**Riemann Hypothesis (`riemann_104`, `riemann_120`, and `riemann_744`):** All three machines (` riemann_104` , Shunia's 120-state machine`riemann_120` , and the 744-state machine`riemann_744` ) halt if and only if Mathlib's`RiemannHypothesis` is false (`Halts candidate ↔ ¬ RiemannHypothesis` ,`Halts riemann120 ↔ ¬ RiemannHypothesis` , and`Halts riemann744 ↔ ¬ RiemannHypothesis` ).
  - 
**ZF Consistency (`zf_380`, `zf_393`, and `zf_432`):** Both` zf_380` and Wade's 393-state machine`zf_393` (incorporating upstream fix`10fcf152b3` ) halt if and only if O'Rear's`zf2.nql` Hilbert system proves$v_0 \in v_0$ (`Halts candidate ↔ ZFInconsistent` ,`Halts zf393 ↔ ZFInconsistent` ), and hence never halt in`ZFSet.{0}` (`not_halts_candidate` ,`not_halts_zf393` ), while`zf_432` halts if and only if the 20-clause deductive system enumerated by`zf_432_wade.tm` derives$v_0 \in v_0$ (`WadeZFInconsistent` ).
3. 

Details on the proof architecture and mathematical background are in [`Cryptids/Semantics/README.md`](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/README.md).

For the reduced machines in [`artifacts/`](https://github.com/google-deepmind/cryptids/blob/main/artifacts), the state-space transformations affect execution step counts as follows:

- **`goldbach_21` (`339,411` vs. `170,966` steps through `n = 40`, `1.99x`):** Eliminating 4 states (` 1, 8, 10, 17` ) reverses the partition enumeration to test larger summands first and routes divisor acceptance through the main counter-tape scanner.
- **`zf_380` (`+0.093%` step overhead at `1,000,000` steps; `0` step overhead during `39,753`-step bootstrap):** Eliminating 13 states (` 65, 68, 266, 278, 322, 335, 341, 353, 359, 361, 379, 387, 389` ) via a 5-cell sliding-window invariant (`validState` ) and non-destructive leftward head bounces (`1, 3, 5, 7, 9` steps) preserves 100% bit-exact tape equality after every macro-step.
- **`riemann_104` (`9,037` vs. `89,775,610` bootstrap steps, `9,934.23x` faster initialization; `498,382` vs. `111,294,346` total steps over the first `1,000` register instructions, `223.31x` speedup):** Eliminating 16 states (`120 -> 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).

Build all Lean 4 proofs:

```
lake exe cache get
lake build
```

Independently audit the proofs against [`Challenge.lean`](https://github.com/google-deepmind/cryptids/blob/main/Challenge.lean) with [` leanprover/comparator`](https://github.com/google-deepmind/cryptids/blob/main/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
```

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

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`](https://github.com/google-deepmind/cryptids/blob/main/CONTRIBUTING.md) for general guidelines.

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.

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