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.

See RELATED_WORK.md for the historical progression across each domain.

All proofs in 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):

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 ,Zf380.lean ,Riemann104.lean ). 2. End-to-end mathematical semantics (Cryptids/Semantics/): - Goldbach (goldbach_21 and goldbach_25): Both goldbach_21 andgoldbach_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 machineriemann_120 , and the 744-state machineriemann_744 ) halt if and only if Mathlib'sRiemannHypothesis is false (Halts candidate ↔ ¬ RiemannHypothesis ,Halts riemann120 ↔ ¬ RiemannHypothesis , andHalts riemann744 ↔ ¬ RiemannHypothesis ). #

ZF Consistency (zf_380, zf_393, and zf_432): Both zf_380 and Wade's 393-state machinezf_393 (incorporating upstream fix10fcf152b3 ) halt if and only if O'Rear'szf2.nql Hilbert system proves$v_0 \in v_0$ (Halts candidate ↔ ZFInconsistent ,Halts zf393 ↔ ZFInconsistent ), and hence never halt inZFSet.{0} (not_halts_candidate ,not_halts_zf393 ), whilezf_432 halts if and only if the 20-clause deductive system enumerated byzf_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.

For the reduced machines in 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 state32 via a 5-cell window invariant (120 -> 121 ), replaces the two-phase89.8M -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 with leanprover/comparator:

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/ — Shared problem definitions (Propositions.lean ), Turing machine specifications (Machines.lean ), halting-equivalence proofs, and end-to-end semantics (Cryptids/Semantics/README.md ).
• Challenge.lean & comparator/ — Self-contained target theorem specification and configuration files forleanprover/comparator .
• artifacts/ — Canonical machine.txt transition tables and reduction write-ups forgoldbach_21 ,zf_380 , andriemann_104 .
• baselines/ — Verbatim upstream source files, canonical 5-tuple tables, andPROVENANCE.md .
• RELATED_WORK.md — Literature lineage and(Q, 2) Turing machine conventions.
• 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 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.

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.