Gemini Argon shrinks Goldbach, Riemann and ZF Turing machines
Result confirmed
Status
- Claim
Result confirmed
- Our reporting
- High confidence
- Verification
- Lean-verified
- Importance
- 3 of 5
- Last verified
- 16 October 2026
Your AI and this story
- GPT-6 Astra162 days after its cutoff
- Claude Opus 5.5101 days after its cutoff
- Gemini 4 Argon192 days after its cutoff
- Grok 4.7131 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 101 days before it.
Key facts
- Goldbach: goldbach_21 reduces the 25-state machine (Yao / Leng 2024) by removing 4 states; Lean proof that Halts M ↔ ¬GoldbachConjecture
- Riemann Hypothesis: riemann_104 reduces Joseph Shunia’s 120-state machine (Sept 2026, arXiv 2609.30306), itself down from O’Rear’s 744 states (2016); Halts M ↔ ¬RiemannHypothesis using Mathlib’s definition
- ZF: zf_380 reduces Andrew Wade’s 393-state machine (Aug 2026), down from 432 (2025); its halting is equivalent to O’Rear’s Hilbert system deriving a contradiction, and the repo proves it never halts in ZFSet
- The repo also has kernel-checked behaviour proofs for some BB(6) holdout machines (potential ‘Cryptids’, subject to community acceptance)
- All proofs checked by the Lean kernel without sorry, using only the standard axioms; reviewed by Eric Wieser; Lean v4.35.0-rc4
- AI role, as stated in the repo: the machines and Lean semantics were ‘developed by Maxence Faldor and Moritz Firsching using Gemini Argon’; no paper or blog post yet
- Context: Yedidia and Aaronson (2016) started this line of work, which bounds where the Busy Beaver function becomes independent of ZF; records are tracked on the bbchallenge wiki
What happened
The repo appeared quietly on DeepMind’s GitHub on Oct 9 and was found by our sweep’s watched-accounts step on Oct 16 (44 stars). It is a formal-verification package, not an announcement: the AI role is one line in the library README and in the history table, which credits each new record to “Faldor & Firsching, using Gemini Argon”.
Why it matters
Smaller machines tighten the known bounds on where the Busy Beaver function stops being computable from standard mathematics. Each reduction comes with machine-checked proofs, so the new records can be checked without trusting the model or the authors. It is also one of the first Gemini 4 Argon research results we have seen.
Sources
3 sources from 2 sites. Numbers match the chips in the text.
3 sources: 2 primary, 1 reaction
Primary
- GitHub: google-deepmind/cryptids (Some Cryptids and Lean proofs)github.com, code
- cryptids: Cryptids/README.md (records table and provenance)github.com, code
Reactions
- bbchallenge wiki: Cryptidswiki.bbchallenge.org, discussion
Changes
- Filed (read the README, library README, RELATED_WORK and PROVENANCE)