Full Lyapunov spectrum of the Sompolinsky–Crisanti–Sommers random neural network derived, a problem named open in 1988; GPT-6 Astra produced the first derivation (David G. Clark, Flatiron)
Event confirmedAwaiting review
The takeaway
A theoretical-neuroscience problem posed in the 1988 paper that founded the field’s chaos theory now has an analytic answer. The first derivation came from GPT-6 Astra working alone for 100 minutes; Claude Opus 5.5 wrote all code and figures.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 11 October 2026
Your AI and this story
- GPT-6 Astra161 days after its cutoff
- Claude Opus 5.5100 days after its cutoff
- Gemini 3.8 Flash191 days after its cutoff
- Grok 4.7130 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 100 days before it.
Key facts
- Paper: ‘Lyapunov spectrum of random neural networks’, David G. Clark (Flatiron Institute), arXiv 2610.12426 [cond-mat.dis-nn], 8 Oct 2026; code and data at github.com/davidclark1/rnn-lyapunov
- Claim (introduction): Sompolinsky et al. ‘named the “evaluation of the whole distribution of Lyapunov exponents” of this network as an open problem nearly four decades ago. The present work solves this problem.’
- Method: a finite-N identity expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics (tangent trajectory chosen by a minimum-norm condition); a cavity method then reduces it at large N to a self-consistent single-site problem. Result: the chaos is extensive
- Use of AI: ‘GPT-6 Astra produced an initial derivation. Working with both models, the author then arrived at the more physically transparent derivation presented in Sec. 2. Claude Opus 5.5 wrote all code and generated all figures, under the direction of the author, and assisted in writing and revision of the text.’
- AI methodology appendix: ‘The author had previously assumed that the present problem was analytically intractable.’ GPT-6 Astra (Max thinking), in a session where it had already derived other results on random recurrent networks, ‘worked independently for 100 minutes’ after the prompt ‘Derive the full N → ∞ Lyapunov spectrum of the SCS model (see Engelken for numerics)… Do not stop until you have nailed the answer.’ Its derivation used a theorem of Deninger equating the sum of positive Lyapunov exponents with a Fuglede–Kadison log-determinant
- Rigour: the exchange of the large-N limit with the long-time and zero-regularizer limits ‘was assumed’ (in both the AI and the final derivation); the theory is checked against known results (maximum exponent, Sompolinsky et al.’s δ → 0 result) and against numerics of Engelken, Wolf and Abbott
- Acknowledgments: Ashok Litwin-Kumar proposed ‘this problem as a target for AI-assisted solution’
- Not found: expert reactions or press coverage as of 11 Oct 2026
What happened
In 1988 Haim Sompolinsky, Andrea Crisanti and Hans-Jürgen Sommers showed that a large network of randomly coupled neurons becomes chaotic, which became a standard model for chaos in neural circuits. Their paper listed the full distribution of Lyapunov exponents (how fast nearby trajectories separate in every direction) as an open problem. Engelken, Wolf and Abbott later computed it numerically, but no analytic theory existed.
David G. Clark’s preprint gives a large-N theory for the whole spectrum. Its “Use of AI” section and appendix describe the process. Ashok Litwin-Kumar suggested the problem as a target for AI. GPT-6 Astra, prompted inside a session where it had already derived other results on random networks, worked alone for 100 minutes and produced a derivation that matched simulations, using a theorem of Deninger on Fuglede–Kadison determinants. Working with GPT-6 Astra and Claude Opus 5.5, Clark then found the more physically transparent cavity-method derivation presented in the paper. Claude Opus 5.5 wrote all code and figures.
Why it matters
It is a long-standing problem in theoretical neuroscience and statistical physics, not pure mathematics, settled with an AI-first derivation and an unusually explicit methodology appendix, including the prompt. It extends the 2026 wave of AI-assisted results beyond mathematics. Like most physics derivations, it is not a rigorous proof: limit exchanges are assumed.
What is disputed or not yet verified
| Verification | Unrefereed preprint; physics-level derivation (limit exchanges assumed) checked against simulations and known special cases |
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Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 3 primary
Primary
Changes
- Filed from the arXiv PDF (abstract, AI methodology appendix, acknowledgments) found by the 2026-10-11 sweep