---
id: "2026-10-16-ai-assisted-results-october-15-listing"
url: "https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/"
as_of: "2026-10-16T12:39:00+02:00"
date: "2026-10-16"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 12
editor: human editor
human_review: null
version: "2026-10-16"
---

As of: 2026-10-16 12:39 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/

# arXiv results with disclosed AI help, 15 Oct 2026 listing

Full title: arXiv results with disclosed AI help, 15 Oct 2026 listing: monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more

Continuation of the mid-October catalogue (closed at about 32 KB). It lists preprints from the 15 Oct 2026 arXiv listing (and a few late 14 Oct items) that settle or advance named problems and disclose an AI contribution, each AI statement quoted from the PDF. Own entries: Smale's mean value conjecture (GPT-6 Astra's Lean disproof and the Jatar–Ng bound M > 1.1018), the Hopf conjecture under T³ symmetry, all imaginary quadratic fields with class number ≤ 2000, the exact rate of Delsarte's LP bound, and LLM-discovered quantum circuits. Highlights here: Esterov's strong monodromy conjecture for high-order poles, where ChatGPT Astra "suddenly invented" a birational inequality; Becker–Lawton's GPT-6.1 Sol counterexamples to Fried's conjecture; and Cúth–Kalenda's negative answer to a Godefroy–Lancien–Zizler question. All items are unrefereed preprints.

## Key facts
- Own entries from this listing: Smale's mean value constant (Jatar & Ng, 2610.21472, building on GPT-6 Astra's Lean disproof); Hopf conjecture with T³-symmetry (Nienhaus & Wulle, 2610.22173); imaginary quadratic fields of class number ≤ 2000 (Kimura, 2610.21511, Claude Opus 5.5) with Kerr's simpler Siegel-zero proof (2610.21099); rate of the Delsarte bound (Salmon, 2610.21263, Opus 5.5 and GPT-5.6 Sol); quantum circuits found by Claude Mythos 5 (Gu, Kemeny, Yelin & Lukin, 2610.22367)
- AI 'invented' the key input: strong local monodromy conjecture for poles of order ≥ n − 1 (Alexander Esterov, 2610.22111): ChatGPT Astra 6.0 was asked to combine three inputs; 'The AI model generated a proof, but, instead of using (3top), suddenly invented (3), with a proof based on deep MMP techniques'; Esterov had started 'with the belief that such an extension was unrealistic'; the raw output is attached to the arXiv record 'as a non-human-verified output'
- Counterexamples from a one-line prompt: ordinary Fried conjecture fails for real-analytic metrics arbitrarily close to hyperbolic ones in dimension 3 (Simon Becker & Benjamin Lawton, 2610.22281): 'S.B. initially prompted GPT-6.1 Sol (Ultra reasoning, in Codex) with "Prove Conjecture 9" … The resulting argument led instead to the counterexample'; Lawton used Claude and Claude Code incl. Lean, which covers 'conditional algebraic and limiting steps' only
- Long-standing question answered: the Lipschitz-free space over a separable Banach space is complemented in its bidual iff the space is isomorphic to Hilbert space, answering negatively a question of Godefroy, Lancien and Zizler for ℓ₁ (Marek Cúth & Ondřej Kalenda, 2610.21818): Sol 5.6 and Astra models were asked 'to suggest solutions to intermediate problems … Through several iterations, this process led to the main ideas of the proof'
- Constructions and counterexamples from models: strong coloring numbers of d-dimensional grids grow as r^(d−1) (Piotr Micek & Torsten Ueckerdt, 2610.21506; 'The construction in Lemma 3 was found by OpenAI's GPT-5.6 Sol'); three families of counterexamples to the Marian–Oprea strange duality conjecture on K3 surfaces (Shizhuo Zhang & Yu Zhao, 2610.21272; 'suggested by GPT-6 Pro and GPT-Astra in ChatGPT')
- Follow-ups to OpenAI's manuscript release: Zhi-Wei Sun shows Hilbert's tenth problem over ℚ is undecidable already with 86 unknowns, starting from 'Hilbert's Tenth Problem over the field Q … was solved negatively by OpenAI' (2610.20691, 14 Oct; no AI statement found); a quantitative planar Falconer distance theorem developing 'the multiscale method of the September 2026 manuscript of OpenAI' (Cheng, Fang, Li, Qu & Xiao, 2610.22222)
- Human results with no AI statement found: the Bardos–Tartar conjecture for 2D Navier–Stokes (Barnes, Liew & Titi, 2610.21530); a counterexample to Plotnick's 1983 conjecture on knot exteriors (Hofmann & Nicholson, 2610.21051); Isaac's Condition (TU) false, Lean 4-verified, single author (Ralf Stephan, 2610.21899)

## What happened

Items below come from the PDFs of the 15 Oct 2026 arXiv listing (papers dated 14–15 Oct), read on 16 Oct. Each AI statement is
quoted or closely paraphrased from the paper. None is refereed.

### Own entries
- [Smale's mean value conjecture](/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/): Jatar & Ng ([2610.21472](https://arxiv.org/abs/2610.21472)) build on GPT-6 Astra's September Lean disproof of the constant-1 form and prove Smale's constant exceeds 1.1018.
- [Hopf conjecture with T³-symmetry](/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/) (Nienhaus & Wulle, [2610.22173](https://arxiv.org/abs/2610.22173)): first proof of the main theorem "obtained with OpenAI's ChatGPT-6-Astra and minimal human input".
- [Imaginary quadratic fields of class number ≤ 2000](/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/) (Kimura, [2610.21511](https://arxiv.org/abs/2610.21511)), built on OpenAI's Siegel-zero proof with Claude Opus 5.5 and Lean, plus Kerr's simpler proof ([2610.21099](https://arxiv.org/abs/2610.21099)).
- [Exact rate of Delsarte's LP bound](/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/) (Salmon, [2610.21263](https://arxiv.org/abs/2610.21263)).
- [Quantum circuits discovered by Claude Mythos 5](/e/2026-10-15-llm-discovered-quantum-circuits-harvard-anthropic-mythos/) (Gu, Kemeny, Yelin & Lukin, [2610.22367](https://arxiv.org/abs/2610.22367)).

### Large AI roles
- **Strong monodromy conjecture for high-order poles** (Alexander Esterov, [2610.22111](https://arxiv.org/abs/2610.22111)). Proves the strong local monodromy conjecture for poles of order at least n − 1 of the naive motivic zeta function of a polynomial in n variables, with explicit eigencycles. The abstract says the proof "combines topological techniques developed by the author with a fact of birational geometry discovered by AI". Section 1.2: given three inputs, ChatGPT Astra 6.0 "generated a proof, but, instead of using (3top), suddenly invented (3), with a proof based on deep MMP techniques". Esterov had believed such an extension "unrealistic"; the model's original output is attached to the arXiv record "as a non-human-verified output". (A smaller monodromy case, the symmetric determinant, was in the mid-October catalogue.)
- **Fried conjecture counterexamples** (Simon Becker & Benjamin Lawton, [2610.22281](https://arxiv.org/abs/2610.22281)). For real-analytic negatively curved metrics arbitrarily close to hyperbolic ones on 3-manifolds, |Ruelle zeta(0)| / torsion² tends to zero along acyclic unitary characters, refuting Fried's identity (stated as Conjecture 9 in a Zworski survey). "S.B. initially prompted GPT-6.1 Sol (Ultra reasoning, in Codex) with 'Prove Conjecture 9' … The resulting argument led instead to the counterexample." Lawton used Claude and Claude Code, including Lean for the "conditional algebraic and limiting steps"; geometric and spectral inputs are not formalized.
- **Godefroy–Lancien–Zizler question** (Marek Cúth & Ondřej Kalenda, [2610.21818](https://arxiv.org/abs/2610.21818)). The Lipschitz-free space over a separable Banach space is complemented in its bidual iff the space is Hilbertian, which answers negatively a "long-standing question" for ℓ₁. Sol 5.6 and Astra suggested solutions to intermediate problems; "Through several iterations, this process led to the main ideas of the proof."
- **Strong coloring numbers of grids** (Piotr Micek & Torsten Ueckerdt, [2610.21506](https://arxiv.org/abs/2610.21506)): scol_r(G(n,d)) = Ω_d(r^(d−1)), matching the volume upper bound. "The construction in Lemma 3 was found by OpenAI's GPT-5.6 Sol."
- **Strange duality on K3 surfaces** (Shizhuo Zhang & Yu Zhao, [2610.21272](https://arxiv.org/abs/2610.21272)): three families of counterexamples to the Marian–Oprea conjecture (the map is either zero or an isomorphism) and a refined statement. "The counterexamples in this paper were suggested by GPT-6 Pro and GPT-Astra in ChatGPT."
- **Jung–Kim conjecture** on centres of walled Brauer algebras (Shunsuke Hirota, [2610.21576](https://arxiv.org/abs/2610.21576)): the idea of Lemma 2.8 and the argument avoiding stable module categories "originated in suggestions from Astra", which also proposed the formula for the arc-algebra centre.
- **Positroids, Theta and Delta conjectures** (Hai Zhu, [2610.21894](https://arxiv.org/abs/2610.21894)): "the author used GPT-5.5 to prove Lemmas 5.7 and 5.10"; the core recursions were "made without AI assistance".
- **Quadratic APN permutations in odd dimension** (Hiroki Fukui, [2610.21169](https://arxiv.org/abs/2610.21169), submitted to IEEE Trans. Inf. Theory): Claude, Claude Code and ChatGPT "proposed and drafted proofs"; Lean code written with Claude Code, checked by the Lean kernel and an independent checker.
- **Nullstellensatz over large fields** (Pablo Cubides Kovacsics & Sebastián Uribe Rincón, [2610.21039](https://arxiv.org/abs/2610.21039)): calls itself "perhaps the authors last attempt to provide a contribution more than 80% human"; still, Claude Opus 4.8 suggested extending Theorem 4.4 to every non-large field and gave early proofs of Theorems 5.11 and 5.15; DeepSeek V4 helped the presentation.
- **Entanglement-assisted spectrum of graphs** (Nidhal Mghirbi & Seth Douglas, [2610.20775](https://arxiv.org/abs/2610.20775), 14 Oct): a continuum of Rényi points beside the Lovász number; generative AI used "throughout … including for mathematical exploration, proof development and critique".
- **Evolution algebras survey and open problems** (Kaygorodov, Martín González & Viruel, [2610.22248](https://arxiv.org/abs/2610.22248)): Harmonic's Aristotle helped with literature search, error checking, experiments and Lean verification of the new observations in Section 2.1.

### Smaller roles
Galois deformation rings in characteristic two (Yufan Luo, [2610.21298](https://arxiv.org/abs/2610.21298); ChatGPT searched for examples that became Theorems 1.7–1.9); interlacing conjecture for Sylvester–Kac singular values (Tang & Shi, [2610.21702](https://arxiv.org/abs/2610.21702); ChatGPT 6 Astra "only for language polishing and for screening counterexamples"); area-vanishing square peg conjecture (Lorenzo Valerio, [2610.21076](https://arxiv.org/abs/2610.21076); Gemini 3.1 Pro and GPT-6 Astra to find errors, "minimal use"); generalized Viterbo conjecture counterexample (Broćić, Serraille & Trifa, [2610.22175](https://arxiv.org/abs/2610.22175); thanks for GPT 6 Astra access, "AI-assisted generalizations" to appear in a later version); asymptotic Erdős–Frankl–Füredi cover-free conjecture (Xinqi Huang & Hong Liu, [2610.22021](https://arxiv.org/abs/2610.22021); AI for literature and editing, and it "suggested a simplification" of a calculation); coarse embeddability characterisations (Li, Zhang & Zhu, [2610.21632](https://arxiv.org/abs/2610.21632); ChatGPT for "exploring mathematical ideas"); Motsch–Tadmor propagation of chaos (Kldiashvili, Rossi & Trussardi, [2610.21828](https://arxiv.org/abs/2610.21828); ChatGPT-5.6 Sol for brainstorming); quantitative planar Falconer (Cheng, Fang, Li, Qu & Xiao, [2610.22222](https://arxiv.org/abs/2610.22222); an AI tool for "local mathematical reasoning"); generalized Erdős–Ginzburg–Ziv constants in rank three (Zhang & Zhang, [2610.21577](https://arxiv.org/abs/2610.21577); routine calculations); King–Tollu–Toumazet conjecture on rectangular Kostka coefficients (Deepanshu Prasad, [2610.21921](https://arxiv.org/abs/2610.21921); Claude for editing only); nilpotent multicorrelation sequences (Andreevski & Hernández, [2610.22076](https://arxiv.org/abs/2610.22076); "No AI tools were used to produce any of the results", only a GPT-6 Astra Max referee-style report).

### Follow-ups to OpenAI's manuscript release
Zhi-Wei Sun, who had brought Hilbert's tenth problem over ℤ down to 10 unknowns, now proves it undecidable over ℚ with 86 unknowns, starting from "Hilbert's Tenth Problem over the field Q of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction" ([2610.20691](https://arxiv.org/abs/2610.20691); no AI statement found). Cheng, Fang, Li, Qu & Xiao give a quantitative planar Falconer theorem by developing "the multiscale method of the September 2026 manuscript of OpenAI" ([2610.22222](https://arxiv.org/abs/2610.22222)). Kimura and Kerr (own entry) build on the Siegel-zero manuscript; Salmon (own entry) on the August "Ten advances". The Hopf paper notes an announced AI counterexample to the Hopf conjecture in dimension 6 "not verified by geometers".

### No AI statement found (for contrast)
The **Bardos–Tartar conjecture** (1973): for 2D Navier–Stokes with periodic boundary conditions, the set of solutions at a positive time is dense in L² (Barnes, Liew & Titi, [2610.21530](https://arxiv.org/abs/2610.21530), dedicated to Claude Bardos); a counterexample to Plotnick's conjecture on knot exteriors (Hofmann & Nicholson, [2610.21051](https://arxiv.org/abs/2610.21051)); Isaac's Condition (TU) for simple normality of √s is false, all results "machine-verified in Lean 4", single author (Ralf Stephan, [2610.21899](https://arxiv.org/abs/2610.21899)); uniform non-homogeneous vector bundles against the Drézet and Ballico–Ellia conjecture (Macaulay2, [2610.20208](https://arxiv.org/abs/2610.20208)).

No Hacker News threads on any of these were found on 16 Oct (Algolia); X was not searched directly.

## Why it matters

The disclosures keep getting more specific: which lemma, which construction, even the one-line prompt ("Prove Conjecture 9") that led a model to a counterexample instead. Two experts (Esterov, Nienhaus–Wulle) say a model produced the key step of a result in their own research program, and the October OpenAI release is already being used as a black box by other authors (Sun, Kimura).

## What is disputed or not yet verified

- Verification: Unrefereed preprints; some with partial Lean formalizations; AI roles self-reported

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 169 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 108 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 199 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 138 days after its cutoff

## Sources

1. [arXiv 2610.22111: The monodromy conjecture for high-order poles, and norms of residue antiderivatives (Esterov)](https://arxiv.org/abs/2610.22111) (arxiv.org, paper)
2. [arXiv 2610.22281: Counterexamples to the ordinary Fried conjecture near hyperbolic metrics (Becker & Lawton)](https://arxiv.org/abs/2610.22281) (arxiv.org, paper)
3. [arXiv 2610.21818: Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual (Cúth & Kalenda)](https://arxiv.org/abs/2610.21818) (arxiv.org, paper)
4. [arXiv 2610.21506: Strong coloring numbers of grids (GPT-5.6 Sol construction)](https://arxiv.org/abs/2610.21506) (arxiv.org, paper)
5. [arXiv 2610.21272: A refinement of the strange duality conjecture on K3 surfaces (GPT-6 Pro counterexamples)](https://arxiv.org/abs/2610.21272) (arxiv.org, paper)
6. [arXiv 2610.21576: A proof of the Jung–Kim conjecture on the centers of walled Brauer algebras (GPT-6 Astra)](https://arxiv.org/abs/2610.21576) (arxiv.org, paper)
7. [arXiv 2610.21894: Algebras connecting positroids, the Theta and Delta conjectures (GPT-5.5 lemmas)](https://arxiv.org/abs/2610.21894) (arxiv.org, paper)
8. [arXiv 2610.21169: Optimal reduced representatives of quadratic APN permutations (Claude Code + Lean)](https://arxiv.org/abs/2610.21169) (arxiv.org, paper)
9. [arXiv 2610.21039: Holomorphy rings and the Nullstellensatz over large fields (Claude Opus 4.8)](https://arxiv.org/abs/2610.21039) (arxiv.org, paper)
10. [arXiv 2610.20691: 86 unknowns for Hilbert's tenth problem over the rationals (Zhi-Wei Sun)](https://arxiv.org/abs/2610.20691) (arxiv.org, paper)
11. [arXiv 2610.22222: A quantitative form of the planar Falconer distance theorem](https://arxiv.org/abs/2610.22222) (arxiv.org, paper)
12. [arXiv 2610.21530: The Bardos-Tartar conjecture for the Navier-Stokes equations](https://arxiv.org/abs/2610.21530) (arxiv.org, paper)

Also cited in the text:
- [2610.21472](https://arxiv.org/abs/2610.21472)
- [2610.22173](https://arxiv.org/abs/2610.22173)
- [2610.21511](https://arxiv.org/abs/2610.21511)
- [2610.21099](https://arxiv.org/abs/2610.21099)
- [2610.21263](https://arxiv.org/abs/2610.21263)
- [2610.22367](https://arxiv.org/abs/2610.22367)
- [2610.20775](https://arxiv.org/abs/2610.20775)
- [2610.22248](https://arxiv.org/abs/2610.22248)
- [2610.21298](https://arxiv.org/abs/2610.21298)
- [2610.21702](https://arxiv.org/abs/2610.21702)
- [2610.21076](https://arxiv.org/abs/2610.21076)
- [2610.22175](https://arxiv.org/abs/2610.22175)
- [2610.22021](https://arxiv.org/abs/2610.22021)
- [2610.21632](https://arxiv.org/abs/2610.21632)
- [2610.21828](https://arxiv.org/abs/2610.21828)
- [2610.21577](https://arxiv.org/abs/2610.21577)
- [2610.21921](https://arxiv.org/abs/2610.21921)
- [2610.22076](https://arxiv.org/abs/2610.22076)
- [2610.21051](https://arxiv.org/abs/2610.21051)
- [2610.21899](https://arxiv.org/abs/2610.21899)
- [2610.20208](https://arxiv.org/abs/2610.20208)

## Changes

- 2026-10-16 (filed): Created from the PDFs of the 15 Oct listing; five results got own entries

## Related

- 2026-10-15: [Hopf's Euler characteristic conjecture proved under T³ symmetry](https://postcutoff.com/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/index.md)
- 2026-10-15: [All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI's Siegel-zero proof made explicit in Lean with Claude Opus 5.5](https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/index.md)
- 2026-10-15: [Exact asymptotic rate of Delsarte's linear programming bound for binary and spherical codes found, with key steps from Claude Opus 5.5 and GPT-5.6 Sol](https://postcutoff.com/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/index.md)
- 2026-10-15: [Harvard–Anthropic team (with Mikhail Lukin) uses Claude Mythos 5 in an evolutionary search to beat state-of-the-art quantum circuits on seven of eight tasks](https://postcutoff.com/e/2026-10-15-llm-discovered-quantum-circuits-harvard-anthropic-mythos/index.md)
- 2026-10-14: [Mid-October 2026: arXiv results with disclosed AI help](https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/index.md)
- 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)
- 2026-09-03: [GPT-6 Astra autonomously disproves Smale's mean value conjecture in Lean](https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/index.md)