On some open problems in commutative algebra resolved by Rethlas

Jiahong Yu; Jiedong Jiang; Liang Xiao; Yixiao Li; Yuefeng Wang; Zeming Sun

arxiv: 2605.25259 · v2 · pith:IORBQYHYnew · submitted 2026-05-24 · 🧮 math.AC

On some open problems in commutative algebra resolved by Rethlas

Jiedong Jiang , Yixiao Li , Zeming Sun , Yuefeng Wang , Liang Xiao , Jiahong Yu This is my paper

Pith reviewed 2026-06-29 22:22 UTC · model grok-4.3

classification 🧮 math.AC

keywords commutative algebraopen problemsautomated reasoningmathematical proofsproblem solvingnatural language

0 comments

The pith

A natural-language automated reasoning system has resolved several open problems in commutative algebra by generating self-contained proofs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reports on open problems in commutative algebra drawn from published lists that have been resolved using an automated reasoning system. The system produces self-contained proofs with no human intervention during their generation. These proofs are then verified by human experts for correctness. A sympathetic reader would care if this shows that automated systems can tackle and settle longstanding mathematical questions in the field.

Core claim

The central claim is that the automated reasoning system has resolved open problems in commutative algebra and related areas by producing self-contained proofs with no human intervention during generation, which are subsequently verified by human experts.

What carries the argument

The natural-language automated reasoning system that generates self-contained mathematical proofs for the stated open problems.

If this is right

Multiple problems from established lists in commutative ring theory are now resolved with provided proofs.
The approach applies to problems in Boij-Söderberg theory as well.
Self-contained proofs are available for each resolved problem for further study.
Verification by experts confirms the validity of the generated results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar automated approaches might be tested on open problems in other branches of mathematics.
The reliance on post-generation verification suggests a hybrid model for mathematical discovery.
Scaling this method could lead to systematic resolution of larger sets of problems.

Load-bearing premise

The problems selected were genuinely open before this work and that expert verification after generation ensures the proofs are mathematically correct.

What would settle it

An expert identifying a mathematical error in one of the generated proofs or showing that a listed problem was already resolved prior to this application.

read the original abstract

We report on a collection of open problems in commutative algebra and related areas that have been resolved (proved or disproved) using the Rethlas natural-language automated reasoning system. The problems are drawn from several published lists, including Open Problems in Commutative Ring Theory (Cahen-Fontana-Frisch-Glaz), Erman-Sam's survey of Boij-S\"oderberg theory. For each problem we record the precise statement and a self-contained proof produced (with no human intervention) by Rethlas and subsequently verified by human experts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims an AI system produced verifiable proofs for open commutative algebra problems listed in prior surveys, with the arguments included for direct checking, but gives no information on the system itself.

read the letter

The main thing to know is that this paper takes published lists of open problems in commutative algebra, runs them through something called Rethlas, and reports that the system output self-contained proofs which humans then verified. The manuscript is said to include both the problem statements and those proofs.

What is new is the application of this natural-language system to problems from the Cahen-Fontana-Frisch-Glaz collection and the Erman-Sam Boij-Söderberg survey. If the generated arguments are correct and actually settle questions that were open, that counts as a data point for AI-assisted resolution of math problems.

The paper does one thing well: by recording the full statements and proofs, it reduces the central claim to ordinary mathematical review rather than an unverifiable black-box assertion. The status of the problems as previously open rests on the cited surveys, which is the normal way such lists are used.

The soft spots are straightforward. The abstract supplies no description of Rethlas, no example of a generated proof, and no indication of proof length or difficulty. Without that, it is impossible to judge whether the system is doing non-trivial work or whether the results are robust. The human verification step is noted but does not replace checking the arguments.

This is mainly for readers interested in automated reasoning tools applied to algebra. A serious referee could evaluate the included proofs against the cited open questions and decide whether they hold. I would send it to peer review rather than desk reject, because the claims are checkable once the full text is examined.

Referee Report

2 major / 2 minor

Summary. The manuscript reports that a collection of open problems in commutative algebra, drawn from published lists such as Cahen-Fontana-Frisch-Glaz and Erman-Sam's survey on Boij-Söderberg theory, have been resolved (proved or disproved) by the Rethlas natural-language automated reasoning system. For each problem the paper records the precise statement together with a self-contained proof generated by Rethlas with no human intervention during generation; these proofs were subsequently verified by human experts.

Significance. If the generated proofs are correct and the problems were genuinely open, the work would demonstrate that natural-language automated reasoning can produce verifiable solutions to open questions in commutative algebra. The explicit inclusion of problem statements and proofs permits standard mathematical review rather than reliance on an opaque black-box claim, which strengthens the contribution.

major comments (2)

[Abstract] Abstract: the central claim that proofs were produced 'with no human intervention' during generation is asserted without any description of the Rethlas system, its input protocol, or safeguards against human guidance, which is load-bearing for the novelty of the automated-resolution result.
[Introduction (or equivalent section listing the problems)] The manuscript relies on the cited published surveys to establish that the selected problems were open, but provides no explicit check or statement confirming that no resolutions have appeared in the literature between the survey dates and the present work.

minor comments (2)

Notation for ring-theoretic objects (e.g., ideal names, module structures) should be standardized across all recorded proofs to aid readability.
The paper should include a brief table summarizing which problems were proved and which were disproved, together with the original source citation for each.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that proofs were produced 'with no human intervention' during generation is asserted without any description of the Rethlas system, its input protocol, or safeguards against human guidance, which is load-bearing for the novelty of the automated-resolution result.

Authors: We agree that the manuscript would benefit from an explicit description of the Rethlas system to support the claim of no human intervention during generation. In the revised version we will add a new subsection (immediately following the abstract) that describes the system architecture, the input protocol consisting solely of the natural-language problem statements drawn from the cited surveys, and the safeguards (interaction logs and absence of follow-up prompts) used to ensure autonomy of the generation process. revision: yes
Referee: [Introduction (or equivalent section listing the problems)] The manuscript relies on the cited published surveys to establish that the selected problems were open, but provides no explicit check or statement confirming that no resolutions have appeared in the literature between the survey dates and the present work.

Authors: We acknowledge that the manuscript does not contain an explicit statement confirming that the problems remained open after the dates of the cited surveys. In the revised manuscript we will insert a short paragraph in the introduction stating that a literature search was conducted via MathSciNet, arXiv, and Google Scholar from the publication dates of the surveys through the submission date of the present work, and that no resolutions were found. The search terms and date range will be recorded for transparency. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript is a report documenting problem statements drawn from external published surveys together with self-contained proofs generated by the external Rethlas system and verified by human experts. No derivation chain, equations, fitted parameters, or self-referential definitions appear. The claim that the problems were previously open rests on standard citation of published lists rather than any internal reduction, and correctness is externally checkable via the included arguments. This is the most common honest finding for a non-derivational report paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Only the abstract is available, so the ledger is based on limited information. The central claim rests on the existence and effectiveness of the Rethlas system, which is postulated without independent evidence or details.

invented entities (1)

Rethlas natural-language automated reasoning system no independent evidence
purpose: Generate self-contained proofs for open mathematical problems without human intervention during generation
The system is introduced in the abstract as the tool used but no description, architecture, or independent evidence of its capabilities is provided.

pith-pipeline@v0.9.1-grok · 5625 in / 1047 out tokens · 42446 ms · 2026-06-29T22:22:21.480795+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages · 1 internal anchor

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D. D. Anderson,Quasi-complete semilocal rings and modules, in:Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer, New York, 2014, pp. 25–37

2014

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Boij and J

M. Boij and J. S¨ oderberg,Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjec- ture, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 85–106

2008

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Cahen and J.-L

P.-J. Cahen and J.-L. Chabert,Integer-Valued Polynomials, Mathematical Surveys and Monographs 48, American Mathematical Society, Providence, RI, 1997

1997

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Cahen, M

P.-J. Cahen, M. Fontana, S. Frisch, S. Glaz,Open problems in commutative ring theory, in:Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, Springer, New York, 2014, pp. 353–375

2014

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David,A characteristic zero non-Noetherian factorial ring of dimension three, Trans

J. David,A characteristic zero non-Noetherian factorial ring of dimension three, Trans. Amer. Math. Soc. 180 (1973), 315–325

1973

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D. E. Dobbs, M. Fontana, S. Kabbaj,Direct limits of Jaffard domains and S-domains, Comment. Math. Univ. St. Pauli 39(2) (1990), 143–155

1990

[7] [7]

Eisenbud and F.-O

D. Eisenbud and F.-O. Schreyer,Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), no. 3, 859–888

2009

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Elliott,Birings and plethories of integer-valued polynomials, Actes des Rencontres du CIRM 2(2) (2010), 53–58

J. Elliott,Birings and plethories of integer-valued polynomials, Actes des Rencontres du CIRM 2(2) (2010), 53–58

2010

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Erman and S

D. Erman and S. V. Sam,Questions about Boij-S¨ oderberg theory, in:Surveys on Recent Developments in Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2017, pp. 285–304. arXiv:1606.01867

work page arXiv 2017

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Open Prob- lems in Commutative Ring Theory

J. D. Farley,Quasi-completeness and localizations of polynomial domains: A conjecture from “Open Prob- lems in Commutative Ring Theory”, Bull. Korean Math. Soc. 53 (2016), no. 6, 1613–1615

2016

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Glaz,Finite conductor rings with zero divisors, in:Non-Noetherian Commutative Ring Theory, MAIA 520, Kluwer Acad

S. Glaz,Finite conductor rings with zero divisors, in:Non-Noetherian Commutative Ring Theory, MAIA 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 251–270

2000

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Glaz,Finite conductor rings, Proc

S. Glaz,Finite conductor rings, Proc. Amer. Math. Soc. 129 (2001), no. 10, 2833–2843

2001

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Jensen,Completions of UFDs with semi-local formal fibers, Comm

D. Jensen,Completions of UFDs with semi-local formal fibers, Comm. Algebra 34 (2006), no. 1, 347–360

2006

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H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong,Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

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Zafrullah,Domains whose ideals meet a universal restriction, arXiv:2006.04135

M. Zafrullah,Domains whose ideals meet a universal restriction, arXiv:2006.04135

work page arXiv 2006

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Zafrullah,t-invertibility and Bazzoni-like statements, J

M. Zafrullah,t-invertibility and Bazzoni-like statements, J. Pure Appl. Algebra 214 (2010), 654–657. Jiedong Jiang, Westlake Institute for Advanced Study, Westlake University, No. 600 Dunyu Road, Sandun town, Xihu district, Hangzhou, Zhejiang, 310030, China. Email address:jiangjiedong@westlake.edu.cn Yixiao Li, Beijing International Center for Mathematica...

2010