On a question of Mauri and Moraga

Jihao Liu

arxiv: 2605.22052 · v1 · pith:S7OZSJWXnew · submitted 2026-05-21 · 🧮 math.AG

On a question of Mauri and Moraga

Jihao Liu This is my paper

Pith reviewed 2026-05-22 03:11 UTC · model grok-4.3

classification 🧮 math.AG

keywords log Calabi-Yau pairsbig divisorsbirational geometrycounterexamplesboundary decompositionalgebraic geometry

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The pith

Log Calabi-Yau pairs whose boundaries decompose into big divisors provide negative answers to both parts of a question posed by Mauri and Moraga.

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

The paper investigates log Calabi-Yau pairs in algebraic geometry where the boundary breaks down into a sum of big divisors. It directly addresses a two-part question from Mauri and Moraga about the properties these pairs should satisfy. By constructing explicit examples, the work shows that neither part of the question holds in general. This refines expectations about how boundaries behave in these pairs and limits the scope of possible classifications. Readers interested in birational geometry would see this as closing off certain optimistic predictions about the structure of such pairs.

Core claim

The central claim is that log Calabi-Yau pairs exist whose boundary is a sum of big divisors yet fail to meet the conditions or conclusions proposed in the Mauri-Moraga question. The paper supplies counterexamples that serve as negative answers to both parts of that question.

What carries the argument

Explicit counterexamples of log Calabi-Yau pairs with boundaries decomposing into big divisors that violate the expected properties.

If this is right

Any attempt to classify log Calabi-Yau pairs must now include examples where the boundary consists of big divisors that do not force the previously expected behavior.
Questions about the decomposition of boundaries in log Calabi-Yau pairs must be reexamined without assuming the properties hold universally.
The minimal model program for pairs with big divisor boundaries requires adjustments to account for these additional cases.
Similar questions about other divisor conditions in log Calabi-Yau pairs may need direct counterexample checks rather than general proofs.

Where Pith is reading between the lines

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

This result may indicate that the big divisor condition alone is insufficient to guarantee toric or other rigid structures in log Calabi-Yau pairs.
It could link to broader problems in birational geometry concerning when divisor decompositions impose strong restrictions on the pair.
Testing analogous questions for non-log Calabi-Yau varieties or different boundary conditions might produce comparable counterexamples.

Load-bearing premise

The constructed counterexamples accurately show that the properties in the question do not hold for these pairs.

What would settle it

The discovery of a log Calabi-Yau pair with boundary decomposing into big divisors that satisfies the properties stated in the Mauri-Moraga question would disprove the negative answers.

read the original abstract

We give negative answers to both parts of a question of Mauri and Moraga on log Calabi-Yau pairs whose boundary decomposes into big divisors. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

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 gives negative answers to Mauri-Moraga's question on log Calabi-Yau pairs but rests entirely on unverified AI-generated proofs.

read the letter

The paper claims negative answers to both parts of Mauri and Moraga's question about log Calabi-Yau pairs whose boundary decomposes into big divisors. It does this by producing counterexamples, and the abstract is direct about the target. That is the core contribution: a targeted negative result in a corner of birational geometry on log pairs. If the arguments hold, it would close off the conjectured properties for this class of pairs and might steer future work away from expecting those behaviors. The paper earns credit for naming the question clearly and stating what it claims to disprove. Negative answers can be useful when they are correct, and the abstract does not overclaim beyond that scope. The work stays within the literature on log Calabi-Yau pairs and big divisors without obvious circularity or invented entities. The stress-test concern lands squarely. The central result is explicitly attributed to generative AI (ChatGPT 5.5 pro and the Rethlas system) with no mention of human verification, formalization in Lean or Coq, or even a walkthrough of the key steps such as constructing log resolutions or confirming bigness and the Calabi-Yau condition. In this area, those checks are delicate and easy to get wrong in ways that look locally plausible. Without any independent check, the counterexamples remain unconfirmed. This is not a minor presentation issue; it is the entire foundation of the claimed result. The paper is mainly for specialists already working on log pairs and questions in the style of Mauri-Moraga who might want to test the counterexamples themselves. A broader algebraic geometry reader would get little without the details and the verification. I would not send this to peer review as written. It needs a human author to verify the AI output and present the arguments with evidence of correctness before it deserves referee time. If that verification happens and the math checks out, then it could be worth engaging; right now the lack of confirmation makes it premature.

Referee Report

1 major / 0 minor

Summary. The paper claims to give negative answers to both parts of a question of Mauri and Moraga on log Calabi-Yau pairs whose boundary decomposes into big divisors. It states that the main result is obtained by generative AI, particularly ChatGPT 5.5 pro and the Rethlas system.

Significance. If the counterexamples are correct, the result would provide a negative resolution to the Mauri-Moraga question, with potential implications for the classification and properties of log Calabi-Yau pairs in birational geometry. No machine-checked proofs, reproducible code, or parameter-free derivations are present to strengthen the assessment.

major comments (1)

[Abstract] Abstract: The central claim of negative answers to both parts of the question rests entirely on the output of generative AI tools, yet the manuscript provides no human verification, independent derivation, or checkable details on key steps such as the existence of log resolutions, confirmation that boundary components are big, or verification that the pairs satisfy the log Calabi-Yau condition while violating the conjectured properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and for highlighting the reliance on generative AI in our work. We address the concerns about verification below and outline revisions to strengthen the presentation of the counterexamples.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim of negative answers to both parts of the question rests entirely on the output of generative AI tools, yet the manuscript provides no human verification, independent derivation, or checkable details on key steps such as the existence of log resolutions, confirmation that boundary components are big, or verification that the pairs satisfy the log Calabi-Yau condition while violating the conjectured properties.

Authors: We acknowledge that the main results were obtained via generative AI (ChatGPT 5.5 pro and the Rethlas system), as explicitly stated in the manuscript. The authors reviewed the AI-generated constructions and confirmed through direct inspection that the pairs admit log resolutions, that the boundary components are big, and that the log Calabi-Yau condition holds while the conjectured properties fail. We agree that the current manuscript lacks sufficient checkable details on these steps. In the revised version we will add an appendix containing explicit descriptions of the pairs, the log resolutions used, and step-by-step verifications of bigness and the log Calabi-Yau property. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; result presented as direct AI-generated counterexamples

full rationale

The paper states that negative answers to the Mauri-Moraga question are obtained via generative AI (ChatGPT 5.5 pro and Rethlas system), with the abstract framing this as a direct provision of counterexamples for log Calabi-Yau pairs. No equations, parameters, or self-citations are exhibited that reduce a claimed prediction or uniqueness result back to its own inputs by construction. The derivation chain is externalized to the AI output rather than internally self-referential or fitted, satisfying the criteria for a self-contained (if externally unverifiable) argument. No load-bearing self-citation chains, ansatz smuggling, or renaming of known results appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available. No explicit free parameters, axioms, or invented entities are stated. The use of generative AI implies unexamined assumptions about the correctness of the AI output.

pith-pipeline@v0.9.0 · 5549 in / 1043 out tokens · 45280 ms · 2026-05-22T03:11:22.290074+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

de Fernex, J

T. de Fernex, J. Koll\'ar, and C. Xu, The dual complex of singularities, in Higher dimensional algebraic geometry: in honor of Professor Yujiro Kawamata's sixtieth birthday, Adv. Stud. Pure Math. 74 (2017), Math. Soc. Japan, Tokyo, 103--129

work page 2017
[2]

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

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Mauri and J

M. Mauri and J. Moraga, Birational complexity and dual complexes, Proc. Lond. Math. Soc. (3) 131 (2025), no. 2, Paper No. e70073

work page 2025
[4]

Reid, Young person's guide to canonical singularities, Algebraic geometry, Bowdoin 1985, Proc

M. Reid, Young person's guide to canonical singularities, Algebraic geometry, Bowdoin 1985, Proc. Symp. Pure Math. 46 (1987), Part 1, 345--414

work page 1985

[1] [1]

de Fernex, J

T. de Fernex, J. Koll\'ar, and C. Xu, The dual complex of singularities, in Higher dimensional algebraic geometry: in honor of Professor Yujiro Kawamata's sixtieth birthday, Adv. Stud. Pure Math. 74 (2017), Math. Soc. Japan, Tokyo, 103--129

work page 2017

[2] [2]

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

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

Mauri and J

M. Mauri and J. Moraga, Birational complexity and dual complexes, Proc. Lond. Math. Soc. (3) 131 (2025), no. 2, Paper No. e70073

work page 2025

[4] [4]

Reid, Young person's guide to canonical singularities, Algebraic geometry, Bowdoin 1985, Proc

M. Reid, Young person's guide to canonical singularities, Algebraic geometry, Bowdoin 1985, Proc. Symp. Pure Math. 46 (1987), Part 1, 345--414

work page 1985