Extremal descendant integrals on moduli spaces of curves: An inequality discovered and proved in collaboration with AI

Abstract: For the pure $ψ$-class intersection numbers $D(\textbf{e})=\langle τ{e_1} \cdots τ{e_n} \rangle_g$ on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves, we determine for which choices of $\textbf{e}=(e_1, \ldots, e_n)$ the value of $D(\textbf{e})$ becomes extremal. The intersection number is minimal for powers of a single $ψ$-class (i.e. all $e_i$ but one vanish), whereas maximal values are obtained for balanced vectors ($|e_i - e_j| \leq 1$ for all $i,j$). The proof uses the nefness of the $ψ$-classes combined with Khovanskii--Teissier log-concavity. Apart from the mathematical content, this paper is also meant as an experiment in collaborations between human mathematicians and AI models: the proof of the above result was found and formulated by the AI models GPT-5 and Gemini 3 Pro. Large parts of the paper were drafted by Claude Opus 4.5, and a part of the argument was formalized in Lean with the help of Claude Code and GPT-5.2. The paper aims for maximal transparency on the authorship of different sections and the employed AI tools (including prompts and conversation logs).

• The paper establishes that ψ-class intersection numbers achieve a minimum when exponents are concentrated and a maximum when they are evenly balanced.
• It employs geometric and combinatorial methods, including log-concavity and majorization arguments, alongside AI assistance for conjecture generation and formal verification in Lean.
• Empirical validations using SageMath and admcycles confirm the theoretical extremal characterizations across various genera and marked points.

Extremal $\psi$-Class Intersection Numbers on Moduli Spaces: An AI-Driven Perspective

Context and Main Results

This work establishes a precise characterization of the extremal (minimal and maximal) pure $\psi$-class intersection numbers $$D(e) = \langle \tau_{e_1} \cdots \tau_{e_n} \rangle_g$$ on the Deligne–Mumford moduli space $\mathcal{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points. These intersection numbers are prominent invariants in algebraic geometry and mathematical physics and lie at the heart of the Witten–Kontsevich theorem relating intersection theory on $\mathcal{M}_{g,n}$ to integrable hierarchies.

The key contribution is the proof, through a combination of geometric and combinatorial techniques, that these intersection numbers attain their minimum when all exponent weight is concentrated on a single marking (i.e., $e = (d,0,\dotsc,0)$ for $d = 3g-3+n$), and their maximum when the exponents are as evenly balanced as possible—specifically, when $|e_i - e_j| \leq 1$ for all $i$, $j$. The proof methodology notably involves the use of AI LLMs, both for conjecture generation and formal proof, and provides a case study in contemporary AI-assisted mathematical discovery.

Theoretical Framework and Techniques

The paper frames the extremal problem on the set $E(g,n)$ of $n$-tuples of non-negative integers summing to $d = 3g-3+n$. For each $e \in E(g,n)$, the associated intersection number $D(e)$ is considered. The central technical ingredients are:

• Nefness and Symmetry of $\psi$-Classes: Each $\psi_i$ class is nef, and the symmetric group $S_n$ acts by permutation on the markings, yielding symmetry in the intersection numbers under coordinate permutations.
• Khovanskii–Teissier Log-concavity: Intersection numbers of nef divisor classes on projective varieties satisfy log-concavity properties. Applied here, this gives (for all $t$ in an appropriate interval):

$(S_t)^2 \geq S_{t-1}S_{t+1},$

where $S_t$ parametrizes the descendant integral upon shifting exponent weights between two marked points.

• Combinatorial Balancing (Majorization Argument): Using log-concavity and palindromicity, the intersection numbers along “slices” (where only two exponents vary, their sum fixed) are shown to be unimodal and to attain maxima at the most balanced point, i.e., when $e_i$ and $e_j$ are as close as possible.
• String and Dilaton Equations: Applied to deduce the minimal value formula by concentrating exponents, reducing $D(e)$ recursively to known one-point intersection numbers.

The essence of the extremal result is formalized in an abstract optimization statement for functions on weak compositions with symmetry, log-concavity, and positivity axioms, and proven via iterative rebalancing and exponent concentration.

Formal and Empirical Validation

The maximality claim for balanced vectors is elementary in genus zero due to the closed formula

$$\langle \tau_{e_1} \cdots \tau_{e_n} \rangle_0 = \frac{(n-3)!}{e_1! \cdots e_n!},$$

where the multinomial coefficient is maximized exactly when the $e_i$ are as close as possible. For $g > 0$, where explicit closed formulas are unavailable, the proof leverages the aforementioned geometric and combinatorial properties, providing a uniform argument applicable in all genera.

Empirical verification is also performed, with computational checks for all $0 \leq g \leq 11$ and $1 \leq n \leq 11$ using SageMath and the admcycles package, confirming the validity of the extremal characterization in practice.

AI Assistance and Formalization

A significant aspect of this study is the use of state-of-the-art LLMs (including OpenAI GPT-5 and Google Gemini 3 Pro) for both the discovery of the balanced maximality conjecture and parts of its proof, documented through interactions and methodology summaries. Notably, the combinatorial optimization core was fully formalized in Lean 4, with all code generated exclusively by AI systems. The approach decouples the generic optimization lemma from the geometric input, clarifying the logical structure and enabling machine verification of the non-geometric part.

Implications and Perspectives

The main result enriches the qualitative theory of descendant integrals, highlighting a “balancing–concentration dichotomy” in their extremal behavior. Practically, the precise knowledge of extremal values of intersection numbers has potential implications for the study of moduli space asymptotics, random structures on $\mathcal{M}_{g,n}$, and optimization problems in enumerative geometry.

The findings also prompt new research directions, notably:

• Explicit Computation for Balanced Configurations: While the one-point intersection numbers have closed formulas, it remains to find analogous concise expressions or generating functions for the balanced maximal cases in general genus.
• Extension to Other Enumerative Invariants: The techniques naturally invite the investigation of extremality for other families of intersection numbers, such as Hodge integrals and double Hurwitz numbers, potentially using AI-aided conjecture and proof methodologies.
• AI Attribution and Mathematical Practice: By meticulously documenting AI contributions, the work sets a standard for transparency in AI-assisted mathematical research, addressing emerging norms for attribution and reproducibility.

Conclusion

This study provides a rigorous characterization of the extremal pure $\psi$-class intersection numbers on the Deligne–Mumford moduli spaces. The extremal structure—minima at concentrated and maxima at balanced exponent tuples—is established by geometric and combinatorial principles (nefness, log-concavity, and symmetry), and formalized through both traditional mathematical writing and Lean verification. The AI-assisted workflow both accelerated the conjecture-to-proof loop and set methodological precedents, signaling future developments in human-AI mathematical collaboration and the systematic study of extremal enumerative invariants.

Reference: "Extremal descendant integrals on moduli spaces of curves: An inequality discovered and proved in collaboration with AI" (2512.14575).