Formalizing $A_1^{(1)}$ Curve Neighborhoods in Lean 4

2026-04-25

AI Summary

This work establishes a combinatorial foundation for quantum Schubert calculus on affine flag manifolds by axiom-free formalization in Lean 4 of the curve neighborhood for type $A_1^{(1)}$. The infinite dihedral group $D_\infty$ is modeled as a Coxeter system, with an explicit length function and degree map defined. Reachable sets are characterized via bounded edge chains, and their maximal vertices precisely describe the curve neighborhood. This constitutes the first complete implementation in a proof assistant of an explicit combinatorial formula for the $A_1^{(1)}$ curve neighborhood, yielding a fully computable and executable construction algorithm that verifies the correctness of neighborhood formulas for arbitrary elements.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Combinatorial curve neighborhoods are somewhat foundational when setting up the quantum Schubert calculus for affine flag manifolds. In the specific case of type $A_1^{(1)}$, you can encode these neighborhoods entirely within the moment graph of the infinite dihedral group $D_\infty$. Building on the framework developed by Mihalcea and Norton, this paper presents a complete, axiom-free formalization of these combinatorial curve neighborhoods in Lean 4. Rather than just wrapping mathematical statements, we formalized $D_\infty$ directly as a Coxeter system to explicitly compute length functions and degree maps. Reachable sets are defined through edge chains bounded by specific degrees, and we ultimately characterize the curve neighborhood by the maximal vertices inside these sets. The core effort here lies in formally verifying the explicit combinatorial formulas for curve neighborhoods of arbitrary elements. Interestingly, by restricting our search space to finite sets, we also managed to extract a fully computable version of these neighborhoods.

Problem

• curve neighborhoods
• quantum Schubert calculus
• affine flag manifolds
• Coxeter system
• combinatorial formulas

Innovation

• curve neighborhoods
• Lean 4 formalization
• Coxeter system
• quantum Schubert calculus
• computable combinatorics