Overview
A machine-guided recurrence boundary method has been developed for affine families of positive-definite Nahm sums. This method integrates a universal coordinate-contiguous relation, a tropical face-limit theorem, and a recurrence–boundary principle. The approach leverages machine learning and reinforcement learning for discovery, while all final proofs are generated through exact symbolic certificates.
Research Context
The research addresses the properties of Nahm sums, specifically focusing on their modularity. A central application of this developed methodology is the proof of Shi and Wang's Conjecture 3.8 concerning the Nahm sums dual to Zagier's twelfth rank-three example (arXiv:2607.23257).
Approach
The methodology, termed the recurrence–boundary method, operates through several interconnected components:
- Universal Coordinate-Contiguous Relation: This component establishes exact algebraic cells for finite certificates.
- Tropical Face-Limit Theorem: This theorem identifies specific parameter directions along which higher-rank Nahm sums degenerate into lower-rank theta or theta–hypergeometric boundary values.
- Recurrence–Boundary Principle: This principle enables the recovery of initial sums from a sufficient number of independent boundary limits.
Machine learning and reinforcement learning are employed exclusively for the discovery phase. Evolutionary symbolic search is utilized to propose recurrences and asymptotic rays derived from exact algebraic data. Concurrently, a Q-learning agent searches for short sequences of legal contiguous-cell identities. It is critical to note that no learned output is directly accepted as proof; instead, every successful candidate discovered through these learning processes is subsequently replaced by an exact symbolic certificate to ensure rigor.
Findings
The machine-guided recurrence boundary theory successfully proved Shi and Wang's Conjecture 3.8 for the Nahm sums dual to Zagier's twelfth rank-three example. The search process yielded several key elements:
- A second-order recurrence was found for a one-parameter family of these Nahm sums.
- A five-cell certificate was identified for this recurrence.
- Two asymptotic rays were discovered, which lead to binary and unary theta series.
These findings provided two linear equations for the two initial sums. Solving the resultant $2\times2$ system reduced the conjecture to two generalized-eta identities. These identities were then certified through valence arguments performed on $\Gamma_1(300)$ and $\Gamma_1(100)$.
Consequently, the dual of Zagier's twelfth example was proven to be modular. Furthermore, every member of the affine family associated with this example was shown to be an explicit $\mathbb{Z}[q,q^{-1}]$-combination of two base products. Complete verification artifacts are available alongside the paper.
Why This Matters
The establishment of the recurrence–boundary method, validated by its application to Shi and Wang's Conjecture, provides a systematic approach for analyzing affine families of positive-definite Nahm sums. The proof of modularity for the dual of Zagier's twelfth rank-three example contributes to the understanding of these mathematical objects.