Lab Report · Number Theory

Some new modular Nahm sums of ranks 3 and 4

arXiv:2606.13590v1 · Zhineng Cao and Liuquan Wang · Number Theory

Announcement tweet + AI — This report summarizes the paper’s scientific intent: finding new modular Nahm sums, proving modularity through Rogers–Ramanujan type identities, and extending the known landscape of Nahm’s problem in ranks 3 and 4.

Overview

Nahm’s problem asks which Nahm sums are modular. This paper discovers six new families of modular Nahm sums in ranks 3 and 4. The authors obtain rank three examples by modifying Zagier’s rank three examples, derive several rank four families through lift-dual operations, and find another rank four family by the constant term method.

To prove modularity, the paper establishes Rogers–Ramanujan type identities that express the Nahm sums as infinite products. Those product forms make the modular structure visible.

Infographic Summary

Infographic titled Demystifying Small-Bias Spaces in Modular Combinatorics, summarizing arXiv 2606.13590 with panels for lab report, observation, classical context, and next steps.
The report focuses on scientific intent: new modular Nahm sums, proof mechanisms, classical context, and next steps.

Engineering Statement

Intent

Identify new modular Nahm sums of ranks 3 and 4 and justify their modularity.

Evidence

Rogers–Ramanujan type identities, Bailey pairs, constant term methods, and lift-dual constructions.

Outputs

Six new modular families, product identities, modular triples, and open directions for dual triples.

Next steps

Study Nahm’s problem, inspect the identities, compare rank 3 and rank 4 constructions, and track future modular examples.

How this paper advances Nahm’s problem

Nahm’s problem asks for modular triples: data defining Nahm sums whose resulting q-series are modular. Rank one is classified, but higher ranks are much more complicated.

This paper adds six new modular families in ranks 3 and 4. Some arise by modifying known rank three examples; others come from lift-dual operations applied to known tadpole Nahm sums. Another family is obtained using the constant term method.

The companion notebook can serve as a reader-facing entry point: it should introduce Nahm sums, modular triples, product identities, and the paper’s proof strategy without attempting to reproduce every identity in full detail.

Explore Further · Next Steps

🧠 Continue learning

  • Review q-series and Rogers–Ramanujan identities.
  • Learn the role of Bailey pairs in proving product identities.
  • Compare rank 3 and rank 4 modular triples.

🤝 Participate locally

  • Use the report as a reading guide.
  • Share the infographic with readers interested in modular forms.
  • Return later with one identity or example to inspect more closely.