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Modulo-9 Kanade–Russell identities and their Nahm-sum duals

For each i∈{1,2,3,4,5}i\in\{1,2,3,4,5\}, prove the corresponding modulo-99 Kanade–Russell Rogers–Ramanujan-type sum–product identity, namely KRisum(q)=KRiproduct(q)\mathrm{KR}^{\mathrm{sum}}_i(q)=\mathrm{KR}^{\mathrm{product}}_i(q). The supplied source does not state the five explicit qq-series and infinite-product expressions.

References

Primary source

arXiv

Additional references

modular-formsnumber-theoryq-series
Posted

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims to complete the modulo-9 Kanade–Russell program, but the result has not been independently verified.

The problem concerns five modulo-99 Kanade–Russell identities, four associated Nahm-sum duals, and a product formula for the fifth dual companion, originally conjectured by Kanade and Russell.

Known results

  • A 2021 finite-version study proved identity (4) and related reflected cases, without resolving the full package.
  • Andrews–van Ekeren–Heluani (2022) proved one relevant identity; other cases were said to follow from their results.
  • 2024 papers continued to describe the modulo-99 identities as conjectural, including an explicitly remaining Nahm-sum example.

September 2026 claimed resolution

On September 8, 2026, an unrefereed preprint claimed proofs of all five identities, four dual identities, and a product formula for the fifth dual companion. If correct, this completes the modulo-99 portion of the program; the claim remains unverified.

Current status (as of September 2026): A preprint claims the complete five-identity and dual-identity package, but no verified complete proof is recorded and the problem remains open pending review.

Sources

Solutions 0

No solutions have been posted yet.