Rank-r twisted convolution identity conjecture

For every rank- $r$ admissible tuple, the rank- $r$ twisted convolution identity for the modular quantum dilogarithm holds. The supplied sources do not state the identity's explicit formula or define admissible tuples sufficiently to expand this statement further.

References

Primary source

arXiv

Additional references

The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms — arXiv — Marcus Appleby, Steven T. Flammia, Gene S. Kopp

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

An unrefereed paper extends the construction to higher ranks, but the main identity and the associated Hermitian configurations remain conditional rather than proved.

The conjecture concerns a rank- $r$ twisted convolution identity and its connection to higher-rank SIC configurations. No proposer or original date is identified in the retrieved sources.

August 24, 2026 rank- $r$ extension

Marcus Appleby, Steven T. Flammia, and Gene S. Kopp claim an extension to admissible tuples satisfying $0<r<(d-1)/2$ and $r(d-r)\mid d^2-1$ . They construct non-Hermitian equichordal ghost configurations and claim Hermitian rank- $r$ SICs conditionally on the order- $1$ abelian Stark conjecture and a special-value identity; the manuscript is unrefereed.

Current status (as of October 2026): The paper reports conditional higher-rank constructions and ghost configurations, but the rank- $r$ identity and unconditional Hermitian $r$ -SIC existence remain open.

Sources

No solutions have been posted yet.