Chen–Murakami reflection conjecture for the continuous quantum 6j-symbol

Establish that the complete asymptotic expansion of the continuous quantum 6j-symbol is invariant under independently reversing any dihedral-angle sign, and clarify the resulting cancellation mechanism for the interpolation considered by Chen and Murakami.

Background

Chen and Murakami proposed a symmetry of the complete asymptotic expansion of a continuous quantum 6j-symbol: independently changing the signs of dihedral angles should leave the expansion unchanged. They connected this symmetry with the cancellation of dominant contributions in quantum state sums.

The paper proves an exponential reflection estimate for a specified midpoint interpolation on compact strict hyperideal sets. However, it explicitly states that no global identification with the interpolation used by Chen and Murakami is assumed, so the broader conjectural statement is not established in the form originally proposed.

References

Chen--Murakami Conjecture 3 and Remark 3 conjecture that the complete expansion of their continuous quantum $6j$-symbol is unchanged by independently reversing a dihedral-angle sign, and relate this symmetry to cancellation. \hyperref[stmt-theorem-2-2]{Theorem 2.2} and \hyperref[sec-A]{Appendix A} establish an exponential reflection estimate for the specified midpoint interpolation on compact strict hyperideal sets. This addresses that proposed mechanism with quantitative remainder control; no global identification with the interpolation in is assumed.

— The Chen-Yang volume conjecture for long integral fillings of fundamental shadow links  (2609.11839 - Shen, 10 Sep 2026) in Section 1.2, “Method and relation to earlier work”