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Prove the general-n formula for tripartite Rényi multi-entropy of U(1)k three-component link states

Prove that for a three-component link state in U(1)k Chern–Simons theory with Gauss linking numbers L_AB, L_BC, and L_CA, the n-th Rényi multi-entropy S_n^{(3)}(A;B;C) equals (1/n)·log[ k^3 / (gcd(k,L_CA,L_AB) · gcd(k,L_AB,L_BC) · gcd(k,L_BC,L_CA) ] + (1 − 2/n)·log[ k / gcd(k,L_CA,L_AB,L_BC) ] for all positive integers n and all k.

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Background

In the Abelian case for three-component links, the authors derive an exact expression for S_2{(3)} and a general-n expression when one linking number vanishes. They then propose a fully symmetric general-n formula for arbitrary linking numbers, Eq. (3.24), based on these rigorous partial results and extensive numerical checks.

The formula connects the tripartite Rényi multi-entropy directly to greatest-common-divisor combinations of the level k and the three pairwise Gauss linking numbers, and would provide a closed-form analytic continuation if proven.

References

We emphasize that the formula~eq-n-multiE-3link above is a conjecture, based on the rigorous results~eq-3-ME2 and~eq-improve.

eq-n-multiE-3link:

Sn(3)(A;B;C)= 1nlog⁡k3gcd⁡(k,LCA,LAB)gcd⁡(k,LAB,LBC)gcd⁡(k,LBC,LCA) +(1−2n)log⁡kgcd⁡(k,LCA,LAB,LBC) .\begin{split} S_n^{(3)} (A;B;C) =&\, \frac{1}{n}\log\frac{k^{3}}{\gcd(k,L_{CA},L_{AB}) \gcd(k,L_{AB},L_{BC}) \gcd(k,L_{BC},L_{CA})}\\ &\, + \Big(1 - \frac{2}{n}\Big) \log\frac{k}{\gcd(k,L_{CA},L_{AB},L_{BC})}\ . \end{split}

eq-3-ME2:

Z4(3)=k−3gcd⁡(k,LCA,LAB)gcd⁡(k,LAB,LBC)gcd⁡(k,LBC,LCA)⇒ S2(3)(A;B;C)=12log⁡k3gcd⁡(k,LCA,LAB)gcd⁡(k,LAB,LBC)gcd⁡(k,LBC,LCA) .\begin{split} &\mathcal{Z}_4^{(3)} = k^{-3} \gcd(k,L_{CA},L_{AB}) \gcd(k,L_{AB},L_{BC}) \gcd(k,L_{BC},L_{CA})\\ \Rightarrow\, & S_{2}^{(3)}(A;B;C) = \frac{1}{2}\log\frac{k^3}{\gcd(k,L_{CA},L_{AB}) \gcd(k,L_{AB},L_{BC}) \gcd(k,L_{BC},L_{CA})}\ . \end{split}

— Multi-entropy from Linking in Chern-Simons Theory (2510.18408 - Yuan et al., 21 Oct 2025) in Section 3.2.1 (Three-component links), around Eq. (3.24)