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Provide a rigorous derivation of the third-order negativity formula for U(1)k three-component link states

Provide a rigorous derivation showing that for a three-component link state in U(1)k Chern–Simons theory the third-order negativity defined by 𝔈_3(B;C) := −(1/2)·log Tr[(ρ_{BC}^{Γ})^3] equals log[ k / gcd(k,L_CA,L_AB,L_BC) ], equivalently that Tr[(ρ_{BC}^{Γ})^3] = k^{−2}·gcd(k,L_CA,L_AB,L_BC)^2.

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Background

The paper introduces a compact formula for the third-order negativity in the Abelian case and gives a non-rigorous derivation supplemented by extensive numerical checks over all three-component links for k ≤ 18.

A formal proof would complete the theoretical foundation of this result and connect it firmly to the replica and stabilizer-state analyses presented in the main text.

References

We also note here that although we do not have a rigorous derivation of eq-third-negativity, it passes lots of numerical test (we test it numerically for all 3-links with $k\le 18$).

eq-third-negativity:

E3(B;C):=−12log⁡Tr(ρBCΓ)3=log⁡kgcd⁡(k,LCA,LAB,LBC) .\mathcal{E}_3(B;C):= -\frac{1}{2}\log\mathrm{Tr}\big(\rho_{BC}^{\Gamma})^3 = \log\frac{k}{\gcd(k,L_{CA}, L_{AB}, L_{BC})}\ .

— Multi-entropy from Linking in Chern-Simons Theory (2510.18408 - Yuan et al., 21 Oct 2025) in Appendix D (Non-rigorous proof of Eq. (4.16))