Eta-Invariant of the Adjoint-Twisted Trefoil Complement: Verification of the Leading APS Boundary Correction to the Fine Structure Constant

We compute the Atiyah–Patodi–Singer (APS) eta-invariant of the trefoil knot complement M=S3∖N(K), twisted by the adjoint representation Ad:SU(2)→SO(3), using the Nicolaescu formula for Seifert-fibered spaces with generalized Dedekind–Rademacher sums. The trefoil complement admits the Seifert fibration {b=−1;(2,1),(3,1)}, and the adjoint representation decomposes into weights {0,+2,−2} under the maximal torus. We obtain the exact result η(Ad)=17/18+37/36+37/36=3=pq−2g−1, where p=2,q=3 are the torus knot parameters and g=1 is the Seifert genus. The Chern–Simons invariant is CS(M,Ad)=1/12=1/(2pq). These results are combined with the independently verified adjoint-twisted Reidemeister torsion ΔAd(t)=t3−1, proven via Fox calculus on the Wirtinger presentation to 10−13 precision. The leading coefficient of the APS boundary correction to the fine structure formula α−1=4π3+π2+π is confirmed: the correction 3π3+π2+π+1 decomposes as η⋅π3+g⋅π2+u⋅π+χ, with each coefficient a known topological invariant of the trefoil.