Theta-weighted instanton sectors ($Z(\vartheta )={\sum }_{k\in \mathbb{Z}}{e}^{ik\vartheta }{Z}_k$)

For integer instanton number $k$, a topological angle can weight each Euclidean gauge-field sector by $e^{ik\vartheta }$ in a stated convention. The sector sum is then $2\pi$-periodic in $\vartheta$. In temporal gauge, the same integer describes a change in the vacuum winding or Chern-Simons number, with compatible orientation conventions. Classical instanton action fixes exponential suppression, while determinants and collective-coordinate integrals supply prefactors.