Subtracted grand potential of the reduced BCS model ($\delta \Omega =V|\Delta {|}^2/\lambda -{\beta }^{-1}{\sum }_{kn}\ln [({\omega }_n^2+E_k^2)/({\omega }_n^2+{\xi }_k^2)]$)

Subtracting the normal-state grand potential fixes the reference constant in the uniform BCS pairing-field effective action. Here ${\omega }_n=(2n+1)\pi /\beta$ has energy units and $E_k^2={\xi }_k^2+|\Delta {|}^2$. The ratio cancels a field-independent determinant phase and produces a convergent high-frequency sum under the same momentum regulator. Differentiation gives the BCS gap equation. For a fixed attractive pairing shell, this is a finite regulated model; an unrestricted continuum contact interaction additionally needs ultraviolet renormalization. Its weak-coupling condensation density $-{\nu }_F{\Delta }^2/2$ uses a single-spin density of states per volume.