= Fermionic pairing-block determinant normalization
A two-component <Nambu spinor> $(\psi_{k\uparrow},\bar\psi_{-k\downarrow})$ for every k accounts for both spin species once after relabelling. Its <Grassmann integral> contributes one <functional determinant>. A fully doubled spin-particle-hole basis instead has redundancy and requires one half in the <trace> logarithm or a corresponding Pfaffian convention. A reliable check is the zero-gap limit: the <grand potential> must become $-2\beta^{-1}\sum_k\ln(1+e^{-\beta\xi_k})$, with exactly two spin states and no invented vacuum offset.
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