We establish an exact Bogoliubov sum rule for any Hermitian single-particle operator $O$: at each momentum, its particle-hole and particle-particle matrix-element weights sum to the single-particle trace ${\mathrm{Tr}}_s\left(O^2\right)$. The result follows solely from Hilbert–Schmidt-norm invariance under a canonical Bogoliubov transformation and contains neither excitation-energy denominators nor occupations; physical response identities therefore require additional assumptions. At zero field, a fully gapped helicity-diagonal state with both helicity sheets present and a spin-orbit splitting asymptotically larger than the gap obeys ${\chi }_{\mu \nu }\left(0\right)/{\chi }_N={\delta }_{\mu \nu }-{\mathrm{\Pi }}_{\mu \nu }+o\left(1\right)$, where ${\chi }_N$ is the normal-state Pauli susceptibility and $\mathrm{\Pi }={\langle {\widehat{\mathbf{n}}}_{\mathbf{k}}{\widehat{\mathbf{n}}}_{\mathbf{k}}\rangle }_{\mathrm{FS}}$ is the Fermi-surface average of the unit spin-locking texture. The eigenvalues of $\mathrm{\Pi }$ form a simplex, while the normalized spin Knight-shift tensor defines an ellipsoid whose semiaxes are the residual principal responses. Full cubic invariance of both the superconducting state and locking texture fixes $\mathrm{\Pi }=\mathbb{I}/3$ and hence $\chi \left(0\right)/{\chi }_N=2\mathbb{I}/3+o\left(1\right)$; cubic crystal symmetry alone does not. For zero-field $s$-wave pairing in the reference-Fermi-surface regime, we obtain the exact closed-form kernel $F_s\left(\lambda \right)=1-asinh\lambda /\left[\lambda \sqrt{1+{\lambda }^2}\right]$, valid for arbitrary $\lambda =\left|\mathbf{g}\right|/\mathrm{\Delta }$. In a finite Zeeman field, the zero-field helicity reduction generally fails, so the equilibrium magnetization and differential response require a self-consistent BdG calculation rather than a field-dependent locking-tensor substitution. Applied to the ${}^{75}\mathrm{As}$ data on ${\mathrm{K}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$, the framework identifies a field-dependent axial suppression pattern at 8–16 T. Both its placement near the $\widehat{c}$-axis vertex and the resulting tension with a specified decoupled-pocket spin-orbit-texture baseline require an additional fixed-tensor $T\to 0$ and zero-field extrapolation; no unique microscopic pairing state follows.

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• Noncentrosymmetric materials
• Superconductors
• Nuclear magnetic resonance