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 Trₛ(O²). 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 χ_(μν)(0)/χ_N=δ_(μν)-Π_(μν)+o(1), where χ_N is the normal-state Pauli susceptibility and Π=⟨n̂ₖn̂ₖ⟩_(rm FS) is the Fermi-surface average of the unit spin-locking texture. The eigenvalues of Π form a simplex, while the normalized spin Knight-shift tensor defines an ellipsoid whose semi-axes are the residual principal responses. Full cubic invariance of both the superconducting state and locking texture fixes Π=I/3 and hence χ(0)/χ_N=2I/3+o(1); 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 Fₛ(λ)=1-asinhλ/[λ√(1+λ²)], valid for arbitrary λ=|g|/Δ. 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 ⁷⁵As data on K₂Cr₃As₃, the framework identifies a field-dependent axial suppression pattern at 8--16~T.