Bogoliubov sum rules and the Knight-shift ellipsoid in noncentrosymmetric superconductors

We show that the residual $T=0$ Knight shift of a noncentrosymmetric superconductor in the strong-locking regime is completely determined by a single Fermi-surface average – the projector ${\mathrm{\Pi }}_{\mu \nu }=⟨{\widehat{n}}_{\mu }{\widehat{n}}_{\nu }{⟩}_{\mathrm{F}\mathrm{S}}$ of the spin-locking direction ${\widehat{n}}_{\mathbf{k}}$ – giving the tensor identity ${\chi }_{\mu \nu }(0)={\chi }_N[{\delta }_{\mu \nu }-{\mathrm{\Pi }}_{\mu \nu }]$ independently of pairing symmetry, gap magnitude, and Fermi-surface shape. Because $\mathrm{T}\mathrm{r}\mathrm{\Pi }=1$, the three principal Knight shifts at $T=0$ lie on a two-dimensional simplex of locking textures, the \emph{Knight-shift ellipsoid}, whose vertices, edges, and interior classify every canonical pairing class. The identity follows from a Bogoliubov sum rule, $\sum \mathrm{\mid }{M}_{ph,O}{\mathrm{\mid }}^2+\sum \mathrm{\mid }{M}_{pp,O}{\mathrm{\mid }}^2={\mathrm{T}\mathrm{r}}_s(O^2)$, valid at every momentum for every Hermitian single-particle operator $O$ as the BdG-doubled form of unitary invariance. Around the central theorem we develop controlled departures (a closed-form $s$-wave SOC interpolation, a finite-field strong-locking identity), a dynamical counterpart (a spin Ferrell–Glover–Tinkham sum rule and a rigorous vanishing-projection theorem for $1\mathrm{/}{T}_1$), and a decoupled-pocket multiband baseline, packaged into six experimental protocols. Applied to the ${}^{75}$As NMR data on K${}_2$Cr${}_3$As${}_3$, the observed ellipsoid sits at the oblate-axial vertex $(0,0,1)$ and saturates the trace bound; the decoupled-pocket SOC-texture baseline is excluded by $\sim 0.5$ in normalized units, requiring a common $\widehat{c}$-axis locking on all three pockets, and the suppression of $1\mathrm{/}{T}_1\parallel \widehat{c}$ is identified, via the vanishing-projection theorem, as a fingerprint of finite-$\mathbf{q}$ ferromagnetic spin-fluctuation gap formation.