What does this equation mean?

Collect the pieces. Three charts, the Newtonian centroid $\mathbf R_N$ , the observer-dependent stress-energy centroid $X^\mu(u,\Sigma)$ , and the canonical Newton–Wigner operator $\mathbf X_{\mathrm{NW}}$ , form a genuine, well-behaved sub-atlas. Define, for any two of them evaluated on the same hypersurface $\Sigma$ , $\delta_{ij}(u,\Sigma) = \frac{\big|X_i^\mu(u,\Sigma) - X_j^\mu(u,\Sigma)\big|_\Sigma}{\rho_M}$. the proper spatial separation within $\Sigma$ between the two charts’ answers, normalized by the spin-to-mass length fixed in the first section. $\delta_{ij}$ is dimensionless by construction, symmetric, zero exactly when the two charts coincide, and it has now been checked, not merely asserted, in both directions this kind of construction is required to pass: it vanishes as v/c $\to$ 0 for any spin, and it vanishes as S $\to$ 0 for any observer or Lorentz frame. It is bounded above by exactly 1 for the observer-transition case, since |$\Delta$$\mathbf x$| $\le$ $\rho_M$ was shown above to saturate rather than merely approach that value. Both checks are this construction’s kill criterion, run rather than assumed: had either limit failed to collapse the disagreement to zero, the atlas would have been reporting confusion between an artefact of convention and an artefact of mismatched apparatus, exactly the failure this kind of object is meant to be judged by.