What does this equation mean?

For a static observer at position $\mathbf x$ in a stationary spacetime, define U($\mathbf x$) := d$\tau$/dt , the local rate of proper time relative to coordinate time. Two clocks compared through any phase-coherent link that depends only on their respective proper frequencies report the ratio $\nu_j$/$\nu_i$ = U($\mathbf x_j$)/U($\mathbf x_i$) — a fact about stationary spacetimes that follows from the timelike Killing vector generating t -translations, and one on which the standard treatments of relativity in a geodetic or navigational setting already rely [ 10 , 14 ] . The dependence on endpoints only, never on the path connecting them, is not an approximation: because $\partial$/$\partial$ t generates a genuine symmetry of a stationary metric, the redshift between any two static observers is fixed entirely by the value of U at each observer’s own location, so a network’s fitted node potential can be compared against a measurement made over any convenient physical route without first specifying which route was used. Define the dimensionless node potential $\phi(\mathbf x)$ := $\ln$ U($\mathbf x$) and the edge observable actually recorded between linked clocks i and j , $y_{ij} := \ln(\nu_j/\nu_i) = \phi(\mathbf x_i) - \phi(\mathbf x_j)$. At weak field this reduces to the familiar $y_{ij}$ $\approx$ ($\Phi_i$ - $\Phi_j$)/$c^2$ for Newtonian potential $\Phi$ , exactly the quantity real fibre-linked clock comparisons already report [ 1 , 7 ] . Collect every measured edge into a vector y , one entry per link, and encode the network’s topology in an incidence matrix B : row e = (i,j) has a +1 in column i , a -1 in column j , and zero elsewhere. The model is y = B$\phi$ + n for measurement noise n with covariance C , and the standard weighted least-squares estimate of the node potentials is