What does this equation mean?

A loop opens, instead, whenever some edge’s own realization is not simply a difference of two corotating nodes’ W . The clearest documented case is the relativistic Sagnac effect: a synchronization or comparison signal that does not itself corotate with the network — a one-way satellite time transfer, or any path analyzed against a non-rotating inertial frame — picks up an explicit timing correction $\Delta$ $t_e$ = (2/$c^2$)\,$\boldsymbol\Omega$ $\cdot$ $\mathbf A_e$ , where $\mathbf A_e$ is the area swept by that specific link’s path, projected onto Earth’s equatorial plane. That expression carries units of time, while an edge datum is a dimensionless log frequency ratio, so what actually enters the edge is $s_e$ = $\Delta$ $t_e$/T for the averaging interval T over which the comparison is gated; this is exactly the term without which the Global Positioning System’s own time transfer disagrees with ground clocks by an amount well outside its design tolerance [ 10 ] . Whenever an edge’s own physical realization departs from plain node-to-node W -differencing in this way, the correct response is to model $s_e$ explicitly on that edge — not to let it distort the fitted $\hat\phi$ , and not to write it off as an unexplained curvature signal. The size of an uncorrected Sagnac term is worth putting a number to, because it is easy to underestimate. Take an illustrative closed network loop enclosing a continental-scale area of 2.5$\times$10^{11}\,$\mathrm{m^2}$ — a square roughly five hundred kilometres on a side, comparable to the geographic span already reached by real international fibre links — at a latitude where the projected area onto Earth’s equatorial plane is close to the full value. With Earth’s sidereal rotation rate $\Omega$ $\approx$ 7.292$\times$10^{-5}\,$\mathrm{rad\,s^{-1}}$ [ 10 ] , the accumulated timing offset is $\Delta$ t = 2$\Omega$ A/$c^2$ $\approx$ 2(7.292$\times$10^{-5})(2.5$\times$10^{11})/(8.988$\times$10^{16}) $\approx$ 4.1$\times$10^{-10}\,$\mathrm s$ , a little over four hundred picoseconds. Gated at the one-second averaging interval conventional for this comparison, the corresponding dimensionless edge bias is s = $\Delta$ t/T $\approx$ 4.1$\times$10^{-10} — eight to nine orders of magnitude larger than any clock’s own fractional instability. A loop of that size realized through any link geometry that does not automatically cancel this term — a one-way relay, most obviously — would swamp every other effect in this article before curvature or a fault ever entered the picture, which is exactly why the term has to be modelled on its own edge rather than left to contaminate the fitted node potentials. A real network built to search for exactly this kind of anomaly is the four-station European fibre link connecting clocks in France, Germany, the United Kingdom, and Italy, which compared measured frequency ratios against those predicted from independently known geopotential differences and searched for any residual daily variation tied to Earth’s motion, rather than to gravity [ 7 ] . A miniature, five-node laboratory array built expressly to test differential redshift at the one-centimetre scale shows the same logic operating at the opposite end of the size range: five atomic ensembles, four independent height differences, and a fractional frequency gradient measured and checked against the value gravity alone predicts [ 2 ] . In neither case does a nonzero comparison certify curvature; in both, it certifies — or fails to certify — that the network’s own bookkeeping accounted for everything that was not a per-node scalar.