Null-channel closure relations in precision clock networks as probes of cosmological scalar fields

Light scalar fields arise generically in extensions of the Standard Model, including dilaton and modulus fields in string theory, ultralight dark matter candidates, and scalar degrees of freedom associated with cosmic defects such as cosmic strings and domain walls. These fields induce species-dependent variations of fundamental constants, producing differential frequency shifts across atomic clocks. We show that networks of N clocks with linearly independent sensitivity coefficients admit exactly N − 1 null-channel observables that eliminate universal gravitational redshift and isolate scalar-induced signals. More generally, in the presence of M independent scalar fields, the null-channel space has dimension N −M , yielding overconstrained closure relations that must be satisfied by any scalar-induced perturbation. These relations provide model-independent consistency tests that distinguish scalar-mediated effects from universal gravitational and instrumental contributions. We demonstrate using realistic optical clock parameters that existing terrestrial clock networks already realize this closure structure. This establishes precision clock networks as a scalable probe of light scalar fields arising in cosmology and string-motivated extensions of gravity, including ultralight dark matter and cosmic defects.