Gravitational Redshift as a Potential Probe: A Discrete Lattice Field Method in the Weak Field

We introduce a method for constructing a calibrated, spatially resolved gravitational potential field on a
discrete lattice in the static weak field. This paper demonstrates the solar field construction using physical
parameters specified independently of the measured redshift. On the Sun the field is built from the solar
mass parameter, radius and distance, on a lattice that keeps the true Sun–Earth geometry. A walk through
successive stored differences returns the endpoint difference to order 10−15. The gradient integral, which
tests the field’s slopes, agrees to 8.9×10−5, and a cancellation identity sets its floor. The weak-field climb
from the photosphere to 1 AU, 633.351m s−1 in velocity units, and the independent HARPS
measurement, 638±6m s−1, agree within the measurement’s uncertainty. A gravitational redshift
between two static locations measures their potential difference directly. The method supplies the step
from such measurements to a calibrated spatial field representation whose constituent values, slopes and
curvature can be examined. Its coordinate map, spatial shape, additive potential reference and amplitude
are specified separately. For a fixed shape and geometry, one nonzero endpoint comparison determines
the amplitude; a connected network of measured comparisons determines nodal potentials up to an
additive constant. The constructed field makes gravitational acceleration and, through the Newtonian
Poisson equation, an effective mass-density diagnostic accessible from its spatial derivatives. A rank
theorem establishes what a network of redshift measurements determines, and synthetic networks recover
the prescribed potentials with the predicted covariance. The method thus provides a common framework
for constructing and examining gravitational potential fields from potential-difference information,
spanning millimetre-scale atomic samples and solar spectroscopy.