Deformation-Field Gravity from a Dilation-Shear/Rotation-Twist Potential Tensor

A gravitational potential-tensor formulation is developed in which the successful Levi-Civita metric sector of general relativity is retained, while the metric, transported connection, torsion, and gravitational distortion tensor are generated from an underlying gravitational potential tensor. The basic variable is the local gravitational potential tensorequation A=OP, equationwhere P is the metric-visible gravitational potential tensor, or equivalently the dilation-shear gravitational potential sector, and O is the connection-visible gravitational potential tensor, or equivalently the rotation-twist gravitational potential sector. The induced metric isequation g=AT A=PT P, equationand its Levi-Civita curvature gives the metric branch of the theory. In this branch the field equation is the Einstein-type equation for g (P), supplemented by a harmonic reference condition that selects the physical coordinates of the Minkowski reference spacetime. The same gravitational potential tensor also induces a transported connection, equation =A^-1 A, equationwhich is generally different from the Levi-Civita connection of g (P). Their differenceequation DA=- called the gravitational distortion tensor. Its antisymmetric lower-index part gives the torsion of the transported connection, equation T^_=2DA^_. equationWhen the reference connection is flat, the transported connection is curvature-free, and the Levi-Civita curvature can be equivalently represented through DA: equation Rg=- (DA+DA DA). equationThus the formulation can be read in the standard geometric language as Levi-Civita metric gravity supplemented by an induced torsion/distortion layer, while its structural principle is that all these objects are generated by the same gravitational potential tensor A=OP. The two factors are coupled sectors of one total deformation field rather than independently specified geometries: P determines the metric layer, while O modifies the connection layer through the P-conjugated term P^-1O^-1 O\, P. The field equations are introduced variationally. The action contains a Levi-Civita metric sector S ₋₂P, a harmonic reference term S ₑ₄₅P, , matter coupling through g (P), an intrinsic or effective rotation-twist sector SOA, and a rotation-current source coupling S ₑ₎ₓA, J. Variation gives the full coupled system EPP, O, , , J=0, C^=0, and EOP, O, J=0, with the usual Einstein--Fock branch and the linearized rotation-twist equation obtained as sectoral reductions. A static spherical application reconstructs the Fock harmonic branch directly as a pure P-field solution. The Minkowski reference spacetime also provides a natural total energy-momentum and angular-momentum conservation structure, with energy-momentum sourcing the P-sector and spin-rotation current sourcing the O-sector. The formulation therefore preserves the standard metric predictions of general relativity while providing a constrained description of torsional, rotation-current, spin, and distortion-tensor effects. A linearized Maxwell--Proca model for the rotation-twist sector is also introduced as a calculable approximation for spin-rotation effects, including gyroscopic precession, polarization transport, phase accumulation, and orientation-dependent responses of spinning or rotating probes. Metric compatibility of the transported connection does not by itself determine the worldlines of ordinary matter. The leading trajectory of a structureless particle is fixed by its matter action, while the transported connection and the distortion tensor become directly relevant for spin, polarization, internal orientation, and other nonminimal probes.