A Geometric Method for Gravity from a Pure Potential Tensor: Induced Metric and Deformation-Comparison Structures

We develop a geometric method for describing gravity in terms of a pure potential tensor \ (P\). The metric and the associated comparison structures are not postulated as independent geometric variables, but are induced by \ (P\). The effective metric is\ g=P^T P, \ (\) is a fixed Minkowskian reference metric. The same potential tensor also determines the deformation-comparison tensor\ _=P^-1^ () _ P, is not an independent affine variable. Its \ (g\) -symmetric part gives the induced nonmetricity, while its lower-index antisymmetric content gives the induced torsion. The resulting framework uses geometry as a representation of the pure potential tensor, rather than taking the metric geometry itself as the primitive gravitational entity. In the minimal matter sector considered here, matter couples to \ (g (P) \), so the primary source is the energy-momentum tensor; direct couplings to spin, dilation, and shear currents are possible nonminimal extensions but are not required in the minimal theory. The static spherical Fock branch is used to reconstruct \ (P\), compute \ (\), and display explicitly the induced torsion and nonmetricity. The first beyond-Fock correction problem is then organized as a perturbative hierarchy. A direct \ (r^-3\) response ansatz is shown to be only diagnostic unless the complete \ (P\) -variation and harmonic-representation terms are retained; the far-zone source must satisfy the linearized Bianchi compatibility condition before the coefficients can be fixed. This paper therefore presents a detailed potential-tensor formulation in which the effective metric, connection, curvature, torsion, and nonmetricity are derived structures of a single pure gravitational potential tensor.