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\[ g=PTη P,\]where \(η\) is a fixed Minkowskian reference metric. The same potential tensor also determines the deformation-comparison tensor\[ Λ_μ=P⁻¹∇(η)_μ P,\]which 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 \(Λ\), 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.