Universal Operator Envelopes and Defect Geometry of Additive Ternary $\Gamma$-Modules

For an additive ternary $\Gamma$-ring $T$, we construct an associative operator rng $\cO(T,\Gamma)$ characterized by a universal property for the two-element, two-index operators occurring in ternary modules, and prove that the Dorroh extension $\cA(T,\Gamma)$ represents the entire module category. The regular module gives a quotient $\cA\twoheadrightarrow\cR_T$ with kernel $K$. We develop the resulting defect geometry through the conormal module $K/K^2$, the graded tower $K^n/K^{n+1}$, higher relation modules, and, for square-zero kernels, a canonical Hochschild extension class. First relative $\Tor$ and $\Ext$ are identified intrinsically with the conormal defect, while nilpotent kernels are shown to be detected completely by their first conormal layer.
Formation of the universal envelope commutes with arbitrary scalar extension, although formation of the regular kernel need not. For two-step tensor systems $T_\mu=E\oplus W$, the envelope and its regular defect are determined by two explicit integral matrices: a visible flattening $\Theta_\mu$ and a middle-relation matrix $B_\mu$. We prove a base-change obstruction exact sequence, define an operator discriminant from maximal-rank minors, and construct determinantal/Fitting strata on the parameter space of tensors; defect dimension is upper semicontinuous and constant on a dense open stratum. Exact examples exhibit non-binary periodic homology in every degree, an infinite arithmetic conormal tower, torsion defects on torsion-free groups, and characteristic jumps. A complete exact census of all $256$ integral $2\times2\times2$ sign tensors is included. Smith-normal-form algorithms and a reproducible notebook provide certificates for every finite computation.