A Conditional Scalar-Gradient Trace-Torsion Branch in PT-Even Einstein-Cartan Geometry

First-order Einstein–Cartan / Palatini geometry treats the coframe and spin connection as independent variables and therefore permits torsion as a geometric sector. This paper studies a scalar-gradient trace-torsion sector and defines a conditional parity-time (PT)-even projected branch sourced by an internal scalar phase $\phi(x)$, equivalently $\epsilon_{\rm ph}(x)$. Physical content is retained only after the stated PT-even observable projection and the parity-even branch restriction. Result C1 is branch-reduced and conditional: after imposing the branch projector, a metric-only linear map from $v_A=\partial_A\phi$ to $T^A{}_{BC}$ selects the pure-trace form $T^{A}{}_{BC}=2\,\eta\,\delta^{A}_{[B}\partial_{C]}\phi$, while axial pseudo-tensor and traceless tensor sectors are outside the retained assumptions. Result C2 is an operator-basis reduction within the C1 truncation: the retained PT-even, two-derivative, quadratic torsion basis is one-dimensional, represented by $I_T$ up to normalization and boundary terms. Dirac–Born–Infeld-like, determinant-like, and topological channels are treated only as bookkeeping routes for possible Wilson-coefficient contributions. Weak-field formulas are included only as a possible diagnostic interface for future work. No empirical galactic fit, complete dark-sector replacement, full scalar dynamics, scalar perturbation stability analysis, or cosmological completion is claimed.