Defect Cartan Gravity I: Emergent Coframes and the Minimal Spin-0+ Torsion Sector

This paper fixes the unique theory branch used by Defect Cartan Gravity. We workin Cartan variables (coframe and Lorentz connection) and impose a parity-even quadraticrestriction engineered to propagate only one additional scalar (spin-0+) connection/torsiondegree of freedom beyond General Relativity in the macroscopic spinless regime. Therestriction avoids the generic spin-2 ghost that accompanies broad curvature-squared sectors.We present: (i) the action and parameter dictionary; (ii) torsion irreducible decompositionand the meaning of the 0+ branch; (iii) full variational structure and field equations incovariant form and in a useful component reduction; (iv) an auxiliary-field (scalar-tensor)rewriting of the R2 sector; (v) the trace-only torsion reduction, showing how the tracetorsion mixes with the scalar curvature sector and how the remaining torsion irreps arenondynamical for spinless sources; (vi) linearization around Minkowski space and isolationof the propagating content; and (vii) concrete health conditions (no ghosts/tachyons) andcontrolled decoupling limits. Papers II and III treat this paper as the definitive theory lockand do not introduce new gravitational degrees of freedom.