Dark matter from the quadratic spinor Lagrangian I: Geometric mass for a gravitationally produced spin-1/2 fermion

The gravitational-wave induced freeze-in of Maleknejad and Kopp (2026) produces dark fermions from a stochastic gravitational-wave background, but requires them to acquire mass by separate means. We develop the Quadratic Spinor Lagrangian (QSL) formulation of general relativity, extended to Einstein--Cartan, as a framework that supplies this mass geometrically. The spinor 1-form built from a single Dirac field is purely spin-1/2 -- its gamma-traceless (spin-3/2) part vanishes identically -- so the propagating excitation is a Dirac fermion, the same content as the produced Weyl fermion. A cosmological spinor condensate sources a vectorial trace torsion $K\propto\dot\chi/\chi$, and an explicit Clifford reduction shows that this torsion gives the fermion a pure Dirac mass $M_{eff}=(1/\sqrt6)\,|\dot\chi/\chi|$, with no pseudoscalar or cross terms. The mass is not a free parameter but is locked to the Hubble rate at production, $M_{eff}\simeq(c_\chi/\sqrt6)H_*$, making the relic abundance a function of essentially the single scale $H_*$ ($\Omega h^2\propto H_*^{5/2}$) and supplying the mass the parent mechanism must postulate. Whether promoting the spinor 1-form to an independent field yields a propagating spin-3/2 candidate is a distinct dynamical question; Paper II shows that it does not -- the QSL channels all propagation into the gravitational sector -- so the composite spin-1/2 Dirac fermion is the unique QSL dark-matter candidate. We discuss the resulting dark-matter phenomenology and its link to asymptotically free scalar-field cosmology.