Trace-Gradient Modified Gravity: Action, Field Equations, Conservation Laws, and Stability

We present the theoretical foundation of Trace-Gradient Modified Gravity (TGMG), a metric theory extending General Relativity by coupling the gravitational sector to the trace of the stress-energy tensor through a dimensionless parameter β. We derive the complete field equations by metric variation of the TGMG action, construct the geometric topology tensor 𝔾μν with explicit component form, and establish the conservation law ∇^μ𝔾μν = 0 by two independent routes: Noether's second theorem and an explicit step-by-step Noether identity proof. Five of the ten Nordtvedt–Will post-Newtonian parameters vanish identically from the conservation structure. Gravitational waves propagate at c_g = c exactly. Stability conditions for the scalar and tensor sectors are derived. This paper provides the theoretical foundation cited as reference [17] in Paper II (DOI: 10.5281/zenodo.20633485), which derives the first astrophysical constraint on β from neutron star tidal deformability and GW170817.