Chain-level decoupling of the physical heterotic $G_2$ deformation complex at the standard embedding
The physical deformation complex of heterotic $G_2$ compactifications couples instanton deformations to geometric fields and their first-order $α'$ corrections. We give an explicit chain-level decoupling of its first-order coefficient complex at the minimal standard embedding, where the gauge bundle is the tangent bundle, the gauge connection is the Levi--Civita connection, and the background $G_2$ structure is torsion-free. The first Atiyah coupling is null-homotopic through a skew covariant derivative extending the induced connection variation. A second homotopy, given by the covariant codifferential of the skew part of an endomorphism-valued cochain, removes the coupling from the bundle sector to the geometric sector. The residual geometric curvature term cancels against the contribution of the first transformation. These identities hold on arbitrary cochains in every canonical degree and yield mutually inverse differential-operator chain maps. Consequently, the cohomology decomposes into instanton and geometric coefficient cohomologies in all degrees. The argument requires only $\mathrm{Hol}(g)\subseteq G_2$. To our knowledge, no previous explicit all-degree splitting removes the internal gauge and geometric couplings of this physically reduced standard-embedding coefficient complex.