Derived deformation theory of heterotic $G_2$ systems near the standard embedding
We study heterotic $G_2$ structures on a fixed string Courant algebroid on a spin seven-manifold $Y$, near a torsion-free standard embedding, to first order in $α'$ and to all formal orders in the deformations. The admissible metric, spinor line and density determine a canonical generalized Dirac functional whose expression in a physical splitting is the heterotic superpotential. We work in the small-flux sector $H^{[0]}=0$. We adjoin this condition and its compatibility identities to obtain a resolved physical theory, and construct a corresponding compatibility extension of the variational critical theory. The two resolutions are locally formally isomorphic as real local formal $Q$-theories. When $Y$ is closed and the gerbe and anomaly sector is fixed, the critical fields of the superpotential coincide with the BPS fields over every local real Artin algebra concentrated in degree zero. The shifted-cotangent Hamiltonian construction makes the variational critical theory cyclic. Mixed-order ellipticity and homotopy transfer give finite-dimensional minimal models, and cyclic transfer yields an effective potential $W_{\mathrm{eff}}$ for the ordinary formal physical solution locus over these degree-zero Artin algebras, after a formal coordinate change. We compute the relative cohomology of this compatibility comparison, which is detected on dg-Artin algebras and forgotten by the variational critical theory, and prove that the pointed relative fibre at the standard embedding is homotopy abelian.