Dual Effective Field Theory formulation of Metric--Affine and Symmetric Teleparallel Gravity

We develop a unified algebraic and effective field theory (EFT) formulation for non--Riemannian extensions of General Relativity with an independent connection. For metric--affine $f(R,Q)$ gravity we show that the connection equations admit an exact matrix solution, whose square--root structure generates a convergent binomial/Neumann expansion in powers of the stress tensor $T_{\mu\nu}$. For the Eddington--inspired Born--Infeld (EiBI) theory we show that the connection can be solved algebraically as well, and that its determinantal field equations produce a parallel Neumann expansion with coefficients fixed by the underlying determinant operator. This allows us to rewrite the Einstein--like equations in the auxiliary metric as an effective Einstein equation for $g_{\mu\nu}$ with a local algebraic correction $(\Delta T)_{\mu\nu}$ that follows from a dual EFT built from the invariants $\{T,\,T^2,\,T_{\mu\nu}T^{\mu\nu},\ldots\}$, organised by a characteristic density scale. We prove a convergence criterion based on the spectral radius of $\hat T^\mu{}_\nu$ and interpret EiBI gravity as a determinantal resummation of the same $T$--tower. Extending the framework to symmetric teleparallel $f(Q)$ gravity, we identify the EFT coefficients in terms of $f_Q$ and $f_{QQ}$ and present a background matching for $f(Q)=Q+\alpha Q^2$. The resulting dual EFT provides a common algebraic language for metric--affine, Born--Infeld and non--metricity gravities.