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 $(\mathrm{\Delta }T{)}_{\mu \nu }$ that follows from a dual EFT built from the invariants $\{T,T^2,T_{\mu \nu }{T}^{\mu \nu },\dots \}$, organised by a characteristic density scale. We prove a convergence criterion based on the spectral radius of ${\widehat{T}}_{\nu }^{\mu }$ 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.