Effective $\Lambda$CDM model emerging from $f(Q,T)$ under a special EOS limit in symmetric cosmology with Bayesian and ANN observational constraints

In this work, we investigate the cosmological consequences of an effective $f(Q)$ model emerging from the more general $f(Q,T)$ gravity theory under the special equation-of-state condition $\rho + p = 0$. Under this limit, the field equations yield the constraint $F(Q,T)H(t)=C$, implying that the function $F=f_Q$ becomes purely dependent on the nonmetricity scalar $Q$, and the background evolution mimics that of the standard $\Lambda$CDM model. We derive the resulting functional forms of $f(Q)$, obtain the corresponding effective cosmological constant, and analyze the physical nature of this reduction. To test the model against observations, we constrain the parameters $H_0$, $\Omega_m$, and $S_8$ using cosmic chronometers (CC), baryon acoustic oscillations (BAO), and Pantheon+ SN Ia datasets. A comparative analysis is performed using both the conventional Bayesian Markov Chain Monte Carlo (MCMC) sampling and a machine-learning based Artificial Neural Network (ANN) emulator. We find that the ANN approach yields tighter posterior constraints while significantly reducing computational time. The model successfully reproduces the observational trends of each dataset and offers insights into the persistent $H_0$ and $S_8$ tensions. Our results indicate that effective nonmetricity-based dark energy scenarios derived from $f(Q,T)$ gravity provide a viable and observationally consistent alternative to $\Lambda$CDM, with future high-precision surveys expected to further distinguish between these frameworks.