We propose the Extended Integrated Symmetry Algebra (EISA) as an exploratory effective field theory (EFT) model for investigating quantum mechanics and general relativity unification, augmented by the Recursive Info-Algebra (RIA) extension incorporating dynamic recursion through variational quantum circuits (VQCs) minimizing Von Neumann entropy and fidelity losses. EISA's triple superalgebra encodes Standard Model symmetries, gravitational norms, and vacuum fluctuations, while RIA optimizes information loops for emergent quantum field dynamics without extra dimensions. Transient processes like virtual pair rise-fall couple to a scalar in a modified Dirac equation, potentially sourcing curvature and phase transitions. The framework's mathematical self-consistency is demonstrated through rigorous verification of super-Jacobi identities, ensuring algebraic closure across all symmetry sectors. Our approach introduces a novel synthesis of quantum information principles with algebraic structures, where recursive optimization drives the emergence of physical laws from fundamental symmetries. The integration of variational quantum circuits provides a powerful computational paradigm for exploring vacuum stability and entropy minimization in extended symmetry spaces. This work establishes a foundation for modeling quantum-gravitational phenomena through a unified algebraic framework that generates dynamics from information-theoretic optimization, offering new pathways for investigating quantum gravity and emergent spacetime.