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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 $\mathcal{A}_{EISA} = \mathcal{A}_{SM} \otimes \mathcal{A}_{Grav} \otimes \mathcal{A}_{Vac}$ 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 $\phi$ 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.