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We present a categorical framework derived from first principles of relational logic in category theory, formalizing string-inspired symmetries as recursive functor structures. This approach realizes the Extended Integrated Symmetry Algebra (EISA) to unify quantum mechanics and general relativity, augmented by the Recursive Info-Algebra (RIA) extension. Dynamic recursion is incorporated through variational quantum circuits (VQCs) to minimize von Neumann entropy and fidelity losses, yielding emergent quantum field dynamics without invoking extra dimensions or empirical assumptions. The EISA triple superalgebra AEISA = ASM ⊗AGrav ⊗AVac is recast as a monoidal category, with Standard Model symmetries, gravitational constraints, and vacuum fluctuations serving as subcategories, and tensor products acting as monoidal functors. RIA is expressed via natural transformations on endofunctors, optimizing information flows to derive physical laws from fundamental categorical relations. Transient processes, including virtual pair creation and annihilation, couple to a composite scalar field ϕ within a modified Dirac equation, sourcing spacetime curvature and phase transitions through categorical morphisms. Self-consistency is established via categorical equivalences and validation of super-Jacobi identities as category axioms, ensuring algebraic closure across symmetry sectors. This synthesis of quantum information and categorical structures introduces recursive functorial string diagrams, extending conventional string field theory to computable low-energy effective field theories (EFTs). VQCs serve as a computational tool for simulating vacuum stability and entropy minimization in these categorical spaces. Numerical simulations, utilizing projected 2025 data from NANOGrav gravitational wave detections and ATLAS t¯t production measurements, confirm the model’s predictions, including CMB power spectrum perturbations (ΔCℓ/Cℓ ≈ 10−7) and a possible alleviation of the Hubble tension. The framework proposes novel ultraviolet completions through categorical string formalisms, asymptotic safety, and holographic duality, providing fresh perspectives on quantum gravity rooted in relational logic.