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\abstract{Based on first principles of Peircean relational logic and category theory axioms, this work develops a categorical framework that formalizes string-inspired symmetries as recursive functorial structures. The framework proposes an Extended Integrated Symmetry Algebra (EISA) as a potential bridge between quantum mechanics and general relativity, further extended by a Recursive Information Algebra (RIA). To incorporate dynamical recursion, variational quantum circuits (VQCs) are employed to minimize von Neumann entropy and fidelity loss. This approach aims to derive emergent quantum field dynamics, including the V-A structure of weak interactions, while minimizing reliance on extra dimensions or ad hoc empirical assumptions. The EISA triple superalgebra $\mathcal{A}{\text{EISA}} = \mathcal{A}{\text{SM}} \otimes \mathcal{A}{\text{Grav}} \otimes \mathcal{A}{\text{Vac}}$ 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. This suggests that the V-A structure of weak interactions can be derived from Peircean relational logic and categorical axioms, potentially ensuring left-handed chirality dominance. Transient processes, including virtual pair creation and annihilation, couple to a composite scalar field $\phi$ 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 recent 2023-2025 data from NANOGrav gravitational wave detections and ATLAS $t\bar{t}$ production measurements, confirm the model's predictions, including CMB power spectrum perturbations ($\Delta C_\ell / C_\ell \approx 10^{-7}$) 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.