Infinity Algebra: A Foundational Framework for Symbolic Computation with Infinite Quantities

Infinity Algebra is a new mathematical framework designed to treat infinity not as a limit, asymptotic value, or conceptual extremum, but as a fully operable and symbolic algebraic entity. This paper introduces the foundational axioms of Infinity Algebra, including explicit rules for infinite addition, subtraction, multiplication, division, inverse operations, and exponentiation. By establishing a closed and internally consistent algebraic system, we enable symbolic manipulation of infinite quantities without reliance on traditional limit-based calculus or set-theoretic extensions. We formally define Infinity Groups, Rings, and Fields that incorporate ∞ as computable operands, allowing for structured algebraic interactions with real and infinite values. These structures serve as the basis for extended domains such as Infinity Calculus, Infinity Linear Algebra, and Infinity-Valued Function Spaces, including ∞-matrices, ∞-tensors, and symbolic derivatives evaluated at ∞. Applications span multiple disciplines: from theoretical mathematics to black hole entropy modeling, from quantum field singularities to symbolic AI systems operating over infinite-depth architectures. The symbolic entropy matrix and ∞-Einstein equations demonstrate how Infinity Algebra can reframe gravitational singularities and divergent field interactions in physics. The framework is supported by a Python-based symbolic computation engine that enables experimentation, visualization, and application of Infinity Algebra to real and abstract problems. Ultimately, this system offers a robust and extensible language for engaging with infinite structure, both as a formal tool and as a philosophical reorientation. Infinity Algebra not only complements existing approaches to divergence but provides a symbolic foundation for future exploration of the infinite in mathematics and physics.