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Calculus of Element-Generated Sets (CEGS): A Symbolic Framework for Mathematics, Physics, and AI

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Description

This monograph introduces the Calculus of Element-Generated Sets (CEGS), a symbolic extension of mathematical analysis. Part I develops the mathematical foundations of CEGS, including symbolic identity sets, symbolic closure, divergence resolution, and functional iteration. Part II applies these tools to physics, with constructions for non-symmetric potentials, displacement and excitation operators, anisotropic Gaussian fields, and constraint projection methods. Part III explores applications in computer science and AI, including symbolic reasoning, algebraic solvers, inverse reasoning operators, and multivalued symbolic computation. Together these developments provide a coherent and deterministic symbolic framework, with implications for mathematics, physical modeling, and intelligent systems.

 

Abstract

The Calculus of Element-Generated Sets (CEGS) develops a symbolic framework that extends traditional mathematical analysis. CEGS introduces symbolic sets and symbolic algebraic operations that preserve structural information beyond what classical functions can express. The framework resolves indeterminate and divergent forms, defines symbolic identity sets, and generalizes functional iteration. This symbolic foundation enables consistent treatment of discontinuities, infinite expansions, and structural equivalences.

Applications of CEGS extend across mathematics, physics, and computer science. In physics, symbolic operators model non-symmetric potential fields, constraint projection, and anisotropic Gaussian structures. The framework also reformulates Fourier synthesis in symbolic terms and provides deterministic approaches to particle physics phenomena where incompleteness arises in conventional models. In computer science and AI, CEGS establishes symbolic reasoning and inverse operators for problem decomposition, explainability, and multivalued reasoning.

CEGS thus provides a unifying symbolic structure for analysis, modeling, and computation, bridging foundational mathematics with applied domains.

Remark

The function used in Chapter 29 and 30 appears to be incorrect and the chapters need an overhaul. However from a didactic point of view the application of the method is correct. Only the functions and images would require a correction.
For a corrected version see the emitting framework about time space oscillations https://zenodo.org/communities/tso-research. Especially TSO & Quantum Mechanics.

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2025-08-19
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References

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