What question did this study set out to answer?

To establish a foundational framework for hyperoperational analysis through realm calculus.

May 3, 2026Open Access

Realm Calculus: Foundations of Hyperoperational Analysis

Key Points

To establish a foundational framework for hyperoperational analysis through realm calculus.
Identified a parameter space for non-Newtonian calculus.
Developed explicit generators for constructing new branches of analysis.
Selected two levels of the Ackermann-Goodstein hierarchy for input perturbation and output measurement.
Demonstrated that every known non-Newtonian calculus is represented in the identified space.
Generated infinitely many new calculi, each possessing a closed-form derivative and integral.
Established a new Fundamental Theorem applicable in this context.

Abstract

Most known non-Newtonian calculus is a special case of a single parameter space. This paper identifies that space and provides explicit generators for constructing new branches of analysis.A realm calculus is built by choosing two levels of the Ackermann-Goodstein hyperoperational hierarchy: one governing how the input is perturbed, one governing how the output change is measured. Every known non-Newtonian calculus (Newton-Leibniz, Grossman-Katz multiplicative, Bashirov bigeometric, the Euler operator) is a named point in this space. The framework generates infinitely many new calculi, each with a closed-form derivative, integral, and Fundamental Theorem. This is Paper I of a series on Hyperoperational Analysis.

Realm Calculus: Foundations of Hyperoperational Analysis

Key Points

Abstract

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