Realm Calculus: Foundations of Hyperoperational Analysis

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.