Realm Calculus II: Calculus Extensions

This paper extends differential and integral calculus to the realm-parameterised framework introduced in *Realm Calculus, Paper I: Foundations of Hyperoperational Analysis*. A realm-aware εε-δδ analysis is developed for realm-nnsize functions, from which the standard theorems of single-variable analysis — continuity, the intermediate, extreme, and mean value theorems, Taylor's theorem with remainder, and L'Hôpital's rule — follow in a uniform parameterised form that reduces to the classical case at n=2n=2. A Cauchy-iterated realm integral provides the corresponding integration theory. The multivariable layer introduces the realm-aware Jacobian, Hessian, and Taylor tensor, with implicit and inverse function theorems and Lagrange-multiplier optimisation. Cross-realm vector operations (Dot[n][n], Cross[n][n]) and cross-realm ordinary differential equations round out the analytic toolkit. Where Kneser principal-branch analysis makes them tractable, results are stated explicitly at realm n=3n=3. The paper is the analytic prerequisite for the geometric and physical applications developed in subsequent papers of the series.