A foundational analysis of local functional calculus

We introduce the local functional calculus, a unifying framework for local functional operators built on arbitrary continuous functions. This framework subsumes many known functional and fractional calculus operators (conformable and non-conformable) and, more generally, any local derivative or integrator with dependent scaling.
As a special case, we develop the local Euler fractional calculus with the exponential kernel, which yields a particularly elegant calculus.
For both the general functional and the Euler fractional cases, we prove the fundamental properties: linearity, product rule, continuity, Rolle’s theorem, the mean value theorem, and the fundamental theorem of calculus.
Crucially, we derive the correct chain rule, which rectifies an error present in the earlier literature on generalized conformable derivatives.
The local functional calculus thus provides a solid, self‑contained foundation for a wide class of local functional calculus operators, extending far beyond the traditional fractional setting. It is easy to show that this work can be extended and applied in statistics. I will present recent advancements in mathematical and cosmological frameworks, drawing from my key studies. Noticing the need for advanced math, I will introduce and explain the exploration of novel tensor-based approaches from Advancing Tensor Theories (ATTs) and Advanced Manifold-Metric Pairs (AMMPs), offering new tools for modeling. I will introduce and explain functorial methods in Functors of Actions Theories (FATs), bridging mathematical rigor with cosmological phenomena and providing a novel way to think about mathematical analysis and its applications in statistics.