Pierros Ntelis

We introduce the local kernel-based calculus, a unifying framework for local differential and integral operators based on an arbitrary positive continuous kernel function. This framework encompasses conformable, non-conformable, and our newly introduced local Euler-kernel derivatives as special cases. The parameter of the kernel is unrestricted and may take negative values, reflecting its role as a genuine parameter rather than an order of fractional differentiation. Within this general setting, we rigorously prove a complete set of foundational theorems: linearity, the product rule, continuity, Rolle's theorem, the mean value theorem, and the fundamental theorem of calculus via the associated integral operator. We also derive a new formulation of the chain rule which expresses the chain rule entirely in terms of the kernel-based derivatives. While algebraically equivalent to the classical form, this representation preserves the intuitive structure of the chain rule without reference to the classical derivative. We further establish the Fundamental Theorem of Local Euler Calculus and its generalisation, the Fundamental Theorem of Local Kernel-Based Calculus, confirming that the derivative and integral operators are genuine inverses, with the classical fundamental theorem recovered as special cases when the kernel reduces to unity. As an important illustration, we develop the local Euler calculus with the exponential kernel in full detail, providing explicit derivative and integral formulas for elementary functions. This special case demonstrates the simplicity and power of the functional approach. Overall, the local kernel-based calculus provides a solid, self-contained foundation that unifies a wide class of local operators and extends far beyond the traditional setting.

This is a playful companion paper of the dot: $\cdot{}$. Is there a single unique primitive that we use to construct ideas, concepts, and symbols in art, science, mathematics, philosophy, and logic? According to the findings of the paper the answer is positive, and this is implemented from the new interpretation of the dot, which represents the idea, the emptiness, and the symbolism. Drawing on ontological tradition, we argue that this dot is the simplest, most primitive, fundamental, foundational analysis we have for creation. Every subsequent concept in philosophy, logic, and mathematics emerges from the repetition of this triadic act. The paper presents a series of "generative proofs" that are meant as phenomenological descriptions, not formal derivations. This is a work of foundational ontology and experimental philosophy. An ancient anonymous once pressed a finger into wet clay or blew red ochre around a hand on a cave wall and left a single mark. That silent, nameless human - the true founder of philosophy, logic, mathematics, and physics - gave the dot its first meaning. In that instant, three inseparable realities awoke together: the conceivable (idea), the pure absence that lets anything be noticed (emptiness), and the visible conceivable mark that lets it be shared (symbol). From this eternal triad springs infinity syntheses itself - not as a number, not as a set, but as the boundless freedom to repeat the same honest act forever. Every thinker since have only ever rearranged the same three silent gifts into ever-new shapes. Infinity is not discovered later. It is the open space already granted by the first dot. Nothing more is needed. Nothing less is possible, yet.

In this document, we introduce a novel formalism for any field theory and apply it to the effective field theories of large-scale structure. The new formalism is based on functors of actions composing those theories. This new formalism predicts the actionic fields. We discuss our findings in a cosmological gravitology framework. We present these results with a cosmological inference approach and give guidelines on how we can choose the best candidate between those models with some latest understanding of model selection using Bayesian inference.