Algebraic Calculus: A Structural Replacement for Limit Theory via the Euler Singularity Kernel

This paper proposes a foundational reconstruction of infinitesimal calculus, replacing the dynamic limit definition ($\lim_{h \to 0}$) with a static structural definition grounded in the properties of Euler's number ($e$). By analyzing the transcendental function $y = x^{1/x}$ at the singularity $x=1$, we derive the fundamental identity $e = (x-y)^{x/y}$, herein referred to as the \textit{Euler Kernel}. We demonstrate that the singular limit of this kernel at $x=1$ yields the standard definition $e = \lim_{t \to 0}(1+t)^{1/t}$, but allows for the redefinition of the differential operator $\mathcal{D}$ as an exponential stability coefficient rather than a geometric slope. Furthermore, we explore the connection between this kernel and the Golden Ratio ($\phi$) via the auxiliary characteristic equation $z^2 - z - 1 = 0$, establishing a link between logarithmic growth and recursive structural stability. Consequently, we show that the classical Wave Equation ($\nabla^2 u - \frac{1}{c^2}\partial_t^2 u = 0$) collapses into a dimensionless algebraic identity of proportionality. The final section of this work extends this framework to General Relativity (GE), proposing that spacetime curvature is not a geometric manifold deformation, but a variation in the local algebraic density of the Euler Kernel