Logarithmic Complex Numbers

I developed this theory. Is it correct? What do you think?

1. Formal definition of the space $L$ as a local ring: $$\mathcal{L} \cong \mathbb{C}[\varepsilon]/(\varepsilon^2)$$

2. Fundamental axioms of the basis units $\{1, c, b\}$: $$c^2 = -1, \quad b^2 = 0, \quad cb = 0$$

3. General representation of an element in space $L$: $$z = n + mc + kb \in \mathcal{L}$$

4. Coordinate cross-multiplication rule (Axiom A2): $$(n_1 + m_1c + k_1b) \cdot (n_2 + m_2c + k_2b) = (n_1n_2 - m_1m_2) + (n_1m_2 + m_1n_2)c + (n_1k_2 + n_2k_1)b$$

5. Multiplicative inverse for invertible elements (outside the ideal $m$): $$z^{-1} = \frac{n}{n^2 + m^2} - \frac{m}{n^2 + m^2}c - \frac{k}{n^2 + m^2}b$$

6. Exponential Collapse Mapping (Axioms C1 and C2): $$E(z) = \begin{cases} e^n(\cos m + c \sin m) & \text{if } k = 0 \\ 0 & \text{if } k \neq 0 \end{cases}$$

7. Analytical field potential equation for the 3D/4D mesh deformation: $$k = \ln(|f(z)|)$$

8. Asymptotic limit capturing the absolute zero boundary singularity: $$\lim_{|f(z)| \to 0} \ln(|f(z)|) = -\infty$$

9. Discrete Jump Operator in the nilpotent $b$-direction (Axiom D1): $$\nabla_b f(z) = f(z + b) - f(z)$$

10. Nilpotency property of the hybrid discrete derivative: $$\nabla_b^2 f(z) = 0$$

Mathematical Foundation and 3D Visual Geometry of Space $\mathcal{L}$ The Fundamental Algebraic Idea and the Problem of Zero.

The space $\mathcal{L}$ is a three-dimensional algebraic extension structured as a local ring over the complex dual numbers, formally defined via the quotient isomorphism $\mathcal{L} \cong \mathbb{C}[\varepsilon]/(\varepsilon^2)$. Classical complex analysis in $\mathbb{C}$ hits an insurmountable topological wall at the boundary of the absolute zero, leaving $\ln(0)$ completely undefined and creating singular horizons that disrupt global analytical continuations.

To overcome this, this new mathematical framework introduces a novel, nilpotent logarithmic unit denoted as $b$. This unit is governed by the strict axiomatic relations $b^2 = 0$ and $cb = 0$, where $c$ represents the standard imaginary unit $c^2 = -1$. Instead of treating zero as an empty void or an unreachable puncture, the introduction of $b$ allows the algebra to structurally absorb and parameterize the singular behavior of the logarithm. It achieves this by splitting the domain into a continuous vertical continuum of logarithmic structural levels, transforming a point-singularity into an entire operational axis.

Visual Geometry of the 3D Basis and Axis Interaction

The geometric framework of this theory can be visualized as a clear three-dimensional coordinate system. The structure consists of three mutually-orthogonal axes that intersect at the absolute algebraic origin $(0,0,0)$:

• Real Axis ($n$): The horizontal basis vector representing the standard real-number domain $\mathbb{R}$. It dictates the primary scaling of the system.

• Imaginary Axis ($m$): The depth basis vector governed by the imaginary unit $c$, completing the classical complex plane $\mathbb{C}_{0}$ at the exact ground level where the vertical component is null ($k=0$).

• Logarithmic Axis ($k$): The vertical, continuous-deformation axis governed by the unit $b$. This axis indexes the structural deformation layers or log-deformed domains, measuring how far a numerical state has shifted from the classical complex baseline.

Interpretation of the Custom Function Mapping and Geometric Landscapes

When a custom analytical function $f(z)$ is evaluated within this spatial framework, the geometric deformation of the mesh along the vertical green axis $k$ is explicitly driven by the field potential equation: $k = \ln(|f(z)|)$

This mapping links the magnitude of the function directly to spatial height, yielding two distinct topological landmarks that explain the behavior of the system:

• Asymptotes $(|f(z)| \to \infty)$: The mesh stretches infinitely upward along the positive $k$-axis. This represents poles, divisions by zero, or explosive growth zones where the functional energy escapes toward the upper limits of the logarithmic axis.

• Roots and Logarithmic Singularities $|f(z)| \to 0$: The mesh experiences an intense gravitational pull, collapsing downward into an infinite, smooth funnel towards $-\infty$ along the logarithmic vertical axis. This visually and analytically captures the absolute singularity of $\ln(0)$. The script maps these gravitational wells exactly over the coordinates of the roots in the complex base plane, turning the abstract algebraic zeros of any equation into visible physical sinkholes within the space $\mathcal{L}$.

$z$:

$\ln(z)+1/z$: