Phase 1: Foundational Proofs of ZIDA

This section establishes the foundational axioms of Zero-Inclusive Division Algebra (ZIDA) and their mathematical consistency. ZIDA is constructed to coherently handle division by zero.

Axioms of ZIDA

• Closure: For all \( a, b \in \mathbb{Z} \), we have \( a + b \in \mathbb{Z} \) and \( a \cdot b \in \mathbb{Z} \).
• Additive Identity: \( a + 0 = a \) for all \( a \in \mathbb{Z} \).
• Multiplicative Identity: \( a \cdot 1 = a \) for all \( a \in \mathbb{Z} \).
• Additive Inverse: \( a + (-a) = 0 \) for all \( a \in \mathbb{Z} \).
• Multiplicative Inverse: For \( a \ne 0 \), \( a \cdot a^{-1} = 1 \).
• Division by Zero: Defined via \( \frac{a}{0} = \zeta_0^{-1} \), where \( \zeta_0^{-1} \) is a consistent algebraic extension.

Proof: Existence of ZIDA Inverse

Let \( a \ne 0 \in \mathbb{Z} \). Then by axiom, \( \exists a^{-1} \in \mathbb{Z} \) such that \( a \cdot a^{-1} = 1 \).

For \( a = 0 \), we define \( \frac{a}{0} = \zeta_0^{-1} \), where \( \zeta_0^{-1} \) is a novel operator in ZIDA, preserving closure and coherence.

Continuity and Limit Behavior in ZIDA

A function \( f: \mathbb{Z} \to \mathbb{Z} \) is ZIDA-continuous at \( x_0 \in \mathbb{Z} \) if \( \forall \epsilon > 0, \exists \delta > 0 \) such that:

\[ |x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < \epsilon \]

In ZIDA, limits near \( x = 0 \) behave consistently, e.g.: \[ \lim_{x \to 0} \frac{1}{x} = \zeta_0^{-1} \]

Proof: ZIDA Derivatives

The derivative is defined by: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] In ZIDA, if \( h = 0 \), then: \[ f'(0) = \zeta_0^{-1} \] preserving consistency even under division by zero.

ZIDA Taylor Expansion Theorem

If \( f \) is ZIDA-differentiable at \( a \in \mathbb{Z} \), then: \[ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \] If any term is undefined due to singularity, a correction term is added: \[ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n + \Theta_0(x) \cdot \zeta_0^{-1} \] where: \[ \Theta_0(x) = \begin{cases} 1, & \text{if } x = a \text{ and } f^{(n)}(a) \text{ is undefined} \\ 0, & \text{otherwise} \end{cases} \]

Explanation of Terms

• \(\zeta_0^{-1}\): Formal inverse of zero in ZIDA.
• \(\Theta_0(x)\): Step function indicating a singularity at \( x = a \).
• \(f^{(n)}(a)\): The \(n\)-th derivative of \(f\) at \(a\).

Proof of ZIDA Axioms and Field-Like Properties

Let \( \mathbb{Z}_\zeta \) denote the ZIDA number system, including \( \zeta_0^{-1} \). Then the system satisfies:

• Closure: \( a \oplus b, a \otimes b, a \oslash b \in \mathbb{Z}_\zeta \) for all \( a, b \in \mathbb{Z}_\zeta \).
• Identity: \( a + 0 = a \), \( a \cdot 1 = a \) for all \( a \in \mathbb{Z}_\zeta \).
• Inverses: Additive and multiplicative inverses exist, including for zero via \( \zeta_0^{-1} \).