Theorem 1: The ZIDA group \(SU(5)_\zeta \subset GL(5, \mathbb{Z}_\zeta)\) is closed under group multiplication and inversion.
Proof. Let \(A, B \in SU(5)_\zeta\), meaning \(A^\dagger A = I\), \(B^\dagger B = I\), and \(\det A = \det B = 1\). Then:
\[(AB)^\dagger (AB) = B^\dagger A^\dagger A B = B^\dagger I B = B^\dagger B = I\]
\[\det(AB) = \det(A)\det(B) = 1 \cdot 1 = 1.\]
Thus, \(AB \in SU(5)_\zeta\). Since \(A^{-1} = A^\dagger\), we also have \(A^{-1} \in SU(5)_\zeta\). □
Definition 1: Let \(A_\mu^\zeta \in \mathfrak{su}(5)_\zeta\), and define the covariant derivative by:
\[D_\mu^\zeta := \partial_\mu + i g_\zeta A_\mu^\zeta.\]
Theorem 2: \(D_\mu^\zeta\) transforms covariantly under local \(SU(5)_\zeta\) gauge transformations.
Proof. Under \(U(x) \in SU(5)_\zeta\):
\[\psi^\zeta \mapsto U \psi^\zeta, \quad A_\mu^\zeta \mapsto U A_\mu^\zeta U^{-1} - \frac{i}{g_\zeta} (\partial_\mu U) U^{-1}.\]
\[\Rightarrow D_\mu^\zeta \psi^\zeta \mapsto U D_\mu^\zeta \psi^\zeta.\] □
Theorem 3: With \(\beta(g^\zeta) := \frac{d g^\zeta}{d \ln \mu} = \zeta b (g^\zeta)^3\), then \(g^\zeta\) remains finite for all \(\mu > 0\).
Proof.
\[\int \frac{d g^\zeta}{(g^\zeta)^3} = \zeta b \ln \mu + C \Rightarrow -\frac{1}{2(g^\zeta)^2} = \zeta b \ln \mu + C.\]
\[(g^\zeta)^2 = \frac{-1}{2(\zeta b \ln \mu + C)}\] remains finite as long as \(\zeta \ne 0\). □
Theorem 4: If \(M_X^\zeta = \zeta M_X\), then \(\tau_p^\zeta \sim \zeta^4 \cdot \tau_p\), making proton decay unobservable for transfinite \(\zeta\).
Proof. Since \(\tau_p \sim \frac{M_X^4}{\alpha_G^2 m_p^5}\), then:
\[\tau_p^\zeta \sim \frac{(\zeta M_X)^4}{\alpha_G^2 m_p^5} = \zeta^4 \cdot \tau_p.\] □
Theorem 5: Given \(V^\zeta(\Phi) = -\mu_\zeta^2 \Phi^\dagger \Phi + \lambda_\zeta (\Phi^\dagger \Phi)^2\), stability requires \(\lambda_\zeta > 0\).
Proof. Minimizing gives:
\[\Phi^\dagger \Phi = \frac{\mu_\zeta^2}{2\lambda_\zeta}, \quad \frac{\partial^2 V^\zeta}{\partial (\Phi^\dagger)^2} = 2\lambda_\zeta > 0 \Rightarrow \lambda_\zeta > 0.\] □