The Complex Boolean Turing Machine (CBTM) characterizes non-deterministic computation using the abstract generator $α$, but the abstractness of $α$ makes it difficult to understand intuitively. In this paper, by concretizing $α$ as the algebraic number $\sqrt{2}$, we introduce the \textbf{Real Boolean Turing Machine (RBTM)} and propose the \textbf{dual-tape perspective}, decomposing each tape into a real tape (storing rational coefficients $a$) and an imaginary tape (storing irrational coefficients $b$). The ``1''s on the imaginary tape intuitively mark the locations of ``new dimensions,'' laying a physical foundation for subsequent dynamic dimension tracking. More importantly, we prove the \textbf{Generator Independence Theorem}: computational power is independent of the specific choice of generator, whether using $\sqrt{2}$, $\sqrt{3}$, or the imaginary unit $i$, the corresponding automata are isomorphic. This reveals that the essence of non-determinism lies in the fact of ``introducing a new element incommensurable with the base field,'' rather than the algebraic identity of the generator. Furthermore, we introduce the \textbf{generator extraction operator} and analyze its limitations within a static framework, highlighting the necessity of introducing a dynamic IVM. The RBTM serves both as a visualized instance of the CBTM and as a bridge to the subsequent dynamic dimension tracking of the Imaginary-part Verification Machine(IVM).

翻译：复杂布尔图灵机（CBTM）利用抽象产生式 $\alpha$ 刻画非确定性计算，但 $\alpha$ 的抽象性使其难以直观理解。本文通过将 $\alpha$ 具体化为代数数 $\sqrt{2}$，引入\textbf{实布尔图灵机（RBTM）}，并提出\textbf{双带视角}，将每条带分解为实带（存储有理系数 $a$）和虚带（存储无理系数 $b$）。虚带上的“1”直观标记了“新维度”的位置，为后续的动态维度追踪奠定物理基础。更重要的是，我们证明了\textbf{产生式无关性定理}：计算能力与产生式的具体选择无关，无论使用 $\sqrt{2}$、$\sqrt{3}$ 还是虚数单位 $i$，相应的自动机均同构。这揭示了非确定性的本质在于“引入与基域不可公度的新元素”这一事实，而非产生式的代数身份。此外，我们引入\textbf{产生式提取算子}，并分析其在静态框架下的局限性，凸显引入动态虚部验证机（IVM）的必要性。RBTM 既是 CBTM 的可视化实例，也是后续动态维度追踪的虚部验证机（IVM）的桥梁。