Dual-Tape Perspective and Generator Independence: The Algebraic Foundation of Real Boolean Turing Machines

The Complex Boolean Turing Machine (CBTM) characterizes non-deterministic computation using the abstract generator $\alpha$, but the abstractness of $\alpha$ makes it difficult to understand intuitively. In this paper, by concretizing $\alpha$ as the algebraic number $\sqrt{2}$, we introduce the $\mathbf{\text{Real}}\mathbf{\text{Boolean}}\mathbf{\text{Turing}}\mathbf{\text{Machine}}\mathbf{\text{(RBTM)}}$ and propose the $\mathbf{\text{dual-tape}}\mathbf{\text{perspective}}$, 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 $\mathbf{\text{Generator}}\mathbf{\text{Independence}}\mathbf{\text{Theorem}}$: 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 $\mathbf{\text{generator}}\mathbf{\text{extraction}}\mathbf{\text{operator}}$ 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).